Supplement to “Optimal Two-sided Invariant Similar Tests for Instrumental Variables Regression” Donald W. K. Andrews1 Cowles Foundation for Research in Economics Yale University
Marcelo J. Moreira Department of Economics Harvard University
James H. Stock Department of Economics Harvard University February 2005 Revised: January 2006
1 Andrews,
Moreira, and Stock gratefully acknowledge the research support of the National Science Foundation via grant numbers SES-0001706 and SES-0417911, SES-0418268, and SBR-0214131, respectively. The authors thank three referees, the co-editor Whitney Newey, Tom Rothenberg, Jean-Marie Dufour, Grant Hillier, Anna Mikusheva, and seminar and conference participants at Harvard/MIT, Michigan, Michigan State, Queen’s, UCLA/USC, UCSD, Yale Statistics Department, 2003 NBER/NSF Conference on Weak Instruments at MIT, 2004 Far-Eastern Econometric Society Meetings in Seoul, and 2004 Canadian Econometrics Study Group Meetings in Toronto for helpful comments.
1
Introduction
This paper contains supplemental material to Andrews, Moreira, and Stock (2006a), hereafter AMS. Tables of the conditional critical values of the CLR test are given at the end of the paper. Section 2 provides details concerning the signinvariant power envelope for similar tests introduced in AMS. Section 3 does likewise for the LU power envelope for invariant similar tests. Section 4 reports additional numerical results to those in AMS. Section 5 establishes consistency of the covariance matrix estimator in AMS. Section 6 gives proofs of Lemmas 1 and 2 of AMS. Section 7 proves the claim made in Comment 2 to Corollary 1 of AMS that when k = 1 the optimal invariant similar test in terms of two-point weighted average power is the Anderson-Rubin test (which is equivalent in this case to the LM and CLR tests). An Appendix describes numerical methods used in Section 4.
2
Power Envelope for Sign-Invariant Tests
Here, we consider similar tests that satisfy a sign invariance condition in addition to the invariance condition of (3.1) of AMS: [S : T ] → [−S : T ].
(2.1)
The corresponding transformation in the parameter space is (β ∗ , λ∗ ) → (β ∗2 , λ∗2 ), where (β ∗2 , λ∗2 ) is defined in (4.1) of AMS. This sign invariance condition is a natural condition to impose to obtain two-sided tests because the parameter vector (β ∗2 , λ∗2 ) is the appropriate “other-sided” parameter vector to (β ∗ , λ∗ ) for the reasons stated in AMS. The maximal invariant under this sign invariance condition (plus the invariance conditions in (3.1) of AMS) is (S S, |S T |, T T ) = (QS , |QST |, QT ).
(2.2)
The LR, LM, and AR test statistics all depend on the data only through this maximal invariant and, hence, satisfy the sign invariance condition (2.1). The density of the maximal invariant (QS , |QST |, QT ) at (qS, qST , qT ) for qST ≥ 0, when the true parameters are (β ∗ , λ∗ ), is 1 [fQ ,Q (qS, qST , qT ; β ∗ , λ∗ ) + fQ1 ,QT (qS, − qST , qT ; β ∗ , λ∗ )]. 2 1 T
(2.3)
Lemma 3(a) of AMS provides an expression for fQ1 ,QT (qS, qST , qT ). Straightforward calculations show that fQ1 ,QT (qS, − qST , qT ; β ∗ , λ∗ ) = fQ1 ,QT (qS, qST , qT ; β ∗2 , λ∗2 )
(2.4)
using (4.1) of AMS. Hence, the density of (QS , |QST |, QT ) when the true parameters are (β ∗ , λ∗ ) equals fQ∗ 1 ,QT (q1 , qT ; β ∗ , λ∗ ). Now, following the same argument as in Section 4 of AMS, this implies that the power envelope for invariant similar tests using
the invariance condition (3.1) of AMS coupled with (2.1) above is the same as the AE two-sided power envelope for invariant similar tests given in AMS.
3
Power Envelope for Locally-Unbiased Tests
3.1
Results
Another approach to constructing a power envelope designed for two-sided alternatives is to impose a necessary condition for unbiasedness–what we call a localunbiasedness (LU) condition. This approach has a long tradition in the statistics literature and is a standard way to derive optimal tests for two-sided alternatives. In exponential families, UMP two-sided tests exist among the class of unbiased tests, see Lehmann (1986, Thm. 4.3, p. 147). This is not the case in the curved exponential family testing problem considered here. Nevertheless, one can construct a power envelope for LU invariant similar tests. We start by determining two necessary conditions for an invariant test (wrt (3.1) of AMS) to be unbiased. The first condition is similarity and the second condition is the requirement that the power function has zero derivative at the null hypothesis. Otherwise, the power function would dip below the size of the test for some alternatives close to the null. We show that the CLR, LM, and AR tests are LU. Theorem 1 An invariant test φ(Q) is unbiased with size α only if Eβ 0 (φ(Q)|QT = qT ) = α and Eβ 0 (φ(Q)QST |QT = qT ) = 0 for almost all qT . Comments. 1. The first condition establishes that all unbiased invariant tests must be similar. The second establishes that the power function of an unbiased test must have zero derivative under H0 . The two conditions together are what we call the LU condition. (Note that the two conditions are only first-order conditions, not sufficient conditions, for a test’s power function to have a local minimum at the null hypothesis.) Obviously, the class of LU tests contains the class of unbiased tests. 2. The two conditions in Theorem 1 are closely related to the conditions used for two-sided alternatives in the classical hypothesis testing theory for exponential families, see Lehmann (1986, Ch. 4). 3. The second condition of Theorem 1 is equivalent to 1/2
Eβ 0 (φ(Q)QST /QT ) = 0.
(3.1)
That is, any LU invariant test statistic φ(Q) must be uncorrelated with the pivotal 1/2 statistic QST /QT under H0 .1 1
The second condition of Theorem 1 clearly implies (3.1). The converse holds by the completeness of QT because by iterated expectations the left-hand side in (3.1) can be written as Eβ 0 h(QT ), where 1/2 h(QT ) = Eβ 0 (φ(Q)QST |QT = qT )/QT .
2
The LR, LM, and AR test statistics depend on the data through (QS , |QST |, QT ). The following result shows that these tests satisfy the second condition of Theorem 1. Corollary 1 Any similar level α test that depends on the observations through (QS , |QST |, QT ) satisfies the LU condition of Theorem 1. Next, we determine the test that maximizes power against any given parameter vector (β ∗ , λ∗ ) among the class of LU invariant tests. We do so using the same conditioning argument as in Section 4 of AMS, but using the generalized NeymanPearson Lemma (see Lehmann (1986, Thm. 3.5, pp. 96-7)) in place of the NeymanPearson Lemma. Define LR(q1 , qT ; β ∗ , λ∗ ) =
ψ(q1 , qT ; β ∗ , λ∗ ) fQ1 ,QT (q1 , qT ; β ∗ , λ∗ ) = , fQT (qT ; β ∗ , λ∗ )fQ1 |QT (q1 |qT ; β 0 ) ψ 2 (qT ; β ∗ , λ∗ )
(3.2)
Theorem 2 The test that maximizes power against (β, λ) = (β ∗ , λ∗ ) among LU invariant tests with significance level α rejects H0 if LR(Q1 , QT ; β ∗ , λ∗ ) > κ1α (QT ; β ∗ , λ∗ ) + QST κ2α (QT ; β ∗ , λ∗ ), where κ1α (QT ; β ∗ , λ∗ ) and κ2α (QT ; β ∗ , λ∗ ) are chosen such that the two conditions in Theorem 1 hold. Comment. The power of the tests LR(Q1 , QT ; β ∗ , λ∗ ) as (β ∗ , λ∗ ) varies maps out the power envelope for LU invariant tests.
3.2
Proofs of Local-Unbiasedness Results
Proof of Theorem 1. By continuity of the power function, which holds by Lehmann (1986, Thm. 2.9, p. 59), any unbiased test φ(Q) is similar. Hence, the first condition of the Theorem holds by Theorem 2 of AMS. Now, for a test to be unbiased, (∂/∂β)Eβ,λ φ(Q1 , QT )|β=β 0 = 0 for all values of λ. By interchanging derivatives and integrals (which is justified by Lehmann (1986, Thm. 2.9, p. 59)) and the chain rule, the left-hand side of this equality equals I1 + I2 , where ] ] ∂fQ1 |QT (q1 |qT ; β 0 , λ) dq1 fQT (qT ; β 0 , λ)dqT and I1 = φ(q1 , qT ) ∂β ] ] ∂fQT (qT ; β 0 , λ) dqT φ(q1 , qT )fQ1 |QT (q1 |qT ; β 0 )dq1 I2 = ∂β ] ∂fQT (qT ; β 0 , λ) dqT = 0, = α (3.3) ∂β 3
where the second last Uequality holds by the condition for similarity and the last equality holds because fQT (qT ; β, λ)dqT = 1 for all β. To compute the derivative of the conditional density of Q1 given QT = qT with respect to β evaluated at β 0 , it is convenient to write the conditional density of Q1 given QT = qT as −(k−2)/2
fQ1 |QT (q1 |qT ; β, λ) = K1 K2−1 exp(−λc2β /2) exp(−qS /2) det(q)(k−3)/2 qT / ∞ ∞ [ [ (λd2β qT /4)j (λξ β (q)/4)j j=0
j!Γ((k − 2) /2 + j + 1)
j=0
×
(3.4)
j!Γ((k − 2/2) + j + 1)
using Lemma 3(a) and (b) of AMS and (4.8) of AMS. Tedious algebraic manipulations show that ∂fQ1 |QT (q1 |qT ; β 0 , λ) λ1/2 = fQ |Q (q1 |qT ; β 0 )qST (det(Ω))−1/2 × ∂β 2(d2β qT )1/2 1 T s s Ik/2 ( λa0 Ω−1 a0 qT )/I(k−2)/2 ( λa0 Ω−1 a0 qT ).
(3.5)
The function Ik/2 (·) arises because
∞ ∞ (λξ β (q)/4)j λ ∂ξ β (q) [ (λξ β (q)/4)s ∂ [ = ∂β j=0 j!Γ((k − 2) /2 + j + 1) 4 ∂β s=0 s!Γ(k/2 + s + 1)
(3.6)
and likewise with ξ β (q) replaced by (d2β qT ). The necessary condition for unbiasedness, (3.3), and (3.5) give s ] Ik/2 ( λa0 Ω−1 a0 qT ) s 0= h(qT )fQT (qT ; β 0 , λ) dqT , where I(k−2)/2 ( λa0 Ω−1 a0 qT ) ] φ(q1 , qT )qST fQ1 |QT (q1 |qT ; β 0 )dq1 . h(qT ) =
(3.7)
By completeness of QT under H0 , see the proof of Theorem 2 of AMS, it must be the case that h(qT ) is zero for almost all qT and all λ ≥ 0, which yields the second condition of the Theorem. Proof of Corollary 1. Any test that depends on (QS , Q2ST , QT ) can be written as φ(QS , S22 , QT ), where S2 = QST /(QS QT )1/2 . By Lemma 3(e) and (f) of AMS, QS , S2 , and QT are independent under H0 and S2 has a distribution that is symmetric about zero. Hence, we have 1/2
1/2
Eβ 0 (φ(QS , S22 , QT )QST |QT = qT ) = Eβ 0 (φ(QS , S22 , qT )S2 QS )qT ] 1/2 1/2 = Eβ 0 (φ(qS , S22 , qT )S2 )qS fQS (qS )dqS · qT = 0 4
(3.8)
for all qT , where the last equality holds because φ(qS , S22 , qT )S2 is an odd function of S2 and S2 is symmetrically distributed about zero. Proof of Theorem 2. By the same argument as in Section 4 of AMS, it suffices to find the test that maximizes power against the single alternative conditional density fQ1 |QT (q1 |qT ; β ∗ , λ∗ ) conditional on QT = qT . Given the restriction to LU tests, we apply the generalized Neyman-Pearson (GNP) Lemma, see Lehmann (1986, Thm. 3.5, pp. 96-7), rather than the Neyman-Pearson Lemma. The GNP Lemma implies that the optimal (conditional) test rejects when LR(Q1 , qT ; β ∗ , λ∗ ) > κ1α (qT ; β ∗ , λ∗ )+ κ2α (qT ; β ∗ , λ∗ )QST for some κ1α (qT ; β ∗ , λ∗ ) and κ2α (qT ; β ∗ , λ∗ ) that are chosen such that the two conditions of Theorem 1 hold. It remains to verify the conditions needed to apply the generalized NeymanPearson Lemma. Let M be the set of points (E (φ(Q1 , QT )|QT = qT ) , E (φ(Q1 , QT )QST |QT = qT ))
(3.9)
as φ ranges over all possible critical functions. It suffices to show that (α, 0) is an interior point of M, see Lehmann (1986, Thm. 3.5(iv), p. 97). The set M is convex because the conditional expectation operator is linear. Moreover, M contains (α, 0) by considering the LM test. It also contains points (α, u+ α) 1/2 + with uα > 0 by considering the one-sided LM test which rejects H0 when QST /QT > cα . This follows because the derivative of the conditional power function of this test is an increasing linear transformation of ] 1/2 (3.10) 1 qST /qT > cα qST fQ1 |QT (q1 |qT ; β 0 )dq1 ,
− which is strictly positive. Likewise, M also contains points (α, u− α ) with uα < 0 1/2 by considering the test which rejects H0 when −QST /QT > cα by an analogous argument. This completes the verification that (α, 0) lies in the interior of M.
4
Numerical Results: Model with Known Covariance Matrix
This section reports numerical results for power envelopes and power functions of two-sided invariant similar tests. A representative subset of these results are reported in AMS. Throughout, we focus on tests with significance level 5% and, without loss of generality, on the case β 0 = 0. As discussed in AMS, the remaining parameters characterizing the distribution of the tests are λ, k, ρ = corr(v1i , v2i ), and the alternative, β. (The distribution of Q and thus the power depends on the sample size only through λ.) Results are reported here for λ/k = 0.5, 1, 2, 4, 8, and 16, which spans the range 5
from weak to strong instruments, for ρ = .95, .5, and .2, and for k = 2, 5, 10, and 20. Numerical issues are discussed in the Appendix. The results are summarized in √Figures S-1 - S-6. As in AMS, the horizontal axis is the scaled “local” alternative, λβ. Figure S-1 presents the asymptotically efficient two-sided power envelope (for invariant similar tests) and the power envelope for LU invariant tests, for k = 5. Figure S-2 presents the asymptotically efficient two-sided power envelope and the power functions of the two-sided CLR, LM, and AR tests for k = 2. These power envelopes and power functions are plotted for k = 5, 10, and 20 in Figures S-3, S-4, and S-5, respectively. Figure S-6 presents the asymptotically efficient two-sided power envelope and the power functions of two POIS2 tests, labeled POIS2a and POIS2b. One approach to testing when there is not a UMPI test is to consider a POI test that has a power function tangent to the power envelope at a certain value. If the power functions remain sufficiently close to the power envelope against alternatives other than that for which the test is point optimal, then that particular POI test provides a good practical choice (cf. King (1988)). Specifically, the POIS2a test is the LR∗ test against (β ∗ = 0.8, λ∗ = 5), and the POISb test is the LR∗ test against (β ∗ = 1.45, λ∗ = 5). When λ∗ = 5 and ρ = .95, the power functions of the POIS2a and POIS2b tests are tangent to the power envelope at approximately 25% power and 75% power, respectively. Like the power envelope itself, these tests depend on ρ; these tests are infeasible if ρ is unknown although a feasible version of these tests could be computed by plugging in a consistent estimator of ρ. Summary of findings. 1. In theory, the 2-sided asymptotically efficient power envelope is no higher than the LU power envelope. Numerically, it appears from Figure S-1 that the two power envelopes are essentially the same for most values of ρ, λ, and β. √ In the cases considered, the greatest difference occurs in Figure S-1, panel (c) for λβ ∼ = 2.2, where the difference between the two power envelopes is approximately 0.05. 2. In Figures S-2 - S-5, the 2-sided CLR test has a power function that is essentially on the power envelope in every case. 3. Figures S-2 - S-5 extend previous findings that the LM statistic can have a non-monotone power function and has poor power properties compared with the CLR statistic. 4. Figures S-2 - S-5 show that the AR statistic can have power well below the AE power envelope and well below that of the CLR statistic. The power gap increases with k. This gap is present for both weak and strong instruments. 5. The strong instrument asymptotic results in AMS indicate that the LM and CLR have power functions that coincide with the AE power envelope when λ is large. As a numerical matter, in the cases considered in Figures S-2 - S-5, for almost all values of k and ρ, this coincidence occurs by λ/k = 16, sometimes by λ/k = 8. 6
6. Comparing the AE two-sided power envelope across all the panels√of Figures S-2 - S-5 reveals that the AE power envelope, viewed as a function of λβ, takes √ on similar values regardless of ρ, k, or λ/k. Said differently, λβ evidently is the appropriate “local” parameterization. 7. The power functions of the POIS2a and POIS2b tests are close to the AE two-sided power envelope. For ρ = .5, the POIS2a performs slightly better than the POIS2b. The very good performance of the POIS2a suggests that further work on a feasible version of this test (in which ρ is estimated) is a promising way to develop a new test, based on the theory of tangency testing, that has power approaching the AE two-sided power envelope. A theoretical advantage of this test over the CLR test is that it is asymptotically admissible among sign invariant tests (under weak instrument asymptotics). The practical disadvantage of this test, relative to the CLR test, is that it is numerically more difficult to work with because it involves Bessel functions, whereas the CLR test does not. Moreover, the results in Figures S-2 - S-5 indicate that there is little room for improvement over the CLR test.
5
Consistency of the Covariance Matrix Estimator In AMS, the covariance matrix Ω (∈ R2×2 ) (defined in Assumption 2) is estimated
via
e n = (n − k − p)−1 Ve Ve , where Ve = Y − PZ Y − PX Y, Ω (5.1) where k and p are the dimensions of Zi and Xi , respectively. Let Vei denote the i-th row of Ve written as a column 2-vector. Under Assumptions 1-3, the variance estimator is consistent. e n →p Ω. Lemma 1 Under Assumptions 1-3, Ω
Comment. The convergence in the Lemma occurs uniformly over all true parameters β, C, γ, and ξ no matter what the parameter space is. This can be seen by inspection of the proof of the Lemma. Proof of Lemma 1. Using the definition Y = Zπa + Xη + V, we obtain Ve = V − PZ V − PX V. This and PZ PX = 0 gives n−1 Ve Ve − Ω = (n−1 V V − Ω) − n−1 V PZ V − n−1 V PX V.
(5.2)
The first summand on the right-hand side of (5.2) converges in probability to zero by Assumption 2. The second summand satisfies 0 ≤ n−1 V PZ V ≤ n−1 V PZ V = n−1 (n−1/2 V Z)(n−1 Z Z)−1 (n−1/2 Z V ) →p 0, (5.3) where the second inequality holds because the span of Z is contained in the span of Z and the convergence to zero holds by Assumptions 1 and 3. The third summand of (5.2) converges in probability to zero by an analogous argument. 7
6
Proofs of Lemmas 1 and 2 of AMS Here, we state Lemmas 1 and 2 of AMS and provide proofs of these Lemmas. The two equation reduced-form model can be written in matrix notation as Y = Zπa + Xη + V, where Y = [y1 : y2 ], V = [v1 : v2 ], a = (β, 1) , and η = [γ : ξ].
(6.1)
The distribution of Y ∈ Rn×2 is multivariate normal with mean matrix Zπa + Xη, independence across rows, and covariance matrix Ω for each row. The parameter space for θ = (β, π , γ , ξ ) is taken to be R × Rk × Rp × Rp .
Lemma AMS-1. For the model in (6.1), (a) Z Y and X Y are sufficient statistics for θ, (b) Z Y and X Y are independent, (c) X Y has a multivariate normal distribution that does not depend on (β, π ) , (c) Z Y has a multivariate normal distribution that does not depend on η = [γ:ξ], (d) Z Y is a sufficient statistic for (β, π ) . Lemma AMS-2. For the model in (6.1), (a) S ∼ N(cβ μπ , Ik ), (b) T ∼ N(dβ μπ , Ik ), and (c) S and T are independent.
Proof of Lemma AMS-1. Let Z = [Z1 : · · · : Zn ] and X = [X1 : · · · : Xn ] . The distribution of Y is multivariate normal with EY = Zπa + Xη,
(6.2)
independence across rows, and covariance matrix Ω for each row. Hence, the density of Y evaluated at the n × 2 matrix y = [y1 :· · ·:yn ] is # $ n [ 1 (Yi − aπ Zi − η Xi ) Ω−1 (Yi − aπ Zi − η Xi ) (2π)−n/2 |Ω|−n/2 exp − 2 i=1 # % n n [ [ 1 −n/2 −n/2 −1 = (2π) |Ω| exp − Y Ω Yi − 2π ( Zi Yi )Ω−1 a 2 i=1 i i=1 &$ n n [ [ −2tr(( Xi Yi )Ω−1 η ) + (aπ Zi − η Xi ) Ω−1 (aπ Zi − η Xi ) . (6.3) i=1
i=1
If a density can be factorized as pθ (x) = fθ (T (x))h(x), then T (X) is a sufficient statistic for θ. In consequence, given that Ω is known, Zi and Xi are fixed and known, a = (β, 1) , and Sn Snη = [γ : ξ], sufficient statistics for θ = (β, π , γ , ξ ) are i=1 Zi Yi = Z Y and i=1 Xi Yi = X Y and part (a) of the lemma holds. 8
To prove part (b) of the lemma, note that Z Y and X Y are (jointly) multivariate normal random matrices and Z X = 0. For any m1 , m2 ∈ R2 , we have n n [ [ Zi Yi m1 , Xi Yi m2 ) cov(Z Y m1 , X Y m2 ) = cov( i=1
=
n [ i=1
i=1
Zi Xi cov(Yi m1 , Yi m2 ) = Z X · m1 Ωm2 = 0,
(6.4)
where the second equality uses independence across i and the third equality uses the assumption that the covariance matrix Ω of Yi does not depend on i. Hence, Z Y and X Y are independent. The distribution of X Y is multivariate normal with variances and covariances that depend on X and Ω, but not on θ, and with mean X EY = X (Zπa + Xη) = X Xη
(6.5)
because X Z = 0. Hence, the distribution of X Y does not depend on (β, π) and part (c) of the lemma holds. The distribution of Z Y is multivariate normal with variances and covariances that depend on Z and Ω, but not on θ, and with mean Z EY = Z (Zπa + Xη) = Z Zπa
(6.6)
because Z X = 0. Hence, the distribution of Z Y does not depend on (γ, ξ) and part (d) of the lemma holds. Part (e) of the lemma follows from parts (b)-(d). Proof of Lemma AMS-2. The k-vector S is multivariate normal with mean ES = (Z Z)−1/2 Z EY b0 · (b0 Ωb0 )−1/2 = (Z Z)−1/2 Z (Zπa + Xη)b0 · (b0 Ωb0 )−1/2 = cβ μπ
(6.7)
using (6.2), Z X = 0, and a β 0 = β − β 0 . We have n n n [ [ [ var(Z Y b0 ) = var( Zi Yi b0 ) = Zi Zi var(Yi b0 ) = Zi Zi b0 Ωb0 = Z Zb0 Ωb0 . i=1
i=1
i=1
(6.8)
Hence, from the definition of S, var(S) = Ik and part (a) of the lemma holds. The k-vector T is multivariate normal with mean ET = (Z Z)−1/2 Z Y Ω−1 a0 · (a0 Ω−1 a0 )−1/2 = (Z Z)−1/2 Z (Zπa + Xη)Ω−1 a0 · (a0 Ω−1 a0 )−1/2 = dβ μπ . 9
(6.9)
From (6.8) with b0 replaced by Ω−1 a0 , we have var(Z Y Ω−1 a0 ) = Z Za0 Ω−1 a0 . Hence, from the definition of T, var(T ) = Ik and part (b) of the lemma holds. The random vectors S and T are independent because they are non-stochastic functions of Z Y b0 and Z Y Ω−1 a0 , respectively, and the latter are jointly multivariate normal with covariance given by n n [ [ cov(Z Y b0 , Z Y Ω a0 ) = cov( Zi Yi b0 , Zi Yi Ω−1 a0 ) −1
i=1
=
n [
Zi Zi cov(Yi b0 , Yi Ω−1 a0 ) =
i=1
n [
i=1
Zi Zi b0 ΩΩ−1 a0 = 0,
(6.10)
i=1
using b0 a0 = 0. Hence, part (c) of the lemma holds.
7
Results for the Just-Identified Model
Here we prove the claim given in Comment 2 following Corollary 1 of AMS that ψ(q1 , qT ; β ∗ , λ∗ ) + ψ(q1 , qT ; β ∗2 , λ∗2 ) is increasing in S 2 when k = 1. (It is strictly increasing unless S = 0.) This claim leads to the result that the AR, LM, and CLR tests (which are equivalent when k = 1) maximize average power against (β ∗ , λ∗ ) and (β ∗2 , λ∗2 ) for all (β ∗ , λ∗ ) in the class of invariant similar tests. That is, these tests are UMP two-sided invariant similar tests. By the definition given in Corollary 1 of AMS, we have t ψ(q1 , qT ; β, λ) = exp(−λ(c2β + d2β )/2)(λξ β (q))−(k−2)/4 I k−2 λξ β (q) . (7.1) 2
When k = 1, we have
I−1/2 (x) = x−1/2 (2/pi)1/2 (exp(x) + exp(−x))/2 = x−1/2 (2/pi)1/2 cosh(x),
(7.2)
see Comment 2 to Lemma 3 of AMS. When k = 1, QST = S · T. Using this and equations (4.1) and (4.7) of AMS, we have λ∗ ξ β ∗ (q) = λ∗ (cβ ∗ S + dβ ∗ T )2 , λ∗2 ξ β ∗2 (q) = λ∗ (cβ ∗ S − dβ ∗ T )2 , and
λ∗ (c2β ∗ + d2β ∗ ) = λ∗2 (c2β ∗2 + d2β ∗2 ).
(7.3)
Combining (7.1)-(7.3) gives ψ(q1 , qT ; β ∗ , λ∗ ) + ψ(q1 , qT ; β ∗2 , λ∗2 ) 1 = exp(−λ∗ (c2β ∗ + d2β ∗ )/2)(2/pi)1/2 × 4 √ √ cosh( λ∗ (cβ ∗ S + dβ ∗ T )) + cosh( λ∗ (cβ ∗ S − dβ ∗ T )) 10
(7.4)
using the fact that cosh(·) is symmetric about zero. Define h(x) = exp(x + K) + exp(−x − K) + exp(x − K) + exp(−x + K),
(7.5)
where K is a constant. The function h(x) only depends on x through |x| because it is symmetric in x. The same is true for K, so without loss of generality assume K ≥ 0. We show that h(x) is increasing in |x| by showing that its derivative is non-negative for x ≥ 0. Combining this with (7.4) and the definition of cosh(·) gives the desired result that the left-hand side of (7.4) is increasing in |S|. The derivative of h(x) is h (x) = exp(x + K) − exp(−x − K) + exp(x − K) − exp(−x + K).
(7.6)
Since x ≥ 0 and K ≥ 0, there are two cases to consider: (i) 0 ≤ K ≤ x and (ii) 0 ≤ x ≤ K. For case (i), we have exp(x + K) − exp(−x − K) ≥ 0 and exp(x − K) − exp(−x + K) ≥ 0 because x + K ≥ 0, x − K ≥ 0, and exp(·) is increasing. For case (ii), we have exp(x + K) − exp(−x + K) ≥ 0 and exp(x − K) − exp(−x − K) ≥ 0 because x ≥ 0 and exp(·) is increasing. Hence, h (x) is non-negative. The inequalities are strict if x > 0.
11
8
Appendix: Numerical Methods
1. General numerical notes (a) All numerical results are based on 5, 000 Monte Carlo draws of Q except: CLR, LM, and AR power functions are computed using 10, 000 draws; and the asymptotically efficient (AE) two-sided power envelopes for k = 2 and k = 10 are based on 2, 000 draws. All computations were done in GAUSS. (b) The statistics ψ and ψ2 have a range of many orders of magnitude so they were computed in logarithms. The statistic ψ 2 is a function of QT only so in theory the denominator term in the LR∗ statistic can be absorbed into the conditional critical value function. However, the conditional critical values of ln(ψ(Q1 , qT ; β ∗ , λ∗ ) + ψ(Q1 , qT ; β ∗2 , λ∗2 )) turn out to depend strongly on qT whereas the conditional critical values of this term minus ln(ψ 2 (Q1 , qT ; β ∗ , λ∗ ) + ψ 2 (Q1 , qT ; β ∗2 , λ∗2 )) typically depend less strongly on qT . Hence, numerical accuracy is improved by computing the LR∗ statistic (and its critical values) as the difference of the two log terms. (c) Bessel functions were computed in logarithms using the GAUSS function mbesselei. 2. Computation of conditional critical values for similar tests (a) For the conditional similar tests involving Bessel functions, conditional critical value functions were computed on a grid of 150 values of QT (125 of which were equispaced on a log scale between the 0.5% percentile of a central chi-squared(k) distribution and QT = 1, 000, plus 25 additional points). The 5% critical values were stored in a lookup table which was then accessed (with linear interpolation) to compute rejection rates under the alternative. (b) The algorithm for numerical evaluation of the p-values for the CLR statistic is described in Andrews, Moreira, and Stock (2006b). 3. Power envelope for LU Invariant tests. According to Theorem 1, the POI LU test is constructed as the conditional test of the LR form in (3.2), against a given point alternative, where the conditional critical value function is of the form given in Theorem 2 and κ1 and κ2 are chosen to satisfy the conditions of Theorem 1. Specifically, consider the problem of construction of the LU test that is POI against a given (fixed) value of the alternative (β ∗ , λ∗ ) for a given value of ρ. Denote this LR test by φ. The numerical task is to find κ1 and κ2 such that E(φ|QT = qT ) = α, E(φQST |QT = qT ) = 0.
(8.1) (8.2)
Note that κ1 and κ2 are functions of qT . This POI LU test is implemented by constructing two lookup tables, one for κ1 and one for κ2 (both as a function of qT ). At 12
each value of qT , κ1 and κ2 can be computed by solving the equations (8.1) and (8.2). These two equations were solved using the following algorithm: (1) For a given value of qT , compute 5, 000 Monte Carlo draws of Q under the null hypothesis. (2) Select a value of κ2 . Given this value, compute κ1 as the .05 percentile of LR − QST κ2 ; that is, κ1 is chosen to satisfy (8.1) (whereS the expectation is replaced by the summation over the 5, 000 draws). (3) Construct MC-Draws SφQST . Repeat steps (2) and (3) for different κ2 with the objective of minimizing | MC-Draws φQST |. The minimization was done using a line search and the minimized value of |corr(φ, QST )| was always less than .001. This was repeated for each value of qT on the standard grid of qT (discussed in item #2 above) to construct the lookup tables for κ1 , κ2 . To construct the power envelopes, the null rejection frequency based on 5, 000 Monte Carlo draws was calculated for each point alternative, (β ∗ , λ∗ ), for each value of ρ and λ/k considered.
13
References Andrews, D. W. K., M. J. Moreira, and J. H. Stock (2006a): “Optimal Two-sided Invariant Similar Tests for Instrumental Variables Regression with Weak Instruments,” Econometrica, 74, forthcoming. ––– (2006b): “Performance of Conditional Wald Tests in IV Regressions with Weak Instruments,” Journal of Econometrics, forthcoming. King, M.L. (1988), “Towards a Theory of Point Optimal Testing,” Econometric Reviews, 6, 169-218. Lehmann, E. L. (1986): Testing Statistical Hypotheses, 2nd ed. New York: Wiley.
14
Figure S-1. Asymptotically efficient two-sided power envelopes for invariant similar tests (AEPE) and power envelopes for locally unbiased (LU) invariant similar tests, k = 5
Figure S-1, ctd.
Figure S-1, ctd.
Figure S-2. Asymptotically efficient two-sided power envelopes for invariant similar tests (AEPE) and power functions for the two-sided CLR, LM, and AR tests, k = 2
Figure S-2, ctd.
Figure S-2, ctd.
Figure S-3. Asymptotically efficient two-sided power envelopes for invariant similar tests (AEPE) and power functions for the two-sided CLR, LM, and AR tests, k = 5
Figure S-3, ctd.
Figure S-3, ctd.
Figure S-4. Asymptotically efficient two-sided power envelopes for invariant similar tests (AEPE) and power functions for the two-sided CLR, LM, and AR tests, k = 10
Figure S-4, ctd.
Figure S-4, ctd.
Figure S-5. Asymptotically efficient two-sided power envelopes for invariant similar tests (AEPE) and power functions for the two-sided CLR, LM, and AR tests, k = 20
Figure S-5, ctd.
Figure S-5, ctd.
Figure S-6. Asymptotically efficient two-sided power envelopes for invariant similar tests (AEPE) and 2-sided POI (tangency) tests at 25% power (POIS2a) and 75% power (POIS2b), k = 5
Figure S-6, ctd.
Figure S-6, ctd.
Tables of 10%, 5%, and 1% Conditional Critical Values for the CLR Statistic, k = 2 – 25 The following tables provide critical values of Moreira’s (2003) conditional likelihood ratio (CLR) test statistic for a given value of the conditioning statistic, QT, and the number of instruments, k. The table is presented as a function of ln(QT/k) (first column of each table). The remaining columns give the 10%, 5%, and 1% critical values for the value of k indicated in the column header. The rows for which ln(QT/k) is given by “-∞” and “+∞” provide the critical values for the limiting cases that QT = 0 and QT → ∞, respectively (these distributions are chi-squared with k and 1 degrees of freedom, respectively). Accurate critical values given the observed value of QT can be computed by linear interpolation. Estimated numerical accuracy: the actual quantile associated with these critical values is estimated to be within .001 percentage points of the stated critical value (this is the largest discrepancy; for most entries the discrepancy is less than .0002). These computations are based on numerical integration of the conditional distribution using the method described in Andrews, D.W.K., M. Moreira, and J.H. Stock, “Performance of Conditional Wald Tests in IV Regression,” forthcoming, Journal of Econometrics.
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
k=2 10% 4.61 4.60 4.60 4.60 4.60 4.60 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.58 4.58 4.58 4.58 4.57 4.57 4.56 4.56 4.56 4.55 4.55 4.54 4.53 4.52 4.52 4.51 4.50 4.49 4.47 4.46 4.45 4.43 4.41 4.39 4.37 4.35 4.32 4.30 4.27 4.24 4.20 4.17 4.13 4.08 4.04 3.99 3.94 3.88 3.83 3.77 3.71 3.65
k=2 5% 5.99 5.98 5.98 5.98 5.98 5.98 5.98 5.98 5.98 5.98 5.98 5.97 5.97 5.97 5.97 5.96 5.96 5.96 5.96 5.95 5.95 5.94 5.94 5.93 5.93 5.92 5.91 5.90 5.89 5.88 5.87 5.86 5.85 5.83 5.82 5.80 5.78 5.76 5.73 5.71 5.68 5.65 5.62 5.58 5.55 5.50 5.46 5.41 5.36 5.31 5.25 5.19 5.13 5.07 5.00
k=2 1% 9.21 9.20 9.20 9.20 9.20 9.20 9.20 9.20 9.20 9.20 9.19 9.19 9.19 9.19 9.19 9.18 9.18 9.18 9.17 9.17 9.17 9.16 9.16 9.15 9.14 9.14 9.13 9.12 9.11 9.10 9.09 9.08 9.07 9.05 9.03 9.01 9.00 8.97 8.95 8.93 8.90 8.87 8.83 8.80 8.76 8.71 8.67 8.62 8.57 8.51 8.45 8.39 8.32 8.25 8.17
k=3 10% 6.25 6.24 6.24 6.23 6.23 6.23 6.23 6.23 6.22 6.22 6.22 6.21 6.21 6.21 6.20 6.20 6.19 6.18 6.18 6.17 6.16 6.15 6.14 6.13 6.12 6.10 6.09 6.07 6.05 6.03 6.01 5.99 5.96 5.93 5.90 5.86 5.82 5.78 5.73 5.68 5.63 5.57 5.50 5.43 5.35 5.27 5.18 5.09 4.99 4.88 4.77 4.65 4.53 4.40 4.28
k=3 5% 7.81 7.80 7.80 7.80 7.80 7.79 7.79 7.79 7.79 7.78 7.78 7.78 7.77 7.77 7.77 7.76 7.75 7.75 7.74 7.73 7.73 7.72 7.71 7.69 7.68 7.67 7.65 7.64 7.62 7.60 7.57 7.55 7.52 7.49 7.46 7.42 7.39 7.34 7.29 7.24 7.19 7.13 7.06 6.99 6.91 6.82 6.73 6.63 6.53 6.42 6.30 6.18 6.05 5.91 5.78
k=3 1% 11.34 11.33 11.33 11.33 11.33 11.33 11.32 11.32 11.32 11.31 11.31 11.31 11.30 11.30 11.30 11.29 11.28 11.28 11.27 11.26 11.25 11.25 11.24 11.22 11.21 11.20 11.18 11.17 11.15 11.13 11.10 11.08 11.05 11.02 10.99 10.95 10.91 10.87 10.82 10.77 10.71 10.65 10.58 10.51 10.43 10.34 10.24 10.14 10.03 9.92 9.79 9.66 9.52 9.38 9.22
k=4 10% 7.78 7.76 7.76 7.75 7.75 7.75 7.75 7.74 7.74 7.73 7.73 7.72 7.72 7.71 7.71 7.70 7.69 7.68 7.67 7.66 7.65 7.63 7.62 7.60 7.58 7.56 7.54 7.51 7.48 7.45 7.42 7.38 7.34 7.30 7.25 7.19 7.13 7.07 7.00 6.92 6.83 6.74 6.64 6.53 6.41 6.29 6.15 6.00 5.85 5.68 5.51 5.33 5.14 4.95 4.76
k=4 5% 9.49 9.47 9.47 9.46 9.46 9.46 9.45 9.45 9.45 9.44 9.44 9.43 9.43 9.42 9.41 9.41 9.40 9.39 9.38 9.37 9.35 9.34 9.32 9.31 9.29 9.27 9.24 9.22 9.19 9.16 9.13 9.09 9.05 9.00 8.95 8.90 8.84 8.77 8.70 8.62 8.54 8.44 8.34 8.23 8.11 7.98 7.84 7.69 7.53 7.36 7.18 6.99 6.79 6.58 6.38
k=4 1% 13.28 13.26 13.26 13.25 13.25 13.25 13.24 13.24 13.24 13.23 13.23 13.22 13.22 13.21 13.21 13.20 13.19 13.18 13.17 13.16 13.14 13.13 13.11 13.10 13.08 13.06 13.03 13.01 12.98 12.95 12.92 12.88 12.84 12.79 12.74 12.69 12.63 12.56 12.49 12.41 12.32 12.23 12.12 12.01 11.89 11.75 11.61 11.45 11.29 11.11 10.92 10.71 10.50 10.28 10.05
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
3.59 3.52 3.46 3.40 3.34 3.29 3.23 3.18 3.14 3.09 3.06 3.02 2.99 2.96 2.94 2.91 2.89 2.87 2.86 2.84 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
4.93 4.86 4.79 4.72 4.65 4.59 4.52 4.46 4.41 4.36 4.31 4.26 4.22 4.19 4.15 4.12 4.10 4.07 4.05 4.03 4.01 4.00 3.98 3.97 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.84
8.10 8.02 7.94 7.86 7.77 7.69 7.62 7.54 7.47 7.40 7.34 7.28 7.22 7.17 7.12 7.08 7.04 7.00 6.97 6.94 6.91 6.89 6.87 6.85 6.83 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.65 6.65 6.65 6.64 6.64 6.64 6.63
4.15 4.03 3.91 3.80 3.69 3.58 3.49 3.41 3.33 3.26 3.20 3.15 3.10 3.06 3.02 2.99 2.96 2.94 2.91 2.89 2.88 2.86 2.84 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
5.64 5.50 5.36 5.23 5.10 4.98 4.87 4.77 4.67 4.59 4.51 4.44 4.38 4.33 4.28 4.24 4.20 4.16 4.13 4.10 4.08 4.06 4.03 4.02 4.00 3.98 3.97 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.84
9.07 8.91 8.75 8.60 8.44 8.29 8.15 8.02 7.89 7.78 7.67 7.58 7.49 7.41 7.34 7.27 7.21 7.16 7.11 7.07 7.03 6.99 6.96 6.93 6.90 6.88 6.85 6.83 6.81 6.80 6.78 6.77 6.76 6.74 6.73 6.72 6.72 6.71 6.70 6.70 6.69 6.68 6.68 6.67 6.67 6.67 6.67 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.65 6.65 6.64 6.63
4.58 4.39 4.22 4.06 3.90 3.77 3.64 3.54 3.44 3.36 3.28 3.22 3.17 3.12 3.07 3.04 3.00 2.97 2.94 2.92 2.90 2.88 2.86 2.85 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
6.17 5.97 5.77 5.58 5.40 5.24 5.09 4.95 4.83 4.72 4.63 4.55 4.47 4.41 4.35 4.30 4.25 4.21 4.18 4.14 4.11 4.09 4.06 4.04 4.02 4.00 3.99 3.97 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.89 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.85 3.85 3.84
9.82 9.58 9.35 9.12 8.90 8.69 8.50 8.32 8.16 8.01 7.88 7.76 7.65 7.55 7.46 7.39 7.31 7.25 7.19 7.14 7.09 7.05 7.01 6.97 6.94 6.91 6.88 6.86 6.84 6.82 6.80 6.79 6.77 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.65 6.63
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
k=5 10% 9.24 9.21 9.21 9.20 9.20 9.20 9.19 9.19 9.18 9.18 9.17 9.16 9.16 9.15 9.14 9.13 9.12 9.10 9.09 9.07 9.06 9.04 9.02 8.99 8.97 8.94 8.91 8.88 8.84 8.80 8.75 8.70 8.65 8.59 8.52 8.45 8.37 8.28 8.19 8.08 7.97 7.84 7.71 7.56 7.40 7.22 7.04 6.84 6.62 6.40 6.16 5.92 5.66 5.41 5.16 4.91
k=5 5% 11.07 11.04 11.04 11.04 11.03 11.03 11.03 11.02 11.02 11.01 11.00 11.00 10.99 10.98 10.97 10.96 10.95 10.94 10.92 10.91 10.89 10.87 10.85 10.83 10.80 10.78 10.75 10.71 10.67 10.63 10.59 10.54 10.48 10.42 10.36 10.28 10.20 10.11 10.02 9.91 9.80 9.67 9.53 9.38 9.22 9.04 8.85 8.65 8.43 8.19 7.95 7.69 7.42 7.15 6.87 6.60
k=5 1% 15.09 15.06 15.06 15.05 15.05 15.05 15.04 15.04 15.03 15.03 15.02 15.01 15.01 15.00 14.99 14.98 14.97 14.95 14.94 14.92 14.91 14.89 14.87 14.85 14.82 14.79 14.76 14.73 14.69 14.65 14.60 14.55 14.50 14.44 14.37 14.30 14.22 14.13 14.03 13.92 13.80 13.68 13.54 13.38 13.22 13.04 12.84 12.63 12.40 12.16 11.90 11.63 11.34 11.04 10.73 10.42
k=6 10% 10.64 10.61 10.61 10.60 10.60 10.59 10.59 10.58 10.58 10.57 10.56 10.55 10.54 10.53 10.52 10.51 10.49 10.48 10.46 10.44 10.42 10.40 10.37 10.34 10.31 10.28 10.24 10.20 10.15 10.10 10.04 9.98 9.91 9.83 9.75 9.66 9.56 9.45 9.33 9.19 9.05 8.89 8.72 8.53 8.33 8.11 7.87 7.61 7.34 7.06 6.76 6.44 6.12 5.80 5.49 5.18
k=6 5% 12.59 12.56 12.55 12.55 12.55 12.54 12.54 12.53 12.52 12.52 12.51 12.50 12.49 12.48 12.47 12.45 12.44 12.42 12.41 12.39 12.37 12.34 12.32 12.29 12.26 12.22 12.18 12.14 12.10 12.04 11.99 11.92 11.86 11.78 11.70 11.60 11.50 11.39 11.27 11.14 10.99 10.83 10.66 10.47 10.26 10.04 9.79 9.53 9.26 8.96 8.65 8.32 7.98 7.64 7.29 6.95
k=6 1% 16.81 16.78 16.77 16.77 16.76 16.76 16.75 16.75 16.74 16.73 16.73 16.72 16.71 16.70 16.69 16.68 16.66 16.65 16.63 16.61 16.59 16.56 16.54 16.51 16.48 16.44 16.40 16.36 16.31 16.26 16.21 16.14 16.07 16.00 15.91 15.82 15.72 15.61 15.49 15.35 15.20 15.04 14.86 14.67 14.46 14.23 13.98 13.72 13.43 13.12 12.80 12.45 12.09 11.71 11.33 10.94
k=7 10% 12.02 11.98 11.97 11.97 11.96 11.96 11.95 11.94 11.94 11.93 11.92 11.91 11.90 11.88 11.87 11.85 11.84 11.82 11.80 11.77 11.75 11.72 11.69 11.65 11.62 11.57 11.53 11.48 11.42 11.36 11.29 11.22 11.13 11.04 10.94 10.83 10.71 10.58 10.43 10.27 10.10 9.90 9.69 9.47 9.22 8.95 8.66 8.35 8.02 7.67 7.31 6.93 6.54 6.15 5.77 5.41
k=7 5% 14.07 14.03 14.02 14.02 14.01 14.01 14.00 13.99 13.99 13.98 13.97 13.96 13.95 13.93 13.92 13.90 13.89 13.87 13.85 13.82 13.80 13.77 13.74 13.70 13.67 13.62 13.58 13.53 13.47 13.41 13.34 13.27 13.18 13.09 12.99 12.88 12.76 12.62 12.48 12.32 12.14 11.95 11.74 11.50 11.25 10.98 10.69 10.37 10.03 9.67 9.29 8.90 8.49 8.07 7.66 7.25
k=7 1% 18.48 18.44 18.43 18.43 18.42 18.42 18.41 18.40 18.39 18.39 18.37 18.37 18.35 18.34 18.33 18.31 18.29 18.28 18.26 18.23 18.21 18.18 18.15 18.11 18.07 18.03 17.99 17.93 17.88 17.82 17.75 17.67 17.59 17.50 17.40 17.29 17.16 17.03 16.88 16.72 16.54 16.35 16.13 15.90 15.64 15.37 15.07 14.74 14.39 14.02 13.62 13.20 12.76 12.31 11.85 11.39
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
4.67 4.45 4.24 4.05 3.89 3.75 3.62 3.51 3.42 3.34 3.27 3.21 3.15 3.10 3.06 3.03 2.99 2.96 2.94 2.91 2.89 2.87 2.86 2.84 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.75 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71 2.71
6.33 6.07 5.83 5.61 5.41 5.23 5.07 4.93 4.81 4.71 4.61 4.53 4.46 4.40 4.34 4.29 4.24 4.20 4.17 4.13 4.10 4.08 4.06 4.03 4.02 4.00 3.98 3.97 3.96 3.95 3.94 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.85 3.84
10.11 9.81 9.52 9.24 8.98 8.75 8.53 8.34 8.17 8.01 7.88 7.75 7.64 7.54 7.46 7.37 7.30 7.24 7.18 7.13 7.08 7.04 7.00 6.96 6.93 6.90 6.88 6.85 6.83 6.81 6.80 6.78 6.77 6.75 6.74 6.73 6.72 6.72 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.65 6.63
4.89 4.62 4.38 4.16 3.98 3.82 3.68 3.56 3.46 3.37 3.30 3.23 3.18 3.13 3.08 3.04 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.85 2.83 2.82 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71 2.71
6.62 6.31 6.03 5.77 5.54 5.34 5.16 5.01 4.88 4.76 4.66 4.57 4.49 4.43 4.37 4.31 4.26 4.22 4.18 4.15 4.12 4.09 4.07 4.04 4.02 4.01 3.99 3.98 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.85 3.84
10.56 10.19 9.84 9.51 9.21 8.93 8.69 8.48 8.29 8.11 7.96 7.83 7.71 7.60 7.50 7.42 7.34 7.27 7.21 7.15 7.10 7.06 7.01 6.98 6.94 6.91 6.89 6.86 6.84 6.82 6.80 6.79 6.77 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.63
5.07 4.76 4.49 4.25 4.04 3.87 3.72 3.60 3.49 3.40 3.32 3.25 3.19 3.14 3.09 3.05 3.02 2.99 2.96 2.93 2.91 2.89 2.87 2.85 2.84 2.83 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71 2.71
6.87 6.51 6.18 5.89 5.64 5.42 5.23 5.06 4.92 4.80 4.70 4.60 4.52 4.45 4.39 4.33 4.28 4.23 4.19 4.16 4.13 4.10 4.07 4.05 4.03 4.01 3.99 3.98 3.97 3.95 3.94 3.93 3.92 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.85 3.84
10.94 10.50 10.09 9.72 9.39 9.08 8.82 8.58 8.38 8.19 8.03 7.89 7.76 7.64 7.54 7.45 7.37 7.30 7.23 7.17 7.12 7.07 7.03 6.99 6.96 6.92 6.90 6.87 6.85 6.83 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.63
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
k=8 10% 13.36 13.31 13.31 13.30 13.30 13.29 13.28 13.28 13.27 13.26 13.25 13.23 13.22 13.21 13.19 13.17 13.15 13.13 13.10 13.08 13.05 13.01 12.98 12.94 12.89 12.85 12.79 12.73 12.67 12.59 12.51 12.43 12.33 12.22 12.11 11.98 11.83 11.68 11.51 11.32 11.11 10.89 10.64 10.37 10.08 9.76 9.42 9.06 8.67 8.25 7.82 7.37 6.92 6.47 6.02 5.61
k=8 5% 15.51 15.46 15.45 15.45 15.44 15.44 15.43 15.42 15.41 15.40 15.39 15.38 15.37 15.35 15.33 15.32 15.30 15.27 15.25 15.22 15.19 15.16 15.12 15.08 15.04 14.99 14.94 14.88 14.81 14.74 14.66 14.57 14.47 14.37 14.25 14.12 13.98 13.82 13.65 13.46 13.25 13.02 12.78 12.51 12.21 11.89 11.54 11.17 10.77 10.35 9.90 9.43 8.95 8.46 7.98 7.51
k=8 1% 20.09 20.04 20.04 20.03 20.03 20.02 20.01 20.00 19.99 19.99 19.98 19.96 19.95 19.93 19.92 19.90 19.88 19.86 19.83 19.80 19.77 19.74 19.71 19.67 19.62 19.57 19.52 19.46 19.39 19.32 19.24 19.15 19.05 18.95 18.83 18.70 18.56 18.40 18.23 18.04 17.83 17.60 17.35 17.07 16.77 16.45 16.10 15.71 15.30 14.86 14.39 13.90 13.38 12.85 12.32 11.78
k=9 10% 14.68 14.63 14.62 14.62 14.61 14.60 14.59 14.59 14.58 14.56 14.55 14.54 14.52 14.51 14.49 14.47 14.44 14.42 14.39 14.36 14.32 14.29 14.25 14.20 14.15 14.09 14.03 13.96 13.89 13.81 13.71 13.61 13.50 13.38 13.25 13.10 12.93 12.75 12.56 12.34 12.11 11.85 11.56 11.25 10.92 10.55 10.16 9.74 9.29 8.81 8.31 7.79 7.27 6.75 6.25 5.78
k=9 5% 16.92 16.87 16.86 16.85 16.85 16.84 16.83 16.82 16.81 16.80 16.79 16.77 16.76 16.74 16.72 16.70 16.68 16.65 16.62 16.59 16.56 16.52 16.48 16.43 16.38 16.33 16.27 16.20 16.12 16.04 15.95 15.85 15.74 15.61 15.48 15.33 15.17 14.99 14.79 14.57 14.33 14.07 13.79 13.48 13.14 12.77 12.37 11.94 11.48 10.99 10.47 9.93 9.38 8.82 8.27 7.74
k=9 1% 21.67 21.61 21.61 21.60 21.59 21.58 21.58 21.57 21.56 21.55 21.53 21.52 21.50 21.49 21.47 21.45 21.42 21.40 21.37 21.34 21.31 21.27 21.23 21.18 21.13 21.07 21.01 20.95 20.87 20.79 20.70 20.59 20.48 20.36 20.23 20.08 19.91 19.73 19.53 19.32 19.07 18.81 18.52 18.21 17.86 17.49 17.08 16.64 16.17 15.66 15.12 14.55 13.96 13.35 12.73 12.13
k =10 10% 15.99 15.93 15.92 15.91 15.91 15.90 15.89 15.88 15.87 15.85 15.84 15.82 15.81 15.79 15.77 15.74 15.72 15.69 15.66 15.62 15.58 15.54 15.49 15.44 15.39 15.32 15.25 15.18 15.09 15.00 14.90 14.78 14.66 14.52 14.37 14.20 14.02 13.81 13.59 13.35 13.08 12.79 12.47 12.12 11.73 11.32 10.88 10.40 9.88 9.34 8.77 8.19 7.60 7.01 6.45 5.93
k =10 5% 18.31 18.25 18.24 18.23 18.23 18.22 18.21 18.20 18.19 18.17 18.16 18.14 18.13 18.11 18.08 18.06 18.04 18.01 17.98 17.94 17.90 17.86 17.81 17.76 17.71 17.64 17.57 17.50 17.41 17.32 17.22 17.10 16.98 16.84 16.69 16.52 16.33 16.13 15.91 15.66 15.39 15.10 14.78 14.42 14.04 13.62 13.17 12.68 12.16 11.60 11.01 10.40 9.78 9.15 8.53 7.95
k =10 1% 23.21 23.15 23.14 23.14 23.13 23.12 23.11 23.10 23.09 23.07 23.06 23.05 23.03 23.01 22.99 22.97 22.94 22.91 22.88 22.84 22.80 22.76 22.71 22.67 22.61 22.54 22.48 22.40 22.31 22.22 22.11 22.00 21.88 21.74 21.58 21.42 21.23 21.03 20.80 20.56 20.29 19.99 19.67 19.31 18.92 18.50 18.04 17.54 17.00 16.43 15.81 15.17 14.50 13.81 13.12 12.45
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
5.22 4.87 4.57 4.31 4.10 3.91 3.76 3.63 3.52 3.42 3.34 3.27 3.21 3.15 3.10 3.06 3.03 2.99 2.96 2.94 2.91 2.89 2.87 2.86 2.84 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.75 2.75 2.75 2.74 2.74 2.73 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
7.07 6.67 6.30 5.99 5.71 5.48 5.28 5.11 4.96 4.83 4.72 4.63 4.54 4.47 4.40 4.34 4.29 4.24 4.20 4.17 4.13 4.10 4.08 4.05 4.03 4.01 4.00 3.98 3.97 3.96 3.94 3.93 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
11.26 10.77 10.31 9.90 9.53 9.20 8.92 8.67 8.45 8.26 8.08 7.93 7.80 7.68 7.57 7.48 7.39 7.32 7.25 7.19 7.13 7.08 7.04 7.00 6.96 6.93 6.90 6.88 6.85 6.83 6.81 6.80 6.78 6.77 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.63
5.35 4.97 4.64 4.37 4.14 3.95 3.79 3.65 3.54 3.44 3.36 3.28 3.22 3.16 3.11 3.07 3.03 3.00 2.97 2.94 2.92 2.90 2.88 2.86 2.84 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.75 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
7.25 6.80 6.41 6.07 5.78 5.53 5.32 5.15 4.99 4.86 4.75 4.65 4.56 4.48 4.41 4.35 4.30 4.25 4.21 4.17 4.14 4.11 4.08 4.06 4.04 4.02 4.00 3.98 3.97 3.96 3.95 3.94 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
11.55 11.00 10.49 10.05 9.65 9.31 9.00 8.74 8.51 8.31 8.13 7.97 7.83 7.70 7.59 7.50 7.41 7.33 7.26 7.20 7.14 7.09 7.05 7.01 6.97 6.94 6.91 6.88 6.86 6.83 6.82 6.80 6.78 6.77 6.76 6.74 6.73 6.72 6.72 6.71 6.70 6.70 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.63
5.46 5.05 4.70 4.41 4.17 3.97 3.81 3.67 3.55 3.45 3.37 3.29 3.23 3.17 3.12 3.08 3.04 3.00 2.97 2.94 2.92 2.90 2.88 2.86 2.85 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
7.40 6.92 6.50 6.14 5.83 5.57 5.36 5.17 5.02 4.88 4.76 4.66 4.57 4.49 4.42 4.36 4.31 4.26 4.22 4.18 4.14 4.11 4.09 4.06 4.04 4.02 4.00 3.99 3.97 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
11.80 11.20 10.66 10.17 9.75 9.39 9.07 8.80 8.56 8.35 8.16 8.00 7.85 7.73 7.62 7.51 7.42 7.34 7.27 7.21 7.15 7.10 7.05 7.01 6.98 6.94 6.91 6.88 6.86 6.84 6.82 6.80 6.78 6.77 6.76 6.75 6.73 6.73 6.72 6.71 6.70 6.70 6.69 6.68 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
k =11 10% 17.28 17.21 17.20 17.19 17.18 17.17 17.16 17.15 17.14 17.13 17.11 17.09 17.07 17.05 17.03 17.00 16.97 16.94 16.91 16.87 16.83 16.78 16.73 16.67 16.61 16.54 16.46 16.37 16.28 16.18 16.06 15.94 15.80 15.64 15.47 15.29 15.08 14.86 14.61 14.34 14.04 13.71 13.35 12.96 12.54 12.07 11.57 11.04 10.46 9.85 9.22 8.56 7.90 7.25 6.63 6.06
k =11 5% 19.68 19.61 19.60 19.59 19.58 19.57 19.56 19.55 19.54 19.52 19.51 19.49 19.47 19.45 19.43 19.40 19.37 19.34 19.31 19.27 19.23 19.18 19.13 19.07 19.01 18.94 18.86 18.77 18.68 18.58 18.46 18.33 18.20 18.04 17.87 17.69 17.48 17.25 17.00 16.73 16.43 16.10 15.74 15.35 14.92 14.45 13.95 13.40 12.82 12.19 11.54 10.85 10.15 9.45 8.77 8.13
k =11 1% 24.72 24.66 24.65 24.64 24.63 24.63 24.61 24.60 24.59 24.57 24.56 24.54 24.52 24.50 24.48 24.45 24.42 24.39 24.36 24.32 24.28 24.23 24.18 24.12 24.06 23.99 23.91 23.82 23.73 23.62 23.51 23.38 23.24 23.09 22.92 22.73 22.53 22.30 22.05 21.78 21.48 21.14 20.78 20.38 19.95 19.47 18.96 18.40 17.80 17.16 16.47 15.75 15.01 14.24 13.48 12.73
k =12 10% 18.55 18.48 18.47 18.46 18.45 18.44 18.43 18.41 18.40 18.38 18.37 18.35 18.33 18.30 18.28 18.25 18.22 18.18 18.14 18.10 18.06 18.00 17.95 17.88 17.81 17.74 17.65 17.56 17.45 17.34 17.21 17.07 16.92 16.75 16.57 16.36 16.13 15.89 15.61 15.31 14.98 14.62 14.23 13.79 13.32 12.81 12.26 11.66 11.03 10.35 9.65 8.92 8.19 7.47 6.80 6.18
k =12 5% 21.03 20.95 20.94 20.94 20.93 20.92 20.90 20.89 20.88 20.86 20.84 20.82 20.80 20.78 20.75 20.73 20.69 20.66 20.62 20.58 20.53 20.48 20.42 20.36 20.29 20.21 20.13 20.03 19.93 19.82 19.69 19.55 19.40 19.23 19.04 18.84 18.61 18.36 18.09 17.78 17.45 17.09 16.69 16.26 15.78 15.27 14.71 14.10 13.45 12.76 12.04 11.28 10.51 9.74 8.99 8.29
k =12 1% 26.22 26.14 26.13 26.13 26.12 26.10 26.10 26.08 26.07 26.05 26.03 26.01 25.99 25.97 25.95 25.92 25.88 25.85 25.81 25.77 25.72 25.67 25.61 25.55 25.48 25.40 25.32 25.22 25.12 25.00 24.88 24.74 24.59 24.42 24.23 24.02 23.79 23.55 23.27 22.97 22.64 22.27 21.87 21.43 20.95 20.43 19.86 19.25 18.58 17.87 17.11 16.31 15.49 14.64 13.81 12.99
k =13 10% 19.81 19.73 19.72 19.71 19.70 19.69 19.68 19.66 19.65 19.63 19.61 19.59 19.57 19.54 19.52 19.48 19.45 19.41 19.37 19.32 19.27 19.22 19.15 19.08 19.01 18.92 18.83 18.73 18.62 18.49 18.35 18.20 18.03 17.85 17.65 17.42 17.17 16.90 16.60 16.28 15.91 15.52 15.09 14.61 14.10 13.54 12.93 12.27 11.58 10.83 10.06 9.26 8.46 7.68 6.95 6.28
k =13 5% 22.36 22.28 22.27 22.26 22.25 22.24 22.23 22.21 22.20 22.18 22.16 22.14 22.12 22.09 22.07 22.04 22.00 21.96 21.92 21.87 21.82 21.77 21.70 21.63 21.56 21.47 21.38 21.28 21.17 21.04 20.90 20.75 20.58 20.40 20.19 19.97 19.72 19.45 19.15 18.82 18.46 18.06 17.63 17.15 16.63 16.06 15.45 14.79 14.07 13.32 12.52 11.69 10.84 10.00 9.19 8.44
k =13 1% 27.69 27.61 27.60 27.59 27.58 27.57 27.56 27.54 27.53 27.51 27.49 27.47 27.44 27.42 27.39 27.36 27.33 27.29 27.25 27.20 27.15 27.09 27.03 26.96 26.88 26.80 26.71 26.61 26.49 26.37 26.23 26.08 25.91 25.72 25.52 25.29 25.05 24.77 24.47 24.14 23.78 23.38 22.94 22.46 21.93 21.36 20.74 20.06 19.33 18.55 17.72 16.85 15.95 15.02 14.11 13.24
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
5.55 5.11 4.75 4.45 4.20 4.00 3.83 3.69 3.57 3.47 3.38 3.30 3.23 3.18 3.13 3.08 3.04 3.01 2.97 2.95 2.92 2.90 2.88 2.86 2.85 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
7.54 7.02 6.57 6.19 5.88 5.61 5.39 5.20 5.04 4.90 4.78 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.15 4.12 4.09 4.06 4.04 4.02 4.00 3.99 3.97 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
12.03 11.38 10.79 10.29 9.84 9.46 9.13 8.85 8.60 8.38 8.19 8.03 7.88 7.75 7.63 7.53 7.44 7.35 7.28 7.22 7.16 7.11 7.06 7.02 6.98 6.94 6.91 6.89 6.86 6.84 6.82 6.80 6.79 6.77 6.76 6.75 6.73 6.73 6.72 6.71 6.70 6.70 6.69 6.68 6.68 6.68 6.67 6.67 6.66 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
5.63 5.17 4.79 4.48 4.23 4.02 3.85 3.70 3.58 3.48 3.39 3.31 3.24 3.18 3.13 3.08 3.04 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.85 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
7.66 7.11 6.63 6.24 5.92 5.64 5.42 5.22 5.06 4.91 4.79 4.69 4.59 4.51 4.44 4.37 4.32 4.27 4.23 4.19 4.15 4.12 4.09 4.07 4.04 4.02 4.01 3.99 3.98 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
12.23 11.54 10.92 10.38 9.92 9.53 9.19 8.89 8.64 8.41 8.22 8.05 7.89 7.76 7.64 7.54 7.44 7.36 7.29 7.22 7.16 7.11 7.06 7.02 6.98 6.95 6.92 6.89 6.87 6.84 6.82 6.80 6.79 6.77 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.66 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
5.71 5.22 4.83 4.51 4.25 4.03 3.86 3.71 3.59 3.48 3.39 3.31 3.25 3.19 3.13 3.09 3.05 3.01 2.98 2.95 2.93 2.90 2.88 2.87 2.85 2.84 2.82 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
7.77 7.18 6.69 6.28 5.95 5.67 5.44 5.24 5.07 4.93 4.80 4.69 4.60 4.52 4.44 4.38 4.32 4.27 4.23 4.19 4.15 4.12 4.09 4.07 4.05 4.03 4.01 3.99 3.98 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
12.41 11.68 11.03 10.47 9.99 9.58 9.23 8.93 8.67 8.44 8.24 8.07 7.91 7.78 7.65 7.55 7.45 7.37 7.29 7.23 7.17 7.11 7.07 7.02 6.99 6.95 6.92 6.89 6.87 6.84 6.82 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.63
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
k =14 10% 21.06 20.98 20.97 20.96 20.95 20.93 20.92 20.90 20.89 20.87 20.85 20.83 20.80 20.77 20.74 20.71 20.67 20.63 20.59 20.54 20.48 20.42 20.35 20.28 20.19 20.10 20.00 19.89 19.77 19.63 19.48 19.32 19.14 18.94 18.72 18.47 18.20 17.91 17.58 17.23 16.84 16.41 15.94 15.42 14.86 14.25 13.59 12.87 12.11 11.30 10.46 9.59 8.72 7.88 7.09 6.38
k =14 5% 23.68 23.60 23.59 23.58 23.57 23.55 23.54 23.53 23.51 23.49 23.47 23.45 23.42 23.39 23.36 23.33 23.29 23.25 23.21 23.16 23.10 23.04 22.97 22.90 22.81 22.72 22.62 22.51 22.39 22.25 22.10 21.94 21.76 21.56 21.34 21.09 20.82 20.53 20.20 19.84 19.45 19.02 18.55 18.03 17.46 16.85 16.18 15.46 14.68 13.85 12.98 12.08 11.16 10.25 9.38 8.58
k =14 1% 29.14 29.06 29.04 29.03 29.02 29.01 29.00 28.98 28.97 28.95 28.92 28.90 28.88 28.85 28.82 28.79 28.75 28.71 28.67 28.61 28.56 28.49 28.43 28.35 28.27 28.18 28.08 27.97 27.85 27.71 27.56 27.40 27.21 27.01 26.79 26.54 26.27 25.98 25.65 25.30 24.90 24.47 23.99 23.46 22.90 22.27 21.60 20.86 20.07 19.22 18.31 17.36 16.38 15.38 14.40 13.45
k =15 10% 22.31 22.21 22.20 22.19 22.18 22.17 22.15 22.14 22.12 22.10 22.08 22.05 22.02 21.99 21.96 21.93 21.89 21.84 21.79 21.74 21.68 21.61 21.54 21.46 21.37 21.27 21.16 21.04 20.91 20.77 20.60 20.43 20.23 20.02 19.78 19.51 19.23 18.91 18.56 18.17 17.75 17.28 16.78 16.22 15.61 14.95 14.24 13.46 12.64 11.76 10.85 9.91 8.97 8.06 7.22 6.47
k =15 5% 25.00 24.90 24.89 24.88 24.87 24.86 24.84 24.82 24.81 24.79 24.76 24.74 24.71 24.68 24.65 24.61 24.57 24.53 24.48 24.43 24.37 24.30 24.23 24.15 24.06 23.96 23.85 23.73 23.60 23.45 23.29 23.12 22.92 22.70 22.46 22.20 21.91 21.59 21.24 20.86 20.43 19.96 19.45 18.90 18.28 17.62 16.90 16.11 15.27 14.38 13.44 12.46 11.47 10.49 9.55 8.70
k =15 1% 30.58 30.49 30.47 30.46 30.45 30.44 30.42 30.40 30.39 30.37 30.35 30.32 30.29 30.26 30.23 30.19 30.15 30.11 30.06 30.01 29.95 29.88 29.81 29.73 29.64 29.54 29.43 29.31 29.18 29.04 28.87 28.70 28.50 28.28 28.04 27.78 27.49 27.17 26.82 26.43 26.01 25.54 25.02 24.46 23.84 23.17 22.44 21.64 20.78 19.86 18.88 17.86 16.80 15.72 14.67 13.66
k =16 10% 23.54 23.44 23.43 23.42 23.41 23.39 23.38 23.36 23.34 23.32 23.29 23.27 23.24 23.21 23.17 23.13 23.09 23.04 22.99 22.93 22.87 22.80 22.72 22.63 22.54 22.43 22.32 22.19 22.05 21.89 21.72 21.53 21.32 21.09 20.83 20.55 20.24 19.90 19.52 19.11 18.65 18.15 17.61 17.01 16.36 15.65 14.88 14.04 13.16 12.21 11.23 10.22 9.21 8.24 7.34 6.55
k =16 5% 26.30 26.19 26.18 26.17 26.16 26.15 26.13 26.11 26.09 26.07 26.05 26.02 25.99 25.96 25.93 25.89 25.84 25.80 25.74 25.69 25.62 25.55 25.47 25.39 25.29 25.19 25.07 24.94 24.80 24.64 24.47 24.28 24.07 23.84 23.58 23.30 22.99 22.65 22.27 21.86 21.40 20.90 20.35 19.75 19.09 18.38 17.60 16.76 15.86 14.89 13.88 12.83 11.76 10.71 9.72 8.81
k =16 1% 32.00 31.90 31.89 31.88 31.87 31.85 31.83 31.82 31.80 31.77 31.75 31.72 31.70 31.67 31.63 31.59 31.55 31.50 31.45 31.39 31.33 31.25 31.18 31.09 30.99 30.89 30.78 30.65 30.50 30.35 30.17 29.98 29.77 29.54 29.28 29.00 28.69 28.35 27.97 27.55 27.10 26.59 26.04 25.44 24.77 24.05 23.26 22.41 21.48 20.50 19.44 18.34 17.20 16.05 14.92 13.85
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
5.77 5.27 4.86 4.53 4.27 4.05 3.87 3.72 3.60 3.49 3.40 3.32 3.25 3.19 3.14 3.09 3.05 3.01 2.98 2.95 2.93 2.91 2.88 2.87 2.85 2.84 2.82 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
7.86 7.25 6.74 6.32 5.98 5.69 5.46 5.26 5.09 4.94 4.81 4.70 4.61 4.52 4.45 4.38 4.33 4.28 4.23 4.19 4.16 4.12 4.10 4.07 4.05 4.03 4.01 3.99 3.98 3.96 3.95 3.94 3.93 3.92 3.91 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
12.59 11.81 11.13 10.55 10.05 9.63 9.27 8.96 8.70 8.46 8.26 8.08 7.92 7.79 7.66 7.56 7.46 7.38 7.30 7.23 7.17 7.12 7.07 7.03 6.99 6.95 6.92 6.89 6.87 6.85 6.83 6.81 6.79 6.77 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
5.83 5.31 4.89 4.55 4.28 4.06 3.88 3.73 3.61 3.50 3.40 3.32 3.25 3.19 3.14 3.09 3.05 3.02 2.98 2.95 2.93 2.91 2.89 2.87 2.85 2.84 2.82 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
7.95 7.31 6.79 6.36 6.01 5.72 5.47 5.27 5.10 4.95 4.82 4.71 4.61 4.53 4.45 4.39 4.33 4.28 4.23 4.19 4.16 4.13 4.10 4.07 4.05 4.03 4.01 3.99 3.98 3.96 3.95 3.94 3.93 3.92 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
12.74 11.92 11.21 10.61 10.11 9.68 9.31 8.99 8.72 8.48 8.27 8.10 7.94 7.80 7.67 7.56 7.47 7.38 7.30 7.24 7.18 7.12 7.07 7.03 6.99 6.96 6.92 6.89 6.87 6.85 6.83 6.81 6.79 6.77 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.63
5.88 5.34 4.91 4.57 4.30 4.08 3.89 3.74 3.61 3.50 3.41 3.33 3.26 3.20 3.14 3.10 3.05 3.02 2.98 2.96 2.93 2.91 2.89 2.87 2.85 2.84 2.82 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.03 7.37 6.83 6.39 6.03 5.74 5.49 5.28 5.11 4.96 4.83 4.72 4.62 4.53 4.46 4.39 4.33 4.28 4.24 4.20 4.16 4.13 4.10 4.07 4.05 4.03 4.01 3.99 3.98 3.97 3.95 3.94 3.93 3.92 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
12.88 12.03 11.29 10.68 10.16 9.72 9.34 9.02 8.74 8.50 8.29 8.11 7.95 7.81 7.68 7.57 7.47 7.39 7.31 7.24 7.18 7.12 7.07 7.03 6.99 6.96 6.93 6.90 6.87 6.85 6.83 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
k =17 10% 24.77 24.66 24.65 24.64 24.62 24.61 24.59 24.57 24.55 24.53 24.50 24.48 24.45 24.41 24.37 24.33 24.29 24.24 24.18 24.12 24.05 23.97 23.89 23.80 23.70 23.58 23.46 23.32 23.17 23.01 22.82 22.62 22.39 22.15 21.87 21.57 21.24 20.88 20.48 20.03 19.55 19.02 18.43 17.79 17.09 16.33 15.51 14.62 13.66 12.65 11.60 10.51 9.44 8.40 7.45 6.62
k =17 5% 27.59 27.48 27.47 27.46 27.44 27.43 27.41 27.39 27.37 27.35 27.32 27.29 27.26 27.23 27.19 27.15 27.10 27.05 27.00 26.94 26.87 26.79 26.71 26.62 26.52 26.40 26.28 26.14 25.99 25.82 25.64 25.44 25.21 24.96 24.69 24.39 24.06 23.69 23.29 22.85 22.36 21.83 21.24 20.60 19.89 19.13 18.29 17.39 16.43 15.39 14.31 13.18 12.04 10.92 9.87 8.92
k =17 1% 33.41 33.30 33.29 33.28 33.26 33.25 33.23 33.21 33.19 33.17 33.14 33.12 33.08 33.05 33.01 32.97 32.93 32.87 32.82 32.76 32.69 32.61 32.53 32.44 32.34 32.23 32.10 31.97 31.81 31.64 31.46 31.26 31.03 30.78 30.51 30.21 29.88 29.51 29.11 28.66 28.17 27.64 27.05 26.40 25.69 24.92 24.07 23.16 22.17 21.11 19.99 18.80 17.59 16.36 15.16 14.03
k =18 10% 25.99 25.87 25.86 25.85 25.83 25.82 25.80 25.78 25.76 25.73 25.71 25.68 25.65 25.61 25.57 25.53 25.48 25.42 25.36 25.30 25.22 25.15 25.06 24.96 24.85 24.73 24.60 24.45 24.29 24.12 23.92 23.70 23.47 23.20 22.91 22.59 22.24 21.85 21.42 20.95 20.44 19.87 19.25 18.57 17.82 17.01 16.13 15.18 14.16 13.08 11.96 10.80 9.65 8.55 7.55 6.68
k =18 5% 28.87 28.75 28.74 28.73 28.71 28.70 28.68 28.66 28.64 28.62 28.59 28.56 28.53 28.49 28.45 28.41 28.36 28.30 28.24 28.18 28.10 28.03 27.94 27.84 27.73 27.61 27.48 27.33 27.17 27.00 26.80 26.58 26.34 26.08 25.79 25.47 25.12 24.73 24.30 23.83 23.31 22.74 22.12 21.43 20.68 19.87 18.98 18.02 16.99 15.89 14.73 13.53 12.31 11.13 10.01 9.01
k =18 1% 34.81 34.69 34.68 34.66 34.65 34.63 34.61 34.59 34.57 34.55 34.52 34.50 34.46 34.43 34.39 34.34 34.29 34.24 34.18 34.11 34.04 33.96 33.87 33.77 33.66 33.55 33.42 33.27 33.11 32.93 32.74 32.52 32.28 32.01 31.72 31.40 31.05 30.66 30.23 29.76 29.24 28.67 28.04 27.35 26.60 25.77 24.87 23.90 22.84 21.71 20.51 19.25 17.96 16.65 15.38 14.20
k =19 10% 27.20 27.08 27.07 27.06 27.04 27.02 27.00 26.98 26.96 26.93 26.91 26.87 26.84 26.80 26.76 26.71 26.66 26.60 26.54 26.47 26.39 26.31 26.22 26.11 26.00 25.87 25.73 25.58 25.41 25.22 25.01 24.78 24.53 24.25 23.94 23.61 23.23 22.82 22.37 21.87 21.32 20.72 20.06 19.33 18.54 17.68 16.75 15.74 14.66 13.51 12.31 11.08 9.86 8.70 7.65 6.74
k =19 5% 30.14 30.02 30.01 30.00 29.98 29.96 29.94 29.92 29.90 29.87 29.85 29.81 29.78 29.74 29.70 29.65 29.60 29.54 29.48 29.41 29.33 29.25 29.16 29.05 28.94 28.81 28.67 28.52 28.35 28.16 27.95 27.72 27.47 27.19 26.88 26.54 26.17 25.76 25.30 24.80 24.25 23.65 22.99 22.26 21.47 20.60 19.66 18.64 17.54 16.37 15.14 13.86 12.57 11.32 10.15 9.10
k =19 1% 36.19 36.07 36.06 36.04 36.03 36.01 35.99 35.97 35.95 35.92 35.89 35.86 35.83 35.79 35.75 35.70 35.65 35.59 35.53 35.46 35.38 35.30 35.20 35.10 34.99 34.86 34.72 34.56 34.39 34.21 34.00 33.77 33.52 33.23 32.93 32.59 32.21 31.80 31.35 30.85 30.29 29.69 29.02 28.29 27.49 26.62 25.66 24.63 23.51 22.31 21.03 19.69 18.31 16.93 15.60 14.36
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
5.93 5.37 4.93 4.59 4.31 4.09 3.90 3.75 3.62 3.51 3.41 3.33 3.26 3.20 3.14 3.10 3.06 3.02 2.99 2.96 2.93 2.91 2.89 2.87 2.85 2.84 2.83 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71
8.10 7.41 6.86 6.41 6.05 5.75 5.50 5.29 5.12 4.96 4.83 4.72 4.62 4.54 4.46 4.39 4.34 4.28 4.24 4.20 4.16 4.13 4.10 4.07 4.05 4.03 4.01 3.99 3.98 3.97 3.95 3.94 3.93 3.92 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
13.01 12.12 11.37 10.73 10.20 9.75 9.37 9.04 8.76 8.52 8.31 8.12 7.96 7.81 7.69 7.58 7.48 7.39 7.31 7.24 7.18 7.13 7.08 7.03 6.99 6.96 6.93 6.90 6.87 6.85 6.83 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.65 6.65 6.65 6.65 6.63
5.97 5.40 4.95 4.60 4.32 4.10 3.91 3.75 3.62 3.51 3.42 3.33 3.26 3.20 3.15 3.10 3.06 3.02 2.99 2.96 2.93 2.91 2.89 2.87 2.85 2.84 2.83 2.81 2.80 2.79 2.78 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.16 7.46 6.89 6.44 6.07 5.77 5.52 5.30 5.13 4.97 4.84 4.73 4.63 4.54 4.46 4.40 4.34 4.29 4.24 4.20 4.16 4.13 4.10 4.07 4.05 4.03 4.01 3.99 3.98 3.97 3.95 3.94 3.93 3.92 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
13.13 12.21 11.44 10.79 10.24 9.79 9.40 9.06 8.78 8.53 8.32 8.13 7.96 7.82 7.69 7.58 7.48 7.40 7.32 7.25 7.19 7.13 7.08 7.03 7.00 6.96 6.93 6.90 6.87 6.85 6.83 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
6.01 5.42 4.97 4.61 4.33 4.10 3.92 3.76 3.63 3.52 3.42 3.34 3.27 3.20 3.15 3.10 3.06 3.02 2.99 2.96 2.93 2.91 2.89 2.87 2.85 2.84 2.83 2.81 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.22 7.50 6.92 6.46 6.09 5.78 5.53 5.31 5.13 4.98 4.85 4.73 4.63 4.54 4.47 4.40 4.34 4.29 4.24 4.20 4.16 4.13 4.10 4.08 4.05 4.03 4.01 4.00 3.98 3.97 3.95 3.94 3.93 3.92 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.84
13.25 12.29 11.50 10.84 10.28 9.81 9.42 9.09 8.79 8.54 8.33 8.14 7.97 7.83 7.70 7.58 7.49 7.40 7.32 7.25 7.19 7.13 7.08 7.04 7.00 6.96 6.93 6.90 6.87 6.85 6.83 6.81 6.79 6.78 6.77 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
k =20 10% 28.41 28.28 28.27 28.26 28.24 28.22 28.20 28.18 28.15 28.13 28.10 28.06 28.03 27.99 27.94 27.89 27.84 27.78 27.71 27.64 27.56 27.47 27.37 27.26 27.14 27.01 26.86 26.69 26.52 26.32 26.10 25.86 25.59 25.29 24.97 24.61 24.22 23.78 23.30 22.78 22.20 21.56 20.86 20.10 19.26 18.35 17.36 16.29 15.14 13.93 12.65 11.35 10.07 8.84 7.74 6.80
k =20 5% 31.41 31.28 31.27 31.25 31.24 31.22 31.20 31.18 31.15 31.13 31.10 31.06 31.03 30.99 30.94 30.89 30.84 30.78 30.71 30.64 30.56 30.47 30.37 30.26 30.14 30.00 29.86 29.69 29.51 29.32 29.10 28.85 28.59 28.29 27.97 27.61 27.21 26.78 26.30 25.77 25.19 24.55 23.85 23.08 22.24 21.32 20.33 19.25 18.08 16.84 15.54 14.19 12.83 11.50 10.27 9.19
k =20 1% 37.57 37.44 37.43 37.41 37.39 37.38 37.35 37.33 37.31 37.28 37.25 37.22 37.18 37.14 37.10 37.05 36.99 36.93 36.87 36.79 36.71 36.62 36.52 36.41 36.29 36.16 36.01 35.85 35.67 35.47 35.25 35.01 34.74 34.45 34.12 33.76 33.37 32.93 32.45 31.92 31.34 30.70 29.99 29.22 28.37 27.45 26.44 25.34 24.16 22.88 21.53 20.12 18.66 17.20 15.80 14.50
k =21 10% 29.62 29.48 29.47 29.45 29.43 29.41 29.39 29.37 29.34 29.32 29.28 29.25 29.21 29.17 29.12 29.07 29.01 28.95 28.88 28.80 28.72 28.62 28.52 28.40 28.27 28.13 27.98 27.81 27.62 27.41 27.18 26.92 26.64 26.33 25.99 25.61 25.20 24.74 24.24 23.68 23.07 22.40 21.66 20.85 19.97 19.01 17.96 16.83 15.62 14.34 12.99 11.62 10.26 8.98 7.83 6.85
k =21 5% 32.67 32.54 32.52 32.51 32.49 32.47 32.45 32.43 32.40 32.37 32.34 32.30 32.27 32.22 32.18 32.13 32.07 32.00 31.93 31.86 31.77 31.68 31.57 31.46 31.33 31.19 31.03 30.86 30.67 30.46 30.23 29.98 29.70 29.39 29.04 28.67 28.25 27.79 27.29 26.73 26.12 25.45 24.71 23.90 23.01 22.04 20.99 19.85 18.62 17.31 15.93 14.50 13.07 11.68 10.39 9.27
k =21 1% 38.93 38.80 38.78 38.77 38.75 38.73 38.71 38.69 38.66 38.63 38.60 38.56 38.53 38.49 38.44 38.39 38.33 38.26 38.19 38.12 38.03 37.94 37.83 37.72 37.59 37.45 37.29 37.13 36.94 36.73 36.49 36.24 35.96 35.65 35.30 34.93 34.51 34.05 33.54 32.98 32.37 31.69 30.95 30.14 29.25 28.27 27.21 26.05 24.80 23.46 22.03 20.53 19.00 17.47 15.99 14.64
k =22 10% 30.81 30.67 30.66 30.64 30.62 30.60 30.58 30.56 30.53 30.50 30.47 30.43 30.39 30.34 30.29 30.24 30.18 30.11 30.04 29.96 29.87 29.77 29.66 29.54 29.41 29.26 29.09 28.92 28.72 28.50 28.25 27.99 27.69 27.37 27.01 26.61 26.17 25.69 25.16 24.58 23.94 23.23 22.45 21.61 20.68 19.66 18.56 17.37 16.10 14.74 13.32 11.88 10.45 9.11 7.91 6.90
k =22 5% 33.92 33.78 33.77 33.75 33.73 33.71 33.69 33.67 33.64 33.61 33.58 33.54 33.50 33.46 33.41 33.35 33.29 33.22 33.15 33.07 32.98 32.88 32.77 32.65 32.52 32.37 32.21 32.03 31.83 31.61 31.36 31.10 30.80 30.48 30.12 29.72 29.28 28.80 28.27 27.68 27.04 26.33 25.56 24.70 23.77 22.75 21.64 20.44 19.15 17.77 16.31 14.81 13.31 11.85 10.51 9.34
k =22 1% 40.29 40.15 40.13 40.12 40.10 40.08 40.06 40.03 40.00 39.98 39.94 39.91 39.86 39.82 39.77 39.72 39.66 39.59 39.52 39.43 39.34 39.24 39.14 39.01 38.88 38.73 38.57 38.39 38.19 37.97 37.73 37.46 37.17 36.84 36.48 36.08 35.64 35.16 34.63 34.04 33.40 32.69 31.90 31.05 30.11 29.09 27.96 26.75 25.43 24.01 22.51 20.94 19.33 17.72 16.18 14.77
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
6.04 5.45 4.99 4.63 4.34 4.11 3.92 3.76 3.63 3.52 3.42 3.34 3.27 3.20 3.15 3.10 3.06 3.02 2.99 2.96 2.93 2.91 2.89 2.87 2.85 2.84 2.83 2.81 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.28 7.54 6.95 6.48 6.10 5.80 5.54 5.32 5.14 4.98 4.85 4.73 4.63 4.55 4.47 4.40 4.34 4.29 4.24 4.20 4.16 4.13 4.10 4.08 4.05 4.03 4.01 4.00 3.98 3.97 3.95 3.94 3.93 3.92 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.84
13.35 12.37 11.56 10.88 10.32 9.84 9.44 9.10 8.81 8.55 8.34 8.15 7.98 7.83 7.70 7.59 7.49 7.40 7.32 7.25 7.19 7.13 7.08 7.04 7.00 6.96 6.93 6.90 6.87 6.85 6.83 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
6.07 5.47 5.00 4.64 4.35 4.12 3.93 3.77 3.64 3.52 3.43 3.34 3.27 3.21 3.15 3.10 3.06 3.02 2.99 2.96 2.93 2.91 2.89 2.87 2.85 2.84 2.83 2.81 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.33 7.57 6.98 6.50 6.12 5.81 5.55 5.33 5.15 4.99 4.85 4.74 4.64 4.55 4.47 4.40 4.34 4.29 4.24 4.20 4.16 4.13 4.10 4.08 4.05 4.03 4.01 4.00 3.98 3.97 3.96 3.94 3.93 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.84
13.45 12.44 11.61 10.92 10.35 9.87 9.46 9.12 8.82 8.57 8.35 8.15 7.98 7.84 7.71 7.59 7.49 7.40 7.32 7.25 7.19 7.13 7.08 7.04 7.00 6.96 6.93 6.90 6.88 6.85 6.83 6.81 6.79 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
6.10 5.49 5.02 4.65 4.36 4.13 3.93 3.77 3.64 3.53 3.43 3.34 3.27 3.21 3.15 3.10 3.06 3.02 2.99 2.96 2.93 2.91 2.89 2.87 2.86 2.84 2.83 2.81 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.37 7.60 7.00 6.52 6.13 5.82 5.56 5.34 5.15 4.99 4.86 4.74 4.64 4.55 4.47 4.41 4.35 4.29 4.24 4.20 4.17 4.13 4.10 4.08 4.05 4.03 4.01 4.00 3.98 3.97 3.96 3.94 3.93 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.84
13.54 12.51 11.66 10.96 10.38 9.89 9.48 9.13 8.83 8.58 8.35 8.16 7.99 7.84 7.71 7.60 7.50 7.41 7.33 7.26 7.19 7.14 7.09 7.04 7.00 6.96 6.93 6.90 6.88 6.85 6.83 6.81 6.80 6.78 6.76 6.75 6.74 6.73 6.72 6.71 6.70 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
ln(QT/k) -∞ -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
k =23 10% 32.01 31.86 31.84 31.83 31.81 31.79 31.76 31.74 31.71 31.68 31.64 31.60 31.56 31.51 31.46 31.41 31.34 31.27 31.20 31.11 31.02 30.91 30.80 30.67 30.53 30.38 30.21 30.02 29.81 29.58 29.33 29.05 28.74 28.39 28.02 27.60 27.14 26.64 26.08 25.47 24.80 24.06 23.24 22.35 21.38 20.31 19.16 17.91 16.57 15.14 13.65 12.13 10.64 9.23 7.98 6.95
k =23 5% 35.17 35.02 35.01 34.99 34.97 34.95 34.93 34.90 34.87 34.84 34.81 34.77 34.73 34.68 34.63 34.57 34.51 34.44 34.36 34.28 34.18 34.08 33.96 33.84 33.70 33.54 33.37 33.18 32.98 32.74 32.49 32.21 31.90 31.56 31.18 30.77 30.31 29.80 29.25 28.63 27.96 27.22 26.40 25.51 24.53 23.46 22.29 21.03 19.67 18.22 16.69 15.12 13.53 12.01 10.61 9.41
k =23 1% 41.64 41.49 41.47 41.46 41.44 41.42 41.39 41.37 41.34 41.31 41.27 41.24 41.19 41.15 41.09 41.04 40.97 40.91 40.83 40.74 40.65 40.54 40.43 40.30 40.16 40.01 39.84 39.65 39.44 39.21 38.96 38.68 38.37 38.03 37.64 37.23 36.77 36.26 35.70 35.09 34.41 33.67 32.85 31.95 30.97 29.89 28.71 27.43 26.05 24.57 22.99 21.33 19.64 17.96 16.36 14.90
k =24 10% 33.20 33.04 33.02 33.01 32.99 32.97 32.94 32.91 32.88 32.85 32.82 32.78 32.73 32.68 32.63 32.57 32.50 32.43 32.35 32.26 32.16 32.05 31.93 31.80 31.65 31.49 31.31 31.12 30.90 30.66 30.39 30.10 29.78 29.42 29.03 28.59 28.11 27.58 27.00 26.36 25.65 24.88 24.03 23.10 22.07 20.96 19.75 18.44 17.03 15.53 13.97 12.38 10.81 9.35 8.05 6.99
k =24 5% 36.42 36.26 36.24 36.23 36.21 36.18 36.16 36.13 36.10 36.07 36.03 35.99 35.95 35.90 35.85 35.79 35.72 35.65 35.57 35.48 35.38 35.27 35.15 35.02 34.87 34.71 34.53 34.34 34.12 33.88 33.61 33.32 32.99 32.64 32.24 31.81 31.33 30.80 30.22 29.57 28.87 28.09 27.24 26.30 25.28 24.16 22.93 21.61 20.19 18.67 17.06 15.41 13.76 12.16 10.72 9.47
k =24 1% 42.98 42.82 42.81 42.79 42.77 42.75 42.73 42.70 42.67 42.64 42.60 42.56 42.52 42.47 42.41 42.35 42.29 42.21 42.13 42.04 41.94 41.84 41.72 41.58 41.44 41.27 41.10 40.90 40.68 40.44 40.17 39.88 39.56 39.20 38.81 38.37 37.89 37.36 36.77 36.13 35.42 34.64 33.79 32.85 31.82 30.69 29.46 28.12 26.66 25.11 23.46 21.72 19.95 18.19 16.52 15.02
k =25 10% 34.38 34.22 34.20 34.18 34.16 34.14 34.12 34.09 34.06 34.02 33.98 33.94 33.90 33.85 33.79 33.73 33.66 33.58 33.50 33.40 33.30 33.19 33.06 32.92 32.77 32.60 32.42 32.21 31.98 31.73 31.46 31.15 30.81 30.44 30.03 29.57 29.07 28.52 27.91 27.24 26.51 25.70 24.81 23.83 22.77 21.60 20.33 18.96 17.49 15.92 14.29 12.62 10.99 9.46 8.12 7.03
k =25 5% 37.65 37.49 37.47 37.45 37.43 37.41 37.39 37.36 37.33 37.29 37.26 37.21 37.17 37.12 37.06 37.00 36.93 36.85 36.77 36.68 36.57 36.46 36.33 36.20 36.04 35.87 35.69 35.48 35.25 35.00 34.73 34.42 34.08 33.71 33.30 32.84 32.34 31.79 31.18 30.51 29.77 28.96 28.07 27.09 26.02 24.85 23.57 22.19 20.70 19.11 17.43 15.70 13.97 12.31 10.81 9.53
k =25 1% 44.31 44.15 44.14 44.11 44.10 44.07 44.05 44.02 43.99 43.95 43.92 43.88 43.83 43.78 43.72 43.66 43.59 43.51 43.43 43.34 43.24 43.12 43.00 42.86 42.71 42.54 42.35 42.15 41.92 41.66 41.39 41.08 40.74 40.37 39.96 39.50 39.00 38.45 37.84 37.17 36.43 35.61 34.72 33.74 32.66 31.48 30.19 28.79 27.27 25.64 23.91 22.10 20.25 18.42 16.68 15.13
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 +∞
6.13 5.50 5.03 4.66 4.37 4.13 3.94 3.78 3.64 3.53 3.43 3.35 3.27 3.21 3.15 3.11 3.06 3.02 2.99 2.96 2.94 2.91 2.89 2.87 2.86 2.84 2.83 2.82 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.42 7.64 7.02 6.54 6.15 5.83 5.56 5.34 5.16 5.00 4.86 4.74 4.64 4.55 4.48 4.41 4.35 4.29 4.25 4.20 4.17 4.13 4.10 4.08 4.05 4.03 4.01 4.00 3.98 3.97 3.96 3.94 3.93 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.85 3.84
13.63 12.57 11.71 11.00 10.41 9.91 9.50 9.15 8.85 8.59 8.36 8.17 8.00 7.85 7.72 7.60 7.50 7.41 7.33 7.26 7.19 7.14 7.09 7.04 7.00 6.96 6.93 6.90 6.88 6.85 6.83 6.81 6.80 6.78 6.77 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
6.15 5.52 5.04 4.67 4.37 4.14 3.94 3.78 3.65 3.53 3.43 3.35 3.27 3.21 3.16 3.11 3.06 3.03 2.99 2.96 2.94 2.91 2.89 2.87 2.86 2.84 2.83 2.82 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.75 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.46 7.66 7.04 6.55 6.16 5.84 5.57 5.35 5.16 5.00 4.86 4.75 4.64 4.55 4.48 4.41 4.35 4.29 4.25 4.21 4.17 4.13 4.10 4.08 4.05 4.03 4.01 4.00 3.98 3.97 3.96 3.94 3.93 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.84
13.71 12.63 11.75 11.03 10.43 9.93 9.52 9.16 8.86 8.59 8.37 8.17 8.00 7.85 7.72 7.60 7.50 7.41 7.33 7.26 7.19 7.14 7.09 7.04 7.00 6.97 6.93 6.90 6.88 6.85 6.83 6.81 6.80 6.78 6.77 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
6.18 5.54 5.05 4.68 4.38 4.14 3.95 3.79 3.65 3.53 3.43 3.35 3.28 3.21 3.16 3.11 3.06 3.03 2.99 2.96 2.94 2.91 2.89 2.87 2.86 2.84 2.83 2.82 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.75 2.75 2.74 2.74 2.73 2.73 2.73 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71
8.50 7.69 7.06 6.56 6.17 5.85 5.58 5.35 5.17 5.01 4.87 4.75 4.65 4.56 4.48 4.41 4.35 4.29 4.25 4.21 4.17 4.14 4.11 4.08 4.06 4.03 4.01 4.00 3.98 3.97 3.96 3.94 3.93 3.93 3.92 3.91 3.90 3.90 3.89 3.89 3.88 3.88 3.88 3.87 3.87 3.87 3.86 3.86 3.86 3.86 3.86 3.86 3.85 3.85 3.85 3.85 3.84
13.79 12.69 11.79 11.06 10.46 9.95 9.53 9.17 8.86 8.60 8.37 8.18 8.00 7.85 7.72 7.61 7.50 7.41 7.33 7.26 7.20 7.14 7.09 7.04 7.00 6.97 6.93 6.90 6.88 6.85 6.83 6.81 6.80 6.78 6.77 6.75 6.74 6.73 6.72 6.71 6.71 6.70 6.69 6.69 6.68 6.68 6.67 6.67 6.67 6.66 6.66 6.66 6.66 6.65 6.65 6.65 6.63
