under the Growth-Curve Model in High-Dimension and its Error Bound.

成長曲線モデルにおける一様共分散構造の検定に対する 尤度比統計量の高次元漸近展開と誤差評価

Asymptotic Expansion of the Distribution of LR Statistics for Testing the Intraclass Correlation

under the Growth-Curve Model in High-Dimension and its Error Bound.

数学専攻 加藤 直広 KATO Naohiro

1 Introduction

Letxbe ap-dimensional random vector distributed as ap-variate normal distri- bution with an unknown mean vectorµand an unknown covariance matrixΣ, denoted,x∼Np(µ,Σ). Suppose thatx1, . . . ,xn is a sampleN(=n+ 1, n≥p) independent observation vectors onx. Consider the problem of testing the null hypothesis

H:Σ=Σ_I =σ²{(1−ρ)I+ρ11} (1) against all alternatives, where1= (1,1, . . . ,1);σ² andρare parameters which satisfyσ²>0 and−1/(p−1)< ρ <1. The covariance structure (1) is known as the intraclass correlation model. The likelihood ratio criterionλfor testing the hypothesis (1) is given by

λ=

(p−1)^p−1|S|

uv^{p−1 N}₂

, where

u=1

p1S1, v= tr(S)−u.

Here,S denotes the sample covariance matrix based onx₁, . . . ,x_n.

For largeN and fixedp, Box type asymptotic expansion of the null distribu- tion ofλ^∗=λ^2/N is given (see e.g. Siotani et, al. 1985). In this paper, we shall derive asymptotic distribution under the following high-dimensional framework.

A:p→ ∞, N → ∞, p

N →c∈(0,1).

In this paper, we derive the asymptotic null distribution ofλ^∗on the framework A. To demonstrate the accuracy of our approximation, numerical simulations are done. Furthemore, we obtain computable error bound between the exact distribution and the asymptotic distribution.

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2 Main Result

In this section, we derive the asymptotic null distribution ofλ^∗on the framework A. Theh-th moment ofλ^∗ is given by

E(λ^∗h) =K(p−1)^(p−1)h _p−1

j=1Γ[ⁿ₂ +h−^j₂] Γ[(p−1)(ⁿ₂ +h)] , where K = Γ[ⁿ₂(p−1)]/_p−1

j=1Γ[ⁿ₂ −^j₂]. The cumulant-generating function of logλ^∗ is expanded as follows;

log E[exp(itlogλ^∗)] =itµn,p+1

2(it)²σ²_n,p+ ∞ r=3

1

r!(it)^rγr,n,p, (2) where

µn,p= (p−1) log(p−1) +ψ(p−1)(n 2 −1

2)−ψ(n

2(p−1))(p−1), σ_n,p² =ψ_(p−1)(n

2 −1

2)−ψ(n

2(p−1))(p−1)², γr,n,p=ψ^(r−1)_(p−1)(n

2 −1

2)−ψ^(r−1)(n

2(p−1))(p−1)^r.

Here,ψis di-gamma function defined byψ(z) = (d/dz) log Γ(z) andψ(p−1)(a) = _p−1

j=1ψ(a−¹₂(j−1)). Let

zn,p=logλ^∗−µn,p

σ²_n,p .

From (2), the r-th cumulant of zn,p can be expressed as γr,n,p/(σ_n,p² )^r2. The characteristic function of thezn,p can be expressed as follows.

E[exp(itzn,p)] = exp −t²

2

1 + ∞ k=1

(it)^3k k!

∞ j=0

γ_k,j,n,p (it)^j

, where

γ_k,j,n,p=

j1+···+jk=j

γj1+3,n,p· · ·γjk+3,n,p

(j1+ 3)!· · ·(jk+ 3)!σ^j+3kn,p . Let

φs(t) = exp(−t² 2)

1 +

s k=1

(it)^3k k!

s−k

j=0

γ_k,j,n,p(it)^j

. (3)

Inverting (3), we obtain the Edgeworth expansion ofzn,pup to the orderO(m^−s) as

Φ_s(x) = Φ(x)−φ(x) ^s

k=1

1 k!

s−k

j=0

γ_k,j,n,p h3k+j−1(x)

2

where Φ andφare the distribution function of standard normal distribution and its derivatives, respectively; hr(x) denotes the r-th order Hermite polynomial defined by

d dx

_r exp

−x² 2

= (−1)^rhr(x) exp −x²

2

.

3 Numerical Comparison

This section presents the results of numerical simulations to demonstrate the effectiveness of asymptotic normality of zn,p for some value of p and n. In all our simulations, we tookσ² = 1, ρ= 1/2. In Table 1, we list the estimated significance levels forzn,pforN = 100 calculated by using 1,000,000 repetitions with nominal significance levels of 0.01,0.05,0.50,0.95 and 0.99.

Table 1．Actual probabilities ofzn,p forN = 100

0.01 0.05 0.50 0.95 0.99

p=10 0.0046 0.0408 0.5261 0.9366 0.9794 p=20 0.0068 0.0445 0.5127 0.9440 0.9855 p=30 0.0078 0.0463 0.5089 0.9461 0.9870 p=40 0.0081 0.0471 0.5063 0.9471 0.9879 p=50 0.0085 0.0479 0.5046 0.9475 0.9883 p=60 0.0087 0.0477 0.5048 0.9478 0.9884 p=70 0.0089 0.0483 0.5049 0.9481 0.9887 p=80 0.0088 0.0480 0.5044 0.9482 0.9887 p=90 0.0087 0.0480 0.5054 0.9480 0.9886 From Table 1, we find thatzn,p give a good approximation forp≥50.

For large N and fixed p, the chi-square approximation of LR statistic is given. We list significance levels for −2τlog(λ^∗)ⁿ² in Table 2 using the same setting as for the simulations presented in Table 1, where

τ = 1− p(2p³+p²−4p−3) 6n(p−1)(p²+p−4) is Bartlett correction factor.

Table 2．Actual probabilities of−2τlog(λ^∗)ⁿ2 forN = 100

0.01 0.05 0.50 0.95 0.99

p=10 0.0100 0.0502 0.5013 0.9502 0.9902 p=20 0.0107 0.0527 0.5121 0.9533 0.9909 p=30 0.0127 0.0607 0.5419 0.9606 0.9928 p=40 0.0189 0.0821 0.6077 0.9739 0.9958 p=50 0.0385 0.1407 1.000 1.000 1.000 p=60 1.0000 1.0000 1.0000 1.0000 1.0000 p=70 1.0000 1.0000 1.0000 1.0000 1.0000 p=80 1.0000 1.0000 1.0000 1.0000 1.0000 p=90 1.0000 1.0000 1.0000 1.0000 1.0000

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From Table 2, the approximation of−2τlog(λ^∗)ⁿ² is accurate forp≤20.

4 Error Bound

In this section we shall find an upper bound of error between the distribution of zn,p and the asymptotic distribution. The following inequality gives an upper bound.

supx |P(zn,p≤x)−Φ_s(x)| ≤ ^∞

−∞

1

|t||E[exp(itzn,p)]−φs(t)|dt, We obtain the following error bound.

sup

x |P(zn,p≤x)−Φ_s(x)|

<

s k=1

2^3k2 k!

4p²

σ^3kmn(n−1)B[2v/σn,p]^k−^s−k

j=0

γ_k,j,n,p

Γ 3k

2 −Γ

3k 2 ,m²v²

2

+ B[2v/σn,p]^s+12^7s+72 p^2s+2 m^s+1n^s+1(n−1)^s+1σn,p^3s+3

1−8B[2v/σn,p]vp² n(n−1)σ³_n,p

^−3s−3₂

· Γ

3 2(s+ 1)

−Γ 3

2(s+ 1),m²v² 2

1−8B[2v/σn,p]vp² n(n−1)σ_n,p³

+ (^m₂ + [^p+1₂ ]−1)²

m²v²(^p−1₂ {[^p+1₂ ]−1} −1)·

1 + m²v²

σ²_n,p(^m₂ + [^p+1₂ ]−1)²

₋^p−1_{2 {[}^p+1_{2 ]−1}+1}

+ 2

m²v²e^−m22v2 + s k=1

2^3k+j2 k!

s−k

j=0

γ_k,j,n,p

Γ

3k+j 2

−Γ

3k+j 2 ,m²v²

2

.

Here,m=n−p+ 1, 0 < v <(σ_n,p² )¹²/2 is a constant, [·] is Gaussian integer, Γ(z, a) =_∞

a x^z−1e^−xdx is imcompete gamma function and B[v] =− 1

2v − 1 v²− 1

v³log(1−v) + 1 m

1 1−v. We computed the values of the error bound forN = 100 ands= 0.

p 30 50 70 90

0.180594 0.101753 0.0816023 0.123004

Acknowledgement: The author would like to thank Professor Yasunori Fu- jikoshi of Chuo University for continuous instruction. In addition, I am grateful to seniors of Sugiyama laboratory for the help of this study.

References

[1] Siotani, M., Hayakawa, T. and Fujikoshi, Y. (1985). Modern Multivariate Statistical Analysis: A Graduate Course and Handbook, American Sciences Press, Columbus, OH.

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