Improved transformation of ϕ-divergence goodness-of-fit test statistics based on minimum ϕ -divergence estimator for GLIM of binary data

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1 SUT Journal of Mathematics Vol. 52, No. 2 (216), Improved transformation of ϕ-diverence oodness-of-fit test statistics based on minimum ϕ -diverence estimator for GLIM of binary data Nobuhiro Taneichi, Yuri Sekiya and Jun Toyama (Received September 16, 216; Revised November 14, 216) Abstract. Generalized linear models of binary data includin a loistic reression model and a probit model are considered. For testin the null hypothesis that the considered model is correct, the ϕ-diverence family of oodness-offit test statistics C ϕϕ that is based on a minimum ϕ -diverence estimator is considered. The family of statistics C ϕϕ includes a power diverence family of statistics R a,b that is based on a minimum power diverence estimator. The derivation of an expression of a continuous term of asymptotic expansion for the distribution of C ϕϕ under the null hypothesis is shown. Usin the expression, a transformed C ϕϕ statistic that improves the speed of converence to the chi-square limitin distribution of C ϕϕ is obtained. In the case of R a,b, it is numerically shown that the transformed statistics usually perform better than the oriinal statistics with respect to speed of converence to the chi-square limitin distribution and it is also numerically shown that the power of the transformed statistics is almost the same as that of the oriinal statistics. AMS 21 Mathematics Subject Classification. 62E2, 62H1. Key words and phrases. Asymptotic expansion, binary data, ϕ-diverence statistics, eneralized linear model, improved transformation, minimum ϕ - diverence estimator. 1. Introduction We discuss eneralized linear models (Nelder and Wedderburn [1]) in which the response variables are measured on a binary scale. Let N independent random variables Y α, α = 1,..., N correspondin to the number of successes in N different subroups be distributed accordin to binomial distributions B(n α, π α ), α = 1,..., N. If we use a monotone and differentiable function as a link function, we obtain a eneralized linear model for binary data as 193

2 194 N. TANEICHI, Y. SEKIYA AND J. TOYAMA follows. (1.1) (π α ) = x αβ (α = 1,..., N), where x α = (x α1,..., x αp ) (α = 1,..., N) are covariate vectors and β = (β 1,..., β p ) is an unknown parameter vector and p < N. We consider a minimum ϕ -diverence estimator of model (1.1) and also consider a ϕ-diverence oodness-of-fit test statistic based on the estimator. Let y α (α = 1,..., N) be an observed value of Y α (α = 1,..., N), then the minimum ϕ -diverence estimator of model (1.1) is iven by ˆβ ϕ = ar min β Θ D ϕ, where D ϕ = 1 N n α y α 1 y α π α(β)ϕ n α + (1 π α (β))ϕ n α π α (β) 1 π α (β), where ϕ is a real convex function in (, ) satisfyin ϕ (1) = ϕ (1) =, ϕ (1) = 1, ϕ (/) =, ϕ (x/) = lim u ϕ (u)/u, and Θ is an open subset of R p (Pardo [11]). When we choose a convex function {a(a + 1)} 1 {t a+1 t + a(1 t)} (a, 1) (1.2) ϕ a (t) = t lo t + 1 t (a = ) lo t 1 + t (a = 1), as ϕ (t), D ϕ becomes a Kullback diverence measure (Kullback [7]). Then, in this case, estimator ˆβ ϕ becomes the maximum likelihood estimator. Therefore, the maximum likelihood estimator is a special case of the minimum ϕ - diverence estimator. In order to test the null hypothesis (1.3) H : π α = π α (β) = 1 (x αβ) (α = 1,..., N), we consider the family of ϕ-diverence statistics based on the minimum ϕ - diverence estimator Y α (1.4) C ϕϕ = 2 n α ˆπϕ α ϕ n α + (1 ˆπ ϕ ˆπ α ϕ α )ϕ 1 Y α n α 1 ˆπ α ϕ, where ˆπ α ϕ = π α (ˆβ ϕ ) (α = 1,..., N), ˆβ ϕ = ( 1,..., ) is the minimum ϕ -diverence estimator of β under H iven by (1.3) and ϕ satisfies the ˆβ ϕ ˆβ ϕ p

3 IMPROVED TRANSFORMATION FOR GLIM 195 same conditions of ϕ (Pardo [11], Pardo and Pardo [12]). The test statistic C ϕ iven by (7) in Taneichi et al. [21] is written as C ϕ C ϕϕ, and therefore the family of statistics iven by (1.4) includes that of C ϕ. When we choose convex functions ϕ a and ϕ b iven by (1.2) as ϕ and ϕ, respectively, in (1.4), C ϕaϕb becomes a power diverence statistic { ( ) ( (1.5) R a,b = 2 n α I a Yα, ˆπ ϕ b α + I a 1 Y )} α, 1 ˆπ ϕ b α, n α n α where I a (e, f) = {( ) {a(a + 1)} 1 e ef a } 1 ( ) e lo ef ( ) fe f lo (a, 1) (a = ) (a = 1), which is based on the minimum power diverence estimator (Cressie and Read [4], Read and Cressie [14]). Under H, all members of the class of statistics C ϕϕ have a χ 2 N p limitin distribution, assumin the condition that (1.6) n α /n µ α (α = 1,..., N) as n, where n = N n α, < µ α < 1 (α = 1,..., N) and N µ α = 1. Usin the results, we can use C ϕϕ as a oodness-of-fit test statistic for model (1.1). With reard to the oodness-of-fit test for a multinomial distribution, Yarnold [23] obtained an approximation based on asymptotic expansion for the null distribution of Pearson s X 2 statistic. The expansion consists of a term of multivariate Edeworth expansion for a continuous distribution and a discontinuous term. In a fashion similar to that for Pearson s X 2 statistic, approximations based on asymptotic expansions for null distributions of some kinds of multinomial oodness-of-fit statistics have been investiated by Siotani and Fujikoshi [16], Read [13] and Menéndez et al. [9]. Edeworth approximations of the distributions of some kinds of multinomial oodnessof-fit statistics under alternative hypotheses have also been investiated by Taneichi et al. [17, 18], and Sekiya and Taneichi [15]. Taneichi and Sekiya [19] discussed approximations for the distribution of ϕ-diverence statistics for the test of independence in r s continency tables. By usin the above theory of approximation, Taneichi et al. [21] considered a family of ϕ-diverence statistics usin the maximum likelihood estimator C ϕ C ϕϕ and investiated asymptotic approximation of the distribution of statistics for testin the null hypothesis H iven by (1.3). They proposed transformed C ϕ statistics that improve the speed of converence to a chi-square limitin distribution. In this paper, we eneralize the family of statistics C ϕ C ϕϕ based on ϕ-diverence to C ϕϕ and investiate an asymptotic approximation of the distribution of C ϕϕ under H. Also, we propose transformed C ϕϕ statistics.

4 196 N. TANEICHI, Y. SEKIYA AND J. TOYAMA In Section 2, we first describe a local Edeworth approximation for the probability of Y α (α = 1,..., N) under H. Next, we consider an expression of asymptotic expansion for the distribution of C ϕϕ under H. Evaluation for the continuous term of the expression is considered. In Section 3, usin the term of multivariate Edeworth expansion assumin a continuous distribution in the expression in Section 2, we construct transformations for improvin small-sample accuracy of the χ 2 approximation of the distribution of C ϕϕ under H. In Section 4, in the case of Ra,b, performance of the transformed statistic and that of the oriinal statistic are compared numerically. 2. Asymptotic approximation for the distribution of C ϕϕ under H First, we consider a local Edeworth approximation for the probability of Y α (α = 1,..., N) under null hypothesis H iven by (1.3). Let Y α, α = 1,..., N be distributed accordin to a binomial distribution B(n α, π α) α = 1,..., N, where each π α (α = 1,..., N) is represented as π α = 1 (x αβ) (α = 1,..., N) by usin covariate vectors x α = (x α1,..., x αp ) and an unknown parameter vector β. Let (2.1) W α = Y α n α π α nα (α = 1,..., N). Then, W = (W 1,..., W N ) is a lattice random vector that takes values in the set { L = w = (w 1,..., w N ) : w α = y α n α πα (α = 1,..., N), nα } y = (y 1,..., y N ) M, where M = { y = (y 1,..., y N ) : y 1,..., y N are non-neative inteers that } satisfy y α n α (α = 1,..., N). If we consider only for a limitin distribution of C ϕϕ, we can discuss under the assumption iven by (1.6). In this section, since we consider asymptotic expansion of the distribution of C ϕϕ, we need an assumption that states the way of converin n α /n to µ α more strictly than the assumption iven by (1.6). Therefore, we consider the followin Assumption 2.1 instead of the

5 IMPROVED TRANSFORMATION FOR GLIM 197 assumption iven by (1.6). Assumption 2.1. n α (α = 1,..., N), as n, with n α dependin on n in such a way that n α /n = µ α (α = 1,..., N), where < µ α < 1 (α = 1,..., N) and N µ α = 1. With reard to a local Edeworth approximation for the probability of Y α (α = 1,..., N) under H, the followin lemma is shown in Taneichi et al. [21]. Lemma 2.1. For each y = (y 1,..., y N ) M, let w = (w 1,..., w N ), where w α = (y α n α π α)/ n α (α = 1,..., N). Then, under Assumption 2.1, Pr{W = w H } = ( N where (2.2) h (w) = (2π) N/2 Ω 1/2 exp { 1 )h (w) h nα n 1 (w) + 1 n h 2 (w) } + 1 n n h 3 (w) + O(n 2 ), ( 1 ) 2 w Ω 1 w, h 1 (w) = πα 2 µα π α(1 πα) w α h 2 (w) = 1 2 {h 1 (w)} πα + (πα) 2 12 µ α πα(1 πα) πα + 2(πα) 2 4 µ α (πα) 2 (1 πα) 2 w2 α πα + 3(πα) 2 12 µ α (πα) 3 (1 πα) 3 w4 α, µα 1 2π α (π α) 2 (1 π α) 2 w3 α, h 3 (w) = 1 3 {h 1 (w)}3 + h 1 (w)h 2 (w) π α 12 µ α µα (πα) 2 (1 πα) 2 w α 1 1 (1 2π α)(1 πα + (πα) 2 ) 6 µ α µα (πα) 3 (1 πα) 3 wα (1 2π α)(1 2πα + 2(πα) 2 ) 2 µ α µα (πα) 4 (1 πα) 4 wα, 5

6 198 N. TANEICHI, Y. SEKIYA AND J. TOYAMA and (2.3) Ω = dia(π 1 (1 π 1 ),..., π N (1 π N )). For the statistics C ϕ C ϕϕ, Taneichi et al. [21] considered the followin approximation for the distribution of C ϕ under H. Pr{C ϕ x H } J,ϕ 1 (x) + J,ϕ 2 (x), where the J,ϕ 1 (x) term is multivariate Edeworth expansion assumin a continuous distribution and the J,ϕ 2 (x) term, which corresponds to the K 2 term of Taneichi et al. [17] in the case of a multinomial oodness-of-fit test, is a discontinuous term to account for the discontinuity. By usin the continuous term J,ϕ 1 (x), a transformation for C ϕ that improves the speed of converence to a χ 2 limitin distribution is constructed. Let J,ϕϕ 1 (x) be a continuous term of the approximation of Pr{C ϕϕ x H }. Similarly, in this paper, we construct the transformation for C ϕϕ by usin J,ϕϕ 1 (x). With reard to evaluation of the J,ϕϕ 1 (x) term, we obtain the followin theorem. Theorem 2.1. When 1 and ϕ are fourth time continuously differentiable functions and ϕ is a fifth time continuously differentiable function, under Assumption 2.1, the J,ϕϕ 1 (x) term is evaluated as (2.4) J,ϕϕ 1 (x) = Pr{χ 2 N p x} + 1 n 3 j= v,ϕϕ j Pr{χ 2 N p+2j x} + O(n 2 ), where χ 2 f denotes a chi-square random variable with derees of freedom f, v,ϕϕ = 1 24 ( Γ 4), v,ϕϕ 1 = 1 [ Γ 1 ϕ (4) (1) + Γ 2 {ϕ (1) + 1} 2 + (2Γ 1 + Γ 3 )ϕ (1) 24 ] +(Γ 3 + Γ 4 ) +, v,ϕϕ 2 = 1 [ ] Γ 1 ϕ (4) (1) 2Γ 2 {ϕ (1) + 1} 2 (2Γ 1 + Γ 3 )ϕ (1) Γ 3, 24 where v,ϕϕ 3 = 1 24 Γ 2{ϕ (1) + 1} 2, = Γ 5 {ϕ (1) + 1}{ϕ (1) 2ϕ (1) 1}, Γ 1 = 3(A 1 2A 3 + A 6 ), Γ 2 = 5A 2 12A 4 + 9A 7 3B 1 + 6B 2 2B 4 3B 7,

7 IMPROVED TRANSFORMATION FOR GLIM 199 Γ 3 = 2(3A 1 2A 2 6A 3 + 6A 4 + 3A 5 + 3A 6 6A 7 3A 8 3B 3 + 2B 4 + 3B 8 ), A 3 = A 5 = A 8 = Γ 4 = 6A 1 4A 2 6A A 8 3A 9 + 4B 4 12B 5 + 6B 6 3B 9, A 1 = Γ 5 = 3(2A 4 4A 7 + B 1 2B 2 + 2B 4 + B 7 ), 1 3πα + 3(πα) 2 µ α πα(1 πα), A 2 = 1 3π α + 3(π α) 2 (π α) 2 (1 π α) 2 G 1(α) 2 σ αα, A 4 = 1 2π α π α(1 π α) G 2(α)σ αα, A 6 = A 7 = µ α (1 2π α) 2 (π α) 3 (1 π α) 3 G 1(α) 4 σ 2 αα, µ α (1 2π α) (π α) 2 (1 π α) 2 G 1(α) 2 G 2 (α)σ 2 αα, A 9 = B 2 = B 3 = B 4 = B 5 = B 6 = B 7 = B 1 = γ=1 γ=1 γ=1 γ=1 (1 2π α) 2 µ α π α(1 π α), (1 2π α) 2 (π α) 2 (1 π α) 2 G 1(α) 2 σ αα, µ α (1 3π α + 3(π α) 2 ) (π α) 3 (1 π α) 3 G 1 (α) 4 σ 2 αα, 1 2πα 1 2πγ πα(1 πα) πγ(1 πγ) G 1(α)G 1 (γ)σ αγ, µ α π α(1 π α) G 2(α) 2 σ 2 αα, µ α (1 2π α) (π α) 2 (1 π α) 2 1 2π γ π γ(1 π γ) G 1(α) 3 G 1 (γ)σ αα σ αγ, µ α 1 2πγ πα(1 πα) πγ(1 πγ) G 1(α)G 2 (α)g 1 (γ)σ αα σ αγ, γ=1 γ=1 γ=1 µ α (1 2πα) µ γ (1 2πγ) (πα) 2 (1 πα) 2 (πγ) 2 (1 πγ) 2 G 1(α) 3 G 1 (γ) 3 σαγ, 3 µ α µ γ (1 2πγ) πα(1 πα) (πγ) 2 (1 πγ) 2 G 1(α)G 2 (α)g 1 (γ) 3 σαγ, 3 µ α µ γ πα(1 πα) πγ(1 πγ) G 1(α)G 2 (α)g 1 (γ)g 2 (γ)σαγ, 3 µ α (1 2π α) (π α) 2 (1 π α) 2 µ γ (1 2π γ) (π γ) 2 (1 π γ) 2 G 1(α) 3 G 1 (γ) 3 σ αα σ αγ σ γγ,

8 2 N. TANEICHI, Y. SEKIYA AND J. TOYAMA µ α µ γ (1 2πγ) B 8 = πα(1 πα) (πγ) 2 (1 πγ) 2 G 1(α)G 2 (α)g 1 (γ) 3 σ αα σ αγ σ γγ, γ=1 µ α µ γ B 9 = πα(1 πα) πγ(1 πγ) G 1(α)G 2 (α)g 1 (γ)g 2 (γ)σ αα σ αγ σ γγ, γ=1 G i (α) = u (i) (x αβ) (α = 1,..., N, i = 1, 2), u(x) = 1 (x), σ αγ = p p κ l,m x αl x γm l=1 m=1 (α, γ = 1,..., N), κ l,m = µ λ {π λ (1 π λ )} 1 G 1 (λ) 2 x λl x λm (l, m = 1,..., p), λ=1 where u (i) is the i-th derivative of u and κ l,m is the inverse matrix K 1 of K = (κ l,m ). (l, m)-element of the Proof of Theorem 2.1 is shown in Appendix. From Theorem 2.1, we can verify the followin. The coefficients v,ϕϕ (j =, 1, 2, 3) satisfy the relation 3 j= v,ϕϕ j =. The coefficients v,ϕϕ and v,ϕϕ 3 are not dependent on ϕ. When ϕ = ϕ, coefficients coincide with those for the family of statistics C ϕ shown in Theorem 1 of Taneichi et al. [21]. If we apply ϕ a as ϕ and ϕ b as ϕ in Theorem 2.1, we obtain the followin corollary for the statistic R a,b based on power diverence. Corollary 2.1. When the statistic is R a,b iven by (1.5) and 1 is a fourth time continuously differentiable function, under Assumption 2.1, the J,ϕϕ 1 (x) term is evaluated as J,ϕϕ 1 (x) = Pr{χ 2 N p x} + 1 n 3 j= j v,(a,b) j Pr{χ 2 N p+2j x} + O(n 2 ), where v,(a,b) j (j =, 1, 2, 3) are defined as v,ϕϕ j (j =, 1, 2, 3) in the case of ϕ (1) = a 1, ϕ (1) = b 1 and ϕ (4) (1) = (a 1)(a 2), respectively.

9 IMPROVED TRANSFORMATION FOR GLIM Transformed statistics based on the J,ϕϕ 1 (x) term In this section, we first describe the idea of transformation for improvin small-sample accuracy of χ 2 approximation of the distribution of a random variable. Suppose that a nonneative random variable T has an asymptotic expansion such that Pr{T x} = Pr{χ 2 f x} + 1 n m a j Pr{χ 2 f+2j x} + O(n 2 ), j= where m is a positive inteer. Also suppose that the coefficients a j (j =, 1,..., m) do not depend on the parameter n(> ) and must satisfy the relation m j= a j =. For m = 1, in order to increase the accuracy of χ 2 approximation of a random variable T, we consider transformed random variable T B defined by ( (3.1) T B = 1 + 2a ) T. fn Then, it holds that Pr{T B x} = Pr{χ 2 f x} + O(n 2 ). This result is known as a Bartlett adjustment. Lawley [8], Barndorff-Nielsen and Cox [2], and Barndorff-Nielsen and Hall [3] discussed Bartlett adjustment for the lo-likelihood ratio statistic. For m = 3, in order to increase the accuracy of χ 2 approximation of a random variable T, we consider transformed random variable T I defined by [ { (3.2) T I = (nα + β) 2 lo (nα) (nα) 2 T + nα 1 (T 2 + γt 3 ) ( 1 3 T 3 + 3γ 4 T 4 + 9γ2 2 T 5 ) }], where α = f(f + 2){2(a 2 + a 3 )} 1, β = (f + 2)a {2(a 2 + a 3 )} 1 and γ = a 3 {(f +4)(a 2 + a 3 )} 1. Then, it holds that Pr{T I x} = Pr{χ 2 f x} + O(n 2 ). The proof of the results for transformation of T I is iven by Yanaihara [22]. The proof is derived by applyin the idea of Kakizawa [6] to the theory of improved transformation iven by Fujikoshi [5].

10 22 N. TANEICHI, Y. SEKIYA AND J. TOYAMA Applyin the evaluation (2.4) iven by Theorem 2.1 to the above transformed statistics T B iven by (3.1) and T I iven by (3.2), we construct transformations for improvin small-sample accuracy of the χ 2 approximation of the distribution of C ϕ under H. When ϕ and ϕ satisfy (3.3) ϕ (1) = 1, ϕ (4) (1) = 2 and ϕ (1) = 1, equations v,ϕϕ 1 = v,ϕϕ and v,ϕϕ 2 = v,ϕϕ 3 = hold in Theorem 2.1. Then, we can consider Bartlett-type adjustment { } Cϕϕ B = 1 + 2v,ϕϕ C ϕϕ. n(n p) On the other hand, when ϕ does not satisfy (3.3), we can consider the transformed statistic C I ϕϕ = (nα + β)2 lo (1 + ζ), where [ ζ = 1 (nα) 2 C ϕϕ + 1 nα {(C ϕϕ )2 + γ(c ϕϕ ) 3 } + 1 (nα) 2 { 1 3 (C ϕϕ )3 + 3γ 4 (C ϕϕ )4 + 9γ2 2 (C ϕϕ )5} ], α = (N p)(n p+2){2(v,ϕϕ 2 +v,ϕϕ 3 )} 1, β = (N p+2)v,ϕϕ {2(v,ϕϕ 2 +v,ϕϕ 3 )} 1 and γ = v,ϕϕ 3 {(N p + 4)(v,ϕϕ 2 + v,ϕϕ 3 )} 1. Practically, we may use estimate ˆv,ϕϕ j (j =, 2, 3) obtained by substitutin minimum ϕ -diverence estimate ˆβ ϕ for true value β in v,ϕϕ j (j =, 2, 3). Therefore, when ϕ and ϕ satisfy (3.3), we propose the statistic C B ϕϕ that is obtained by substitutin ˆv,ϕϕ for v,ϕϕ in Cϕϕ B, that is, } (3.4) CB ϕϕ = {1 + 2ˆv,ϕϕ C ϕϕ. n(n p) Similarly, when ϕ and ϕ do not satisfy (3.3), we also propose the statistic C ϕϕ I that is obtained by substitutin ˆv,ϕϕ j (j =, 2, 3) for v,ϕϕ j (j =, 2, 3) in Cϕϕ I. In the case of power diverence statistic R a,b = C ϕaϕb usin the minimum power diverence estimator, condition (3.3) is satisfied if and only if a =

11 IMPROVED TRANSFORMATION FOR GLIM 23 and b = (lo likelihood ratio statistic). Then, we consider the transformed statistic iven by (3.4) when a = and b = and put = C ϕ B ϕ. When the link function is a loit link function, statistic B coincides with the statistic D proposed by (3.4) of Taneichi et al. [2]. On the other hand, we consider statistic C ϕ I a,b aϕ b when a or b and put R I = C ϕ I aϕ b (a or b ). We summarize the difference and relation between T B and T I. Transformed statistic T B is a simple monotone transformation of C ϕ constructed by a linear function whose intercept is zero. On the other hand, transformed statistic T I is a monotone transformation of C ϕ constructed by loarithm of quintic function. It is much more complicated than T B. Then, from point of view of stability, T I seems to be inferior to T B. However, for Cressie and Read family of statistics R a,b, T B increases the speed of converence to chi-square distribution only for the statistic in the case of a = b =, that is, the lo-likelihood ratio statistic. Therefore, for improvin the other statistics, statistic T I is developed. R, R, B 4. Performance of transformed statistics In this section, we compare the performance of transformed statistics I (a or b ) with that of the oriinal power diverence statistics R a,b usin the minimum power diverence estimator by the Monte Carlo procedure. The, performance of transformed statistic R B for complementary lo-lo link and probit link P is shown in Fi.1, Fi.2 and Fi.3 of Taneichi et al. [21]. We consider a eneralized linear model iven by (1.1) with p = 2 and x α1 = 1 and x α2 = x α (α = 1,..., N). Let the true values of parameters β 1 and β 2 be β1 and β 2, respectively. Then, the true value of πα (α = 1,..., N) is (4.1) π α = 1 (β 1 + β 2x α ) (α = 1,..., N). As a link function, we consider the family of link functions iven by Aranda- Ordaz [1], { (1 t) c } 1 (t) = c (t) = lo, c that depend on parameter c. c include the loit link 1 and complementary lo-lo link as a limit. We also consider the probit link P (t) = Φ 1 (t), where Φ is the cumulative distribution function of a standard normal distribution. We ive a desin matrix R a,b ( 1 1 X = x 1 x N )

12 24 N. TANEICHI, Y. SEKIYA AND J. TOYAMA and execute the followin procedure. For each α, we enerate n α (α = 1,..., N) binomial random numbers that are distributed accordin to B(1, πα ) (α = 1,..., N). From them, we calculate the number of successes Y α (α = 1,..., N) and the minimum ϕ b - diverence estimates ˆβ ϕ b 1 and ˆβ ϕ b 2 for the parameters β 1 and β 2. Usin the estimates, we calculate the values π α (ˆβ ϕb ) (α = 1,..., N), where ˆβ ϕb = ( ˆβ ϕ b 1, ˆβ ϕ b 2 ), and observed values of the statistics R a,b, Ra,b I (a or b ). This process is repeated D times. Amon D times, let V be the number of times that the observed values of the statistic exceed the upper ε point of the χ 2 distribution with derees of freedom N p, that is, χ 2 N p (ε). The performance of χ2 approximation for the distribution of each statistic can be evaluated on the basis of the index I = V D ε. We consider the followin two true parameters (i) β 1 =.1, β 2 =.1, (ii) β 1 =.1, β 2 =.1, and investiate the performance of the followin four cases of desin matrix when N = 8. (I) (II) (III) X = X = ( ). ( ( X = lo(2.7) lo(3.) lo(3.3) lo(3.6) ) lo(3.9) lo(4.2) lo(4.5) lo(4.8) ). (IV) ( X = lo(2.85) lo(3.5) lo(3.25) lo(3.45) lo(3.65) lo(3.85) lo(4.5) lo(4.25) ).

13 IMPROVED TRANSFORMATION FOR GLIM 25 R.2,.2.,1. R I For each case, we consider a sample desin n 1 = = n 8 = n. We investiate the performance for all combinations of the two true parameters (i) and (ii), four desin matrices (I), (II), (III) and (IV), and the sample desin with n = 2. Some of the results of the investiations are shown in fiures as follows. Fi.1 shows the absolute values of index I when the test statistic is R.2,.2 and models are iven by link functions (complementary lo-lo model), 1/2, 1 (loistic reression model) and P (probit model) in the case of true parameters (i) and (ii), desin matrices (I) (IV), and sinificance level ε =.1,.5 and.1. Fi.2 and Fi.3 show the absolute values of index I when the test statistics are R.,1. and R 1.,1. in the same models and situations as those in the explanation of Fi.1, respectively. From Fi.1 and Fi.2, we find that the performance of transformed statistics I and is better than that of oriinal statistics R.2,.2 and R.,1., respectively, when the models are iven by the link functions (complementary lo-lo model), 1/2, 1 (loistic reression model) and P (probit model) for the two true parameters, all desin matrix cases, and sample desin n = 2. From Fi.3, we find that the performance of transformed statistic R 1.,1. I is better than that of oriinal statistic R 1.,1. when the true parameter is type (i). However, when the true parameter is type (ii), the performance of the transformed statistic is not better than that of the oriinal statistic. Consequently, from Fis.1 3 and other simulation results, we conclude as follows. The performance of Ra,b I ( < a 1, < b 1) is usually better than that of oriinal statistic R a,b ( < a 1, < b 1) when the models are iven by the link functions (complementary lo-lo model), 1/2, 1 (loistic reression model) and P (probit model) under the conditions of the simulation. However, as shown in Fi.3, when the chi-square approximation of the oriinal statistic performs very well, approximation of the transformed statistic sometimes does not perform better than the oriinal statistic. That is, when the chi-square approximation of the oriinal statistic already performs very well, the transformed statistic sometimes cannot improve the performance of chi-square approximation. a,b Next, we compare the power of transformed statistics R I (a or b ) with that of the oriinal statistics R a,b. The power of transformed statistic R.,. B for complementary lo-lo link and probit link P is shown in Fi.6 and Fi.7 of Taneichi et al. [21]. Aainst the null model iven by (4.1), we consider an alternative model: (4.2) H 1 : π α = 1 (β 1 + β 2x α ) + δ α (α = 1,..., 8), where (δ 1, δ 2, δ 3, δ 4, δ 5, δ 6, δ 7, δ 8 ) = (.1,.1,.1,.1,.1,.1,.1,.1).

14 26 N. TANEICHI, Y. SEKIYA AND J. TOYAMA We calculate the simulated averae power P aainst the alternative model (4.2) by usin simulated exact critical values of statistics. We investiate the averae power for all combinations of the two true parameters (i) and (ii), four desin matrices (I) (IV), and sample desin n = 2. In the investiation, the number of repetitions is D = 1 6. Some of the results of the investiations are shown in fiures as follows. Fis.4, 5 and 6 show the power of statistics correspondin to the cases in Fis.1, 2 and 3, respectively. From Fis.4 6 and other simulation results, we conclude that the power aainst H a,b 1 iven by (16) of the transformed statistics R I ( < a 1, < b 1) is not so different from that of the oriinal power diverence statistic R a,b in the models based on link functions c (c =, 1/2, 1) and P (i) (ii) Value for R.2,.2 (ε=.1) Value for R.2,.2 I (ε=.1) Value for R.2,.2 (ε=.5) Value for R.2,.2 I (ε=.5) Value for R.2,.2 (ε=.1) Value for R.2,.2 I (ε=.1) Absolute Value of Index I /2 1 P 1/2 1 P Fiure 1: Absolute value of index I when the oriinal test statistic is R.2,.2 and models are iven by link functions, 1/2, 1 and P for true parameters (i) and (ii) and sample desin n =2:, and are the values for R.2,.2 when ε =.1,.5 and.1, respectively, and, and are the values for R.2,.2 I when ε =.1,.5 and.1, respectively. The 1st column is for desin matrix (I), the 2nd column is for desin matrix (II), the 3rd column is for desin matrix (III), and the 4th column is for desin matrix (IV).

15 IMPROVED TRANSFORMATION FOR GLIM (i) (ii) Value for R.,1. (ε=.1) Value for R.,1. I (ε=.1) Value for R.,1. (ε=.5) Value for R.,1. I (ε=.5) Value for R.,1. (ε=.1) Value for R.,1. I (ε=.1) Absolute Value of Index I.1.5 1/2 1 P 1/2 1 P Fiure 2: Absolute value of index I when the oriinal test statistic is R., (i) (ii) Value for R 1.,1. (ε=.1) Value for R 1.,1. I (ε=.1) Value for R 1.,1. (ε=.5) Value for R 1.,1. I (ε=.5) Value for R 1.,1. (ε=.1) Value for R 1.,1. I (ε=.1) Absolute Value of Index I /2 1 P 1/2 1 P Fiure 3: Absolute value of index I when the oriinal test statistic is R 1.,1..

16 28 N. TANEICHI, Y. SEKIYA AND J. TOYAMA (i) (ii) Value for R.2,.2 (ε=.1) Value for R.2,.2 I (ε=.1) Value for R.2,.2 (ε=.5) Value for R.2,.2 I (ε=.5) Value for R.2,.2 (ε=.1) Value for R.2,.2 I (ε=.1) Power /2 1 P 1/2 1 P Fiure 4: Simulated averae power P aainst an alternative model (4.2) when the oriinal test statistic is R.2,.2 and models are iven by link functions, 1/2, 1 and P for true parameters (i) and (ii) and sample desin n = 2:, and are the values for R.2,.2 when ε =.1,.5 and.1, respectively, and, and are the values for R.2,.2 I when ε =.1,.5 and.1, respectively. The 1st column is for desin matrix (I), the 2nd column is for desin matrix (II), the 3rd column is for desin matrix (III), and the 4th column is for desin matrix (IV) (i) (ii) Value for R.,1. (ε=.1) Value for R.,1. I (ε=.1) Value for R.,1. (ε=.5) Value for R.,1. I (ε=.5) Value for R.,1. (ε=.1) Value for R.,1. I (ε=.1) Power /2 1 P 1/2 1 P Fiure 5: Simulated averae power P aainst an alternative model (4.2) when the oriinal test statistic is R.,1..

17 IMPROVED TRANSFORMATION FOR GLIM (i) (ii) Value for R 1.,1. (ε=.1) Value for R 1.,1. I (ε=.1) Value for R 1.,1. (ε=.5) Value for R 1.,1. I (ε=.5) Value for R 1.,1. (ε=.1) Value for R 1.,1. I (ε=.1) Power /2 1 P 1/2 1 P Fiure 6: Simulated averae power P aainst an alternative model (4.2) when the oriinal test statistic is R 1., Appendix: Proof of Theorem 2.1 By transformation (2.1), statistic C ϕϕ C ϕϕ (W ) = 2 can be rewitten as { ( π n α ˆπ α ϕ (W )ϕ α + W α ( n α ) 1 + (1 ˆπ α ϕ (W ) ) ˆπ α ϕ (W ) ) ( 1 π ϕ α W α ( n α ) 1 ) }. 1 ˆπ α ϕ (W ) If we reard { h (w) h n 1 (w) + 1 n h 2 (w) + 1 } n n h 3 (w) as the continuous density function of W, then we can reard { J,ϕϕ 1 (x) = h (w) h n 1 (w) + 1 n h 2 (w) U ϕϕ (x) + 1 n n h 3 (w) } dw

18 21 N. TANEICHI, Y. SEKIYA AND J. TOYAMA as the distribution function of C ϕϕ (W ), where U ϕϕ (x) = {w = (w 1,..., w N ) : C ϕϕ (w) x}. So, the characteristic function of C ϕϕ (W ) is calculated as (A1) ψ ϕϕ (u) = [exp{iuc ϕϕ (w)}] h (w) { h n 1 (w) + 1 n h 2 (w) + 1 } n n h 3 (w) dw. We can expand C ϕϕ (w) as (A2) C ϕϕ (w) = τ (w)+ 1 n τ,ϕ 1 (w)+ 1 n τ,ϕϕ 2 (w)+ 1 n n τ,ϕϕ 3 (w)+o(n 2 ), where Ξ = (ξ αβ ) is a N N matrix, τ (w) = w (Ω 1 Ξ)w, µα G 1 (α) µβ G 1 (β) ξ αβ = πα(1 πα) π β (1 π β )σ αβ τ,ϕϕ 2 (w) = τ,ϕ 1 (w) = (α, β = 1,..., N), ( 3 N ) Ba+1(α)C 1 1(α) (w) 3 a wα a, a= ( 2 N ) Ba+1(α)C 2 ϕ 2(α) (w)2 a C 1(α) (w) 2a a= ( 1 N ) + Ba+4(α)C 2 ϕ 2(α) (w)1 a C 1(α) (w) 1+2a w α a= ( 1 N ) + Ba+6(α)C 2 ϕ 2(α) (w)1 a C 1(α) (w) 2a wα 2 a= + B8(α)C 2 1(α) (w)wα 3 + B9(α)w 2 α, 4 B 1 1 (α) = µ α 3(π α) 2 (1 π α) 2 { 3π α (1 π α)g 1 (α)g 2 (α) (3 + ϕ (1))(1 2π α)g 1 (α) 3}, B 1 2 (α) = µα (π α) 2 (1 π α) 2 { π α (1 π α)g 2 (α) +(2 + ϕ (1))(1 2π α)g 1 (α) 2},

19 IMPROVED TRANSFORMATION FOR GLIM 211 B 1 3(α) = ( 1 + ϕ (1) ) (1 2π α)g 1 (α) (π α) 2 (1 π α) 2, B 1 4(α) = ϕ (1)(1 2π α) 3 µ α (π α) 2 (1 π α) 2, B 2 1(α) = µ αg 1 (α) 2 π α(1 π α), B2 2(α) = 3B 1 1(α), B3 2(α) = µ α 12(πα) 3 (1 πα) 3 {(π α) 2 (1 πα) 2 (3G 2 (α) 2 + 4G 1 (α)g 3 (α)) 6(3 + ϕ (1))πα(1 πα)(1 2πα)G 1 (α) 2 G 2 (α) +(12 + 8ϕ (1) + ϕ (4) (1))(1 3πα + 3(πα) 2 )G 1 (α) 4 }, B 2 4(α) = 2B 1 2(α), B5 2(α) = µα 3(πα) 3 (1 πα) 3 { (π α) 2 (1 πα) 2 G 3 (α) +3(2 + ϕ (1))πα(1 πα)(1 2πα)G 1 (α)g 2 (α) (6 + 6ϕ (1) + ϕ (4) (1))(1 3πα + 3(πα) 2 )G 1 (α) 3 }, B 2 6(α) = B 1 3(α), B7 2(α) = 1 2(πα) 3 (1 πα) 3 { (1 + ϕ (1))πα(1 πα)(1 2πα)G 2 (α) +(2 + 4ϕ (1) + ϕ (4) (1))(1 3πα + 3(πα) 2 )G 1 (α) 2 }, B 2 8(α) = (2ϕ (1) + ϕ (4) (1))(1 3π α + 3(π α) 2 )G 1 (α) 3 µ α (π α) 3 (1 π α) 3, B9(α) 2 = ϕ(4) (1)(1 3πα + 3(πα) 2 ) 12µ α (πα) 3 (1 πα) 3, ) p C 1(α) (w) = κ m,k M k (w) (α = 1,..., N), C ϕ 2(α) (w) = p Q ϕ M k (w) = m=1 x αm ( p k=1 m=1x αm{ p k= p k 1 =1 M m,k (w)m k (w) + p k 5 =1 p k=1 κ m,k S ϕ k (w) κ m,k 3 κ k 1,k 4 κ k 2,k 5 κ ϕ k 3,k 4,k 5 M k1 (w)m k2 (w) µλ x λk G 1 (λ){π λ (1 π λ )} 1 w λ λ=1 } (α = 1,..., N), (k = 1,..., p), N { i,j (w) = µλ x λi x λj (2 + ϕ (1)) (1 2π λ )G 1(λ) 2 (π λ=1 λ )2 (1 π } λ )2 + G 2(λ) π λ (1 π λ ) w λ (i, j = 1,..., p),

20 212 N. TANEICHI, Y. SEKIYA AND J. TOYAMA κ ϕ i,j,k = { µ λ x λi x λj x λk (3 + ϕ (1)) (1 2π λ )G 1(λ) 3 (π λ )2 (1 π } λ )2 3 G 1(λ)G 2 (λ) π λ (1 π λ ) (i, j, k = 1,..., p), λ=1 S ϕ k (w) = 1 2 {1 + ϕ (1)} λ=1 x λk (1 2π λ )G 1(λ) (π λ )2 (1 π λ )2 w2 λ (k = 1,..., p), G i (α) = u (i) (x αβ) (α = 1,..., N, i = 1, 2, 3), Q ϕ (w) = (Q ϕ i,j (w)) is a p p matrix, M i,j (w) is the (i, j)-element of matrix K 1 Q ϕ (w)k 1, Ω is defined by (2.3), σ αβ and K 1 = (κ i,j ) are defined in Theorem 2.1, and τ,ϕϕ 3 (w) is a homoeneous polynomial of deree 5 with respect to variables w 1,..., w N. Then, from (2.2), (A1) and (A2), we obtain (A3) ψ ϕϕ (u) = (1 2iu) (N p)/2 ( (2π) N/2 Λ {exp 1/2 1 )} 2 w Λ 1 w { } D 1 (w) + 1 n n D 2(w) + 1 n n D 3(w) dw +O(n 2 ), where Λ = (1 2iu) 1 (Ω 2iuΩΞΩ), D 1 (w) = h,ϕ 1 (w) + (iu)τ1 (w), D 2 (w) = h,ϕ 2 (w) + (iu)τ1 (w)h,ϕϕ 1 (w) + (iu)τ2 (w) + 1 { 2 2 (iu)2 τ,ϕ 1 (w)}, and derees of all terms of polynomial D 3 (w) are odd. Therefore, by carryin out the interation of (A3), the characteristic function ψ ϕϕ (u) is expanded as (A4) ψ ϕϕ (u) = (1 2iu) (N p)/ (1 2iu) j v,ϕϕ j + O(n 2 ). n By invertin (A4), we obtain (2.4). We have completed the proof of Theorem 2.1. j= Acknowledments This research is partially supported by the Grants-in-aid for Scientific Research of Japan Society for the Promotion of Science (C)

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