Expected probabilities of misclassification in linear discriminant analysis based on 2-Step monotone missing samples

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1 Expected probabilities of misclassification in linear discriminant analysis based on 2-Step monotone missing samples Nobumichi Shutoh, Masashi Hyodo and Takashi Seo 2 Department of Mathematics, Graduate School of Science, Tokyo University of Science, -3, Kagurazaka, Shinjuku-ku, Tokyo , Japan 2 Department of Mathematical Information Science, Faculty of Science, Tokyo University of Science, -3, Kagurazaka, Shinjuku-ku, Tokyo , Japan Absact As concerns the asymptotic approximation for expected probabilities of misclassification in linear or quadratic discriminant analysis, theoretical results are given under some situations with complete data However it has not been enough to discuss that under missing samples In this paper, we derive that with explicit form in linear discriminant function for two groups consucted by 2-Step monotone missing samples For our purpose, some results of expectations concerning inverted Wishart maices are also obtained Finally the numerical evaluations of our result by Monte Carlo simulations are presented AMS 2000 Mathematics Subject Classification 62H2, 62H30 Key Words and Phrases: linear discriminant function; expected probability of misclassification; asymptotic approximation; monotone missing samples; missing observations Inoduction Linear discriminant analysis is well known as one of statistical procedures to assign p dimensional observation vector x into one of two populations Π and Π 2 If Π g g =, 2 has p dimensional normal disibution with mean vector µ g and common covariance maix Σ which are known where µ µ 2 and Σ is positive definite, linear discriminant function LDF is consucted as follows: Corresponding author address: nshutoh28@yahoocojp W = µ µ 2 Σ x 2 µ µ 2

2 x may be assigned to Π if W > 0, or Π 2 if W < 0 The probability of misclassification in equals Φ 2 exactly, where Φ denotes the cumulative density function of standard normal disibution and 2 µ µ 2 Σ µ µ 2 is Mahalanobis squared distance between Π and Π 2 However it is usual that both of parameters µ g and Σ are unknown Therefore their own estimators are substituted for µ g and Σ respectively in If N g observation vectors x g j g =, 2, j =,, N g which are can be observed completely from Πg, LDF suggested by Wald 944 is as follows: W c = x x 2 S x 2 x x 2, 2 where x g and S are sample mean vector from Π g and pooled sample covariance maix, and they are the maximum likelihood estimators MLEs of µ g and Σ respectively Although it is hard to obtain exact probabilities of misclassification under the situation where parameters are unknown, two types of asymptotic approximations for expected probabilities of misclassification EPMC have been considered Type I approximations are ones of framework such that N and N 2 are large and p is fixed Okamoto 963, 968 gave the approximation of this type by deriving asymptotic expansion formulas of LDF up to terms of the second order with respect to N, N 2, n, where n = N N 2 2 Memon and Okamoto 97 considered it in the case of quadratic discriminant function QDF Siotani and Wang 977 extended Okamoto 963, 968 and Memon and Okamoto 97 up to terms of the third order It has been known that these approximations are good for small p, but are poor as p is large In the face of this problem, Type II approximations are suggested under the framework such that N, N 2, p and n p are large As concerns the asymptotic approximations for EPMC in this type, Raudis 972 considered one of them in LDF for N = N 2 Fujikoshi and Seo 998 extended Raudis 972 to the case of N N 2 and also gave it in QDF Moreover Fujikoshi 2000 gave explicit error bounds for approximation for EPMC proposed by Lachenbruch 968 and discussed with a method of obtaining asymptotic expansion formulas for EPMC and their error bounds Besides the asymptotic approximation for EPMC proposed by Lachenbruch 968 is the one which can fit the situations in both Type I and Type II, so that the asymptotic approximation proposed Lachenbruch 968 can be used in various situations of data set without missing data concerns discussion under elliptical populations, see Wakaki 994 On the other hand, as concerns MLEs consucted by observation vectors including missing data, Srivastava 985 derived MLEs for one and two sample problems without condition of missing patterns by using incidence maix which can remove missing data from each observation vector Although they can not be obtained explicitly, Srivastava and Carter 986 proposed the 2 As

3 procedure to obtain them numerically by using Newton-Raphson method Also MLEs for one sample in the case of 2-Step monotone missing samples from multivariate normal disibution were derived explicitly by Anderson and Olkin 985 The k-step monotone missing samples from Π g are defined as data set having following form: x g x g p x g p x g p k x g p k x g N g x g N g x g N g x g x g N g 2 x g N g x g p N g p x g N g p N g p x g N g 2 p x g N g 2 p x g N g p k x g x g N g p k N g p k x g N g 2 p k x g N g p, 3 where p = p p k, N g = N g N g k and each part of denotes missing data Further Kanda and Fujikoshi 998 inoduced some properties of MLEs in the case of k = 2, 3 and general k They also gave asymptotic expansions for the disibutions of MLEs in the situation when the sample size is large In these days, Batsidis, Zografos and Loukas 2006 considered LDF based on 2-Step monotone missing samples and gave the asymptotic expression for probability density function of the probability of misclassification In this paper, as one of extension for Lachenbruch 968, we propose asymptotic approximation for EPMC where aining data has 2-Step monotone missing samples That is, we extend the results of Lachenbruch 968 in the case of LDF consucted by Batsidis, Zografos and Loukas 2006 Although Kanda and Fujikoshi 2004 derived that under known Σ, we propose that under unknown case as main result in this paper The organization of this paper is as follows In Section 2, we show method for obtaining MLEs in two groups in the case of 2-Step monotone missing samples and give some Lemmas which can support us to give the asymptotic approximation for EPMC considered in Section 3 In Section 3, we consider asymptotic approximation for EPMC in this case by using the results of Section 2 and derive the unbiased estimators of Mahalanobis squared distances which come in useful for obtaining the approximation of EPMC Further we evaluate our results in Section 3 numerically by Monte Carlo simulations in Section 4 Finally we present conclusion in this paper and future problems in Section 5 3

4 2 MLEs Based on 2-Step Monotone Missing Samples In this section, we derive the MLEs under assuming that 2-Step monotone missing samples are obtained from each group, ie, we extend MLEs given by Anderson and Olkin 985 to the case of two groups where k = 2 in 3 Therefore we assume each data set from Π g as follows: x g x g 2 x g p x g p x g p x g 2 x g 22 x g 2 p x g 2 p x g 2p x g x g N g N g 2 x g x g N g N g x g N g x g N g 2 x g x g N g 2 x g p N g p N g p x g N g p, x g N g p where p = p p 2 and N g = N g N g 2 Then we prepare notations and assumption concerning observation vector Let x g j x g j, xg 2j g =, 2, j =,, N g be the p dimensional observation vector from Π g which has complete data, where x g j = x g j,, xg jp j =,, N g and x g 2j = x g j p,, xg g jp j =,, N Now we assume disibution of observation vector: x g j N p µ g, Σ j =,, N g and xg j N p µ g, Σ j = N g,, N g respectively, where µ g = µ g µ g 2 and Σ = Σ Σ 2 Σ 2 Σ 22 µ g and Σ are partitioned according to blocks of data set Therefore µ g is p dimensional vector, µ g 2 is p 2 dimensional vector, Σ is p p maix, Σ 2 = Σ 2 is p p 2 maix and Σ 22 is p 2 p 2 maix respectively Then we consider ansformation of observation vector x g j by multiplying I p 0 Σ 2 Σ I p2 on the left-hand The mean vector η g and covariance maix of ansformed observation vector are η g η g η g 2 = µ g µ g 2 Σ 2 Σ µg 4 and Σ 0 0 Σ 22,

5 where Σ 22 = Σ 22 Σ 2 Σ Σ 2 and Σ has nonsingularity Therefore we consider the new parameters {η, η 2, Ψ} which are one to one corresponding to {µ, µ 2, Σ}, where Ψ Ψ Ψ = 2 Σ Σ = Σ 2 Ψ 2 Ψ 22 Σ 2 Σ Σ 22 Next the likelihood function to obtain MLEs of η, η 2 and Ψ is as follows: Lη, η 2, Ψ = Const Ψ 2 N N 2 Ψ 22 2 e 2 N g y g j 2 e 2 g= j= 2 N g g= j= y g 2j 2 N N ηg Ψ yg j ηg ηg 2 Ψ 22 yg 2j ηg 2, 2 where y g j = xg j and y g 2j = xg 2j Ψ 2x g j If we define the sample mean vectors x g T = x g 2F = N g N g x g j, N g j= N g j= x g 2j xg F = N g and x g L = N g 2 N g j= x g j, N g g xj, j=n g then by 2 and the well known correction of their own coefficients, we can obtain following MLEs and where Ψ = n Ψ 2 = 2 N g g= j= 2 N g g= j= x g j x g j η g = Ψ 22 = { 2 N g x g 2j n g= j= η g η g 2 = Ψ = Ψ Ψ 2 Ψ 2 Ψ 22 xg T xg j xg T, xg F xg j xg F xg 2F xg 2j xg 2F 5 x g T x g 2F Ψ 2 x g F 2, N g x g j g= j= xg F xg 2j xg 2F,

6 2 N g x g 2j g= j= 2 N g x g j g= j= xg 2F xg j xg F xg F xg j xg F 2 N g x g j g= j= xg F xg 2j xg 2F n = N N 2 2 and n = N N 2 2 Also these maices are expressed as follows: Ψ = n Γ Γ2, Ψ 2 = Γ Γ 2 and Ψ 22 = Γ n respectively Γ and Γ 2 are maices such that Γ Γ = Γ 2 Γ 2 Γ 22 and where = n S { 2 g N Γ 2 = n 2 S 2 N g } 2 N g= g x g F xg L xg F xg L, S = N 2 g x g j n g= j= S 2 = 2 N g g xj n 2 g= j=n g x g x g j x g, xg L xg j xg L and n 2 = N 2 N Clearly we can find that S and S 2 are pooled sample covariance maices of x g j g =, 2, j =,, N g and xg j g =, 2, j = N g,, N g respectively Also n is needed to be larger than p by nonsingularity of Γ MLEs derived in this section have natural forms of extension of Anderson and Olkin 985 Here we have Lemmas concerning the disibutions and expectations of 22 Lemma 2 Γ, Γ Γ2, Γ 2 given Γ and Γ 22 have following disibutions respectively: Γ W p n, Σ, Γ Γ2 = n Ψ W p n, Σ, VecΓ 2 Γ N p p 2 VecΓ Σ Σ 2, Σ 22 Γ, Γ 22 = n Ψ 22 W p2 n p, Σ 22, where W p n, Σ denotes Wishart disibution with the parameters n and Σ }, 6

7 Proof: Since Γ has W p n, Σ, it has seen that Γ has W p n, Σ and Γ 22 has W p 2 n p, Σ 22 The conditional disibution of Γ 2 function of Γ, Γ 2 given Γ is derived by joint probability density For a proof, eg, see Siotani, Hayakawa and Fujikoshi 985 Moreover the disibution of Γ 2 can be derived since n 2 S 2 has W p n 2, Σ, the other maix has W p 2, Σ and they are independent Also Γ and Γ2 are independently disibuted, which proves the result Lemma 22 Let C i and D i be p i p i i =, 2 constant maices respectively, then the expectations as follows can be obtained: E Γ = E Γ Γ2 = E Γ 22 = E Γ C Γ = n p Σ, n p Σ, n p Σ 22, n p 2 n p n p n p 3 Σ C Σ n p n p n p 3 { } Σ C Σ C Σ, Σ E Γ Γ2 C Γ Γ2 = E Γ 22 C 2Γ 22 = n p 2 n p n p n p 3 Σ C Σ n p n p n p 3 { Σ C Σ C Σ Σ n p 2 n pn p n p 3 Σ 22 C 2Σ 22 n pn p n p 3 { Σ 22 C 2Σ 22 C 2 Σ 22 }, Σ 22 }, E C Γ D Γ = n p 2 n p n p n p 3 C Σ D Σ 7

8 n p n p n p 3 { } D Σ C Σ D Σ C Σ, respectively Proof: Lemma 22 is derived by the original properties concerning the expectation of inverted Wishart maix The results can be derived by Haff 979 and Marx 98 At the end of this section, we show the other expression of some terms included in V m defined by Section 3 Here we give the definition of Type I beta disibution Definition 23 A p p symmeic random maix Y disibuted as Type I beta disibution with the parameters a and b is one which has the following probability density function: {β p a, b} dety a 2 p deti p Y b 2 p, 0 < Y < I p, where a > p /2, b > p /2 and β p a, b is p dimensional beta function given by β p a, b = Γ paγ p b Γ p a b for the real parts of a and b are larger than p /2 respectively Further the function Γ p a consucting β p a, b is the following function such that p Γ p a = π 4 pp Γ a 2 i, where Γ is Gamma function i= As concerns the details and properties concerning the above definition and the processes of calculating β p a, b and Γ p a, eg, see Gupta and Nagar 2000 Besides Type I beta disibution with the parameters a and b is denoted by B I pa, b in this paper Lemma 24 Let A be a p p constant and symmeic maix Then Γ Γ2 Σ Γ A can be also rewritten with following expression including random maices: β I,p W W Σ 2 A Σ 2 W W 2 2, where β I,p = W W 2 2 W W W 2 2 has Bp I n /2, n 2 2/2, W and W 2 have W p n, I p and W p n 2 2, I p respectively Proof: Note that β I,p has B I p n /2, n 2 2/2 For a proof, see, eg, Gupta and Nagar 2000 Moreover by making use of the properties of ace and Wishart maices, the proof can be completed 8

9 3 Asymptotic Approximation for EPMC Now we consider the asymptotic approximation for EPMC when the aining data has 2-Step monotone missing samples The approximation is obtained by making use of Lachenbruch 968 and results obtained in Section 2 If µ g and Σ are unknown, the aining data has 2-Step monotone samples and x does not have missing data, then LDF is consucted by the same way as 2: W m = µ µ 2 Σ x 2 µ µ 2, 3 where µ, µ 2 and Σ are MLEs obtained by aining data variance of W m given µ, µ 2 and Σ under x Π : Now we find expectation and EW m µ, µ 2, Σ = U m, VarW m µ, µ 2, Σ = V m, where U m = µ µ 2 Σ µ µ 2 D2 m, 32 D 2 m = µ µ 2 Σ µ µ 2, 33 V m = µ µ 2 Σ Σ Σ µ µ 2 34 Further if we define then Z m Z m = V 2 m µ µ 2 Σ x µ, 35 is independent of U m, V m and disibuted as N0, under the situation given µ, µ 2, Σ By using 32 35, 3 can be also rewritten as W m = V 2 m Z m U m 36 Thus W m can be expressed by U m, V m and Z m in the same way as Lachenbruch 968 in spite of 2-Step monotone missing samples As concerns the probability of misclassification where x is assigned to Π 2 when x comes from Π, it is expressed by using 36 as Pr W m 0 x Π = Pr Z m V 2 m U m x Π 37 Since Z m is disibuted as N0, if x comes from Π, 37 can be rewritten easily as ΦV 2 m U m Therefore EPMC of 3 where x is assigned to Π 2 when x comes from Π is obtained as e m 2 = E Um,V m ΦV 2 m U m 38 9

10 Here we consider the asymptotic approximation for EPMC in 3 disibuted as normal disibution exactly, but is closely normal asymptotically asymptotic approximation of the right-hand of 38, we propose In actual, W m is not Thus as an e m 2 Φ{EV m } 2 EUm 39 obtained by substituting their expectations of U m and V m For the case that Σ is known, Kanda and Fujikoshi 2004 proposed 39 where EU m = 2 p N N 2 2 N N 2 p N N 2 N N 2 EV m = 2 p N N 2 N N 2 p 2N N 2 N N 2 under the framework such that N N N positive const and positive const 2 N 2 as N, N 2, N 2 and N 2 2 tend to infinity if p and p 2 are fixed or N N 2 positive const, N N 2 positive const and δ δ = O as N, N 2, N 2, N 2 2, p and p 2 tend to infinity, where δ = µ µ 2 If p and p 2 are fixed, then this approximation corresponds to Type I approximation If p and p 2 are also large, that corresponds to Type II approximation Under unknown Σ, for the purpose of obtaining EU m and EV m, ie, deriving approximation given in 39, we have the following Lemmas Lemma 3 Let VecX N p p 2 VecM, Σ Γ, M = m ij, Σ = σ ij and Γ = γ ij Then the expectations with respect to X are Ex i j x i2 j 2 = σ i i 2 γ j j 2 m i j m i2 j 2, Ex i j x i2 j 2 x i3 j 3 = m i j σ i2 i 3 γ j2 j 3 m i2 j 2 σ i i 3 γ j j 3 m i3 j 3 σ i i 2 γ j j 2 m i j m i2 j 2 m i3 j 3, Ex i j x i2 j 2 x i3 j 3 x i4 j 4 = σ i i 4 γ j j 4 σ i2 i 3 γ j2 j 3 σ i i 2 γ j j 2 σ i4 i 3 γ j4 j 3 σ i i 3 γ j j 3 σ i4 i 2 γ j4 j 2 m i j m i2 j 2 σ i4 i 3 γ j4 j 3 m i j m i3 j 3 σ i4 i 2 γ j4 j 2 m i2 j 2 m i3 j 3 σ i i 4 γ j j 4 m i j m i4 j 4 σ i2 i 3 γ j2 j 3 m i4 j 4 m i2 j 2 σ i i 3 γ j j 3 m i4 j 4 m i3 j 3 σ i i 2 γ j j 2 m i j m i2 j 2 m i3 j 3 m i4 j 4, 0

11 Proof: This Lemma was derived by considering the characteristic function of X eg, see Gupta and Nagar 2000 Moreover Lemma 2 and Lemma 3 with putting X = Γ 2 concerning conditional expectations of Ψ 2 given Γ lead the following Lemma Lemma 32 Let C ij p i p j and D ij p i p j be constant maices respectively, then the conditional expectations given Γ as follows can be obtained: E Ψ 2 Γ = Σ Σ 2, E Ψ 2 C 2 Ψ 2 Γ = Γ C 2Σ 22 Σ Σ 2C 2 Σ Σ 2, E Ψ 2 C 22 Ψ 2 Γ = C 22Σ 22 Γ Σ Σ 2C 22 Σ 2 Σ, E Ψ 2 C Ψ 2 Γ = Γ C Σ 22 Σ 2 Σ C Σ Σ 2, E Ψ 2 C Ψ 22 2 C Ψ 2 Γ = C Γ Σ Σ 2C 22 Σ 22 Γ C Σ Σ 2C 22Σ 22 C 22 Σ 22 Γ C Σ Σ 2 Σ Σ 2C 22 Σ 2 Σ C Σ Σ 2, E Ψ 2 C Ψ 2 2 C Ψ 22 2 Γ = C 22 Σ 22 Σ Σ 2C 2 Γ C 2 Σ Σ 2C 22 Σ 22 Γ Γ C 2Σ 22 C 22 Σ 2 Σ Σ Σ 2C 2 Σ Σ 2C 22 Σ 2 Σ, E Ψ 2 C 22 Ψ 2 C Ψ 2 D 22 Ψ 2 Γ = C 22 Σ 22 D 22 Σ 22 Γ C C 22 Σ 22 D 22 Σ 22 Γ C Γ Σ Σ 2C 22 Σ 22 D 22Σ 2 Σ C Γ Γ C Γ C 22 Σ 22 D 22Σ 22 Γ C Γ D 22 Σ 22 Σ Σ 2C 22 Σ 2 Σ C Γ C 22 Σ 2 Σ C Σ Σ 2D 22 Σ 22 Γ Σ Σ 2C 22 Σ 22 D 22 Σ 2 Σ Γ C Σ Σ 2C 22Σ 22 D 22 Σ 2 Σ C 22 Σ 22 Γ C Σ Σ 2D 22 Σ 2 Σ Σ Σ 2C 22 Σ 2 Σ C Σ Σ 2D 22 Σ 2 Σ, E C Ψ 2 2 D Ψ 2 2 Γ = C 2 Γ D 2Σ 22 C 2 Σ Σ 2 D 2 Σ Σ 2, E = C Ψ 22 2 C Ψ 2 C 22 Σ 2 Σ C 22 Σ 22 C Γ Γ Ψ 2 C 2 C 22 Σ 2 Σ C Σ Σ 2 Γ C 2Σ 22 C Σ Σ 2C 2 Σ Σ 2C 2, C 22 Σ 22 C 2 Γ C Σ Σ 2

12 E C Ψ 22 2 C Ψ 2 D Ψ 22 2 D Ψ 2 Γ = C 22 Σ 22 D 22 Σ 22 Γ C Γ D C 22 Σ 22 Γ C D 22 Σ 22 Γ D C 22 Σ 22 D 22Σ 22 Γ C Γ D C 22 Σ 2 Σ C Σ Σ 2 D 22 Σ 22 Γ D C 22 Σ 2 Σ C Γ D Σ Σ 2D 22Σ 22 C 22 Σ 22 D 22 Σ 2 Σ D Γ C Σ Σ 2 C 22 Σ 2 Σ C Γ D Σ Σ 2D 22 Σ 22 C 22 Σ 22 D 22Σ 2 Σ D Γ C Σ Σ 2 C 22 Σ 22 Γ C D 22 Σ 2 Σ D Σ Σ 2 C 22 Σ 2 Σ C Σ Σ 2 D 22 Σ 2 Σ D Σ Σ 2 Proof: By making use of results of Lemma 2 and Lemma 3, we can obtain the above results For some expectations of the above, the concrete methods for calculating them will be given in Appendix A The other expectations can be also obtained by the methods similarly Furthermore we also give the some results of calculating expectations by Lemma 24, Konno 988 and Marx 98 Lemma 33 The following expectations can be calculated as E Γ Γ2 Σ Γ Σ E µ µ 2 Γ Γ2 Σ Γ µ µ 2 = p c, = c δ 2, where c = n /{n p n p 3n p }, δ 2 = µ µ 2 Σ µ µ 2 Proof: The proof will be completed in Appendix B By Lemma 22, Lemma 32 and Lemma 33, we can obtain the following theorem Theorem 34 If µ g and Σ are unknown, then the expectations of U m and V m defined by 32 and 34 are EU m = 2 u δ 2 p N N 2 N N 2 2 u 3 2 p N N 2 N N 2 p 2N N 2 N N 2 2 p 2u 2 δ 2 p N N 2, N N 2 EV m = v δ 2 p N N 2 N N 2 p 2 v 2 δ 2 p N N 2 N N 2 2

13 where u = u 2 = u 3 = v = v 2 = v 3 = v 3 2 p N N 2 N N 2 p 2N N 2 N N 2 n n p n n p, n n p n p, n n p, n 2 n n p n p n p 3 2p 2 nn n n p n p 3n p n p n 2 n n pn p n p 3, n 2 n pn p n p 3 p 2 n 2 n n pn p 3n p n p n p 3 2n 2 n n pn p n p 3n p n p n p 3 p n 2 n p 2 n pn p n p 3n p n p 3 2n 2 n pn p n p 3n p n p 3, n 2 n n pn p n p 3, Proof: The proof will be completed in Appendix C Thus we can derive the asymptotic approximation for EPMC in 39 with explicit form By Theorem 34, we derive that under the framework such that N N N positive const and 2 N 2 positive const as N, N 2, N 2 and N 2 2 tend to infinity if p and p 2 are fixed or N N 2 positive const, N N 2 positive const, δ δ = O, n p and n p as N, N 2, N 2, N 2 2, p and p 2 tend to infinity In addition, we can find the following corollary as checking on the results in Theorem 34 3

14 Corollary 35 If we put N g = N g g =, 2 and note that p = p p 2, EU m and EV m given by Theorem 34 can be reduced to EU and EV derived in Lachenbruch 968 as follows: n EU = 2n p EV = { 2 N N 2 p N N 2 n 2 n n pn p n p 3 }, { 2 N N 2 p } N N 2 Besides Lachenbruch 968 proposed the asymptotic approximation for EPMC of LDF in 2 by using EU and EV : e c 2 Φ{EV } 2 EU 30 As it were, this corollary implies asymptotic approximation proposed in this paper is one of extension for Lachenbruch 968 Although we can obtain the asymptotic approximation for EPMC, we can not use µ g and Σ actually since they are usually unknown Then it is natural to consider substituting d 2 m µ µ 2 Σ µ µ 2 and D2 m for δ 2 and 2 respectively However under the situation where sample sizes are not very large, the approximation may be affected by the biases of d 2 m and D2 m Therefore we also propose the unbiased estimators of δ 2 and 2 respectively at the end of this section Theorem 36 The unbiased estimators δ 2 and 2 of δ 2 and 2 which are included in the asymptotic approximation for EPMC given in Theorem 34 can be obtained as following forms: δ 2 = b d 2 m p N N 2 N N 2, 3 2 = b 2 D 2 m b 2 b b 3 δ 2 N N 2 p 2N N 2 b b 2 p N N 2 N N 2 p p 2 N N 2 b 3 n N N 2 where b = n/n p, b 2 = n /n p and b 3 = n /n p Proof: The proof will be completed in Appendix D, 32 Besides in the case of complete data, the unbiased estimator of Mahalanobis squared distance has the following form by using D 2 = x x 2 S x x 2 : n p D 2 N n N 2 N N 2 p 33 As concerns the asymptotic approximation proposed in this section, it will be evaluated by Monte Carlo simulations in Section 4 Then the above unbiased estimators will be used in stead of Mahalanobis squared distances 4

15 4 Simulation Studies We are interested in the accuracy of the asymptotic approximation for EPMC proposed in this paper Although we are also interested in comparing it with the other ones, the asymptotic approximations for EPMC in the case of missing samples had not been proposed before Therefore we compare the accuracy of it with the other ones for the case of complete data, eg, Lachenbruch 968 and Fujikoshi and Seo 998 As concerns the asymptotic approximation for EPMC proposed by Lachenbruch 968, see Corollary 35 On the other hand, Fujikoshi and Seo 998 proposed the following the asymptotic approximation for EPMC of LDF presented in 2: where ζ = N N 2 /2 p { 2 2 N N 2 e c 2 Φζ, 4 pn N 2 N N 2 } { 2 pn N 2 N N 2 } /2 Then we start to compare the accuracy of 39, 30 and 4 by Monte Carlo simulations As you know, in actual, the exact Mahalanobis squared distances δ 2 and 2 are unknown respectively Thus we use the unbiased estimator 33 in stead of 2 included in the asymptotic approximations 30 and 4 In similar, the unbiased estimators 3 and 32 are used in stead of δ 2 and 2 in 39 respectively As concerns the settings of sample sizes concerning complete data, dimension and Mahalanobis distance, we chose the parameters which are similar to Fujikoshi and Seo 998 The settings of sample sizes and dimensions concerning missing samples are as follows: N 2, N 2 2 = 0, 0, 20, 20, 20, 40, 40, 40, and p, p 2 =,, 3, 2, 5, 5 The tables in these simulations are separated by the setting of dimensions Besides the values of asymptotic approximations in tables are average values of the ones calculated,000,000 times by Monte Carlo simulations and evaluated the accuracy of ones by them The results are shown in Tables 3 respectively 5 Conclusion and Future Problems In this paper, we extended the MLEs of 2-Step monotone missing samples to that of two groups in Section 2 By making use of the MLEs, we considered LDF based on 2-Step monotone missing samples Moreover by giving the many results concerning the expectations which can 5

16 not be calculated easilyeg, Lemma 33, we proposed the asymptotic approximation for EPMC with an explicit form which is Lachenbruch type of the LDF considered by us As it were, it is one of theoretical extensions of Lachenbruch 968 We proposed not only the asymptotic approximation but also the unbiased estimators which are used in calculating the approximation actually in Section 3 Finally we evaluated our result numerically by comparing with the other approximations in Section 4 By simulations, we find that our result is more accurate than the ones which had been proposed before Although the asymptotic approximation proposed in this paper is limited only in the case of 2-Step monotone missing samples, consucting LDF by missing samples holds good method for making EPMC lower and giving more accurate asymptotic approximation for EPMC These results imply that method proposed in this paper can be considered as one of useful methods for experimenters who have put linear discriminant analysis to practical use In the case of complete data, in general, the results of Fujikoshi and Seo 998 may be more accurate than Lachenbruch 968 As left problems, although it may be easy to be suck with an idea of the extension of the result in this paper to k-step monotone missing samples, the calculations of expectations will be much complicated If we limit the discussion in the case of 2-Step monotone missing samples, also we will be able to derive theoretical error bounds for the asymptotic approximation proposed in this paper by the method which is similar to Fujikoshi 2000 By the results of simulations in the case of complete data, the extensions of the other types of approximations eg, Okamoto 963, 968, Fujikoshi and Seo 998 to the case of 2-Step monotone missing samples will be needed Finally we point many discussions eg, testing equality procedure, MANOVA based on monotone missing samples have been left Acknowledgements We wish to express much gratitude to Professor Yasunori Fujikoshi for suggesting the ideas and giving useful and valuable advices in this paper 6

17 References Anderson, T W and Olkin, I 985 Maximum-likelihood Estimation of the Parameters of a Multivariate Normal Disibution Linear Algebra and Its Applications, 70, 47 7 Batsidis, A, Zografos, K and Loukas, S 2006 Errors in Discrimination with Monotone Missing Data from Multivariate Normal Populations Computational Statistics & Data Analysis, 50, Fujikoshi, Y 2000 Error Bounds for Asymptotic Approximations of the Linear Discriminant Function When the Sample Sizes and Dimensionality are Large Journal of Multivariate Analysis, 73, 7 Fujikoshi, Y and Seo, T 998 Asymptotic Approximations for EPMC s of the Linear and Quadratic Discriminant Functions When the Sample Sizes and the Dimension are Large Random Operators and Stochastic Equations, 6, Gupta, A K and Nagar, D K 2000 Maix Variate Disibutions Chapman and Hall/CRC, Monographs and Surveys in Pure and Applied Mathematics Haff, L R 979 An Identity for the Wishart Disibution with Applivations Journal of Multivariate Analysis, 9, Kanda, T and Fujikoshi, Y 998 Some Basic Properties of the MLE s for a Multivariate Normal Disibution with Monotone Missing Data American Journal of Mathematical and Management Sciences, 8, 6 90 Kanda, T and Fujikoshi, Y 2004 Linear Discriminant Function and Probabilities of Misclassification with Monotone Missing Data Proceedings of The Eighth China-Japan Symposium on Statistics, Konno, Y 988 Exact Moments of the Multivariate F and Beta Disibutions Journal of The Japan Statistical Society, 8, Lachenbruch, P A 968 On Expected Probabilities of Misclassification in Discriminant Analysis, Necessary Sample Size, and a Relation with the Multiple Correlation Coefficient Biomeics, 24, Marx, D G 98 Aspects of the Maic t-disibution Ph D thesis, Free State, RSA University of The Orange Memon, A Z and Okamoto, M 97 Asymptotic Expansion of the Disibution of the Z Statistic in Discriminant Analysis Journal of Multivariate Analysis,, Okamoto, M 963 An Asymptotic Expansion for the Disibution of the Linear Discriminant Function Annals of Mathematical Statistics, 34,

18 Okamoto, M 968 Correction to An Asymptotic Expansion for the Disibution of the Linear Discriminant Function Annals of Mathematical Statistics, 39, Raudis, S 972 On the Amount of Priori Information in Designing the Classification Algorithm Engineering Cybernetics, 0, 7 78 in Russian Siotani, M, Hayakawa, T and Fujikoshi, Y 985 Modern Multivariate Statistical Analysis : A Graduate Course Handbook American Sciences Press, Columbus, Ohio Siotani, M and Wang, R H 977 Asymptotic Expansions for Error Rates and Comparison of the W - procedure and Z-procedure in Discriminant Analysis Multivariate Analysis-IV P R Krishnaiah, ed, North Holland, Srivastava, M S 985 Multivariate Data with Missing Observations Theory and Methods, 4, Communication in Statistics Srivastava, M S and Carter, E M 986 The Maximum Likelihood Method for Non-Response in Sample Survey Survey Methodology, 2, 6 72 Wakaki, H 994 Discriminant Analysis under Elliptical Populations Hiroshima Mathematical Journal, 24, Wald, A 944 On a Statistical Problem Arising in the Classification of an Individual into One of Two Groups Annals of Mathematical Statistics, 5,

19 Appendix A Proof of Lemma 32 We show the proof for calculations E Ψ 2 C Ψ 22 2 C Ψ 2 Γ A and E C Ψ 2 2 D Ψ 2 2 Γ A2 At First, we complete the proof of A A is also expressed as follows: Γ E Γ 2 C 22Γ 2 C Γ 2 Γ, A3 where C = Γ C Γ Therefore i, jth element of A3 is expressed by Γ 2 = Γij 2, Γ 2 = Γij 2, C 22 = C ij 22 and C = C ij as follows: E Γ 2 C 22Γ 2 C Γ 2 Γ ij By Lemma 3 with putting = = p 2 p 2 p p i = i 2 = i 3 = i 4 = p 2 p 2 p p i = i 2 = i 3 = i 4 = E Γ ii 2 Ci i 2 22 Γi 2i 3 2 Ci 3i 4 Γi 4j 2 Γ C i i 2 22 Ci 3i 4 E Γ i i 2 Γi 2i 3 2 Γji 4 2 Γ A4 A4 is also expressed by X = Γ 2, M = Σ 2Σ Γ, Σ = Σ 22 and Γ = Γ, M = m ij, Σ = σ ij, Γ = γ ij, M = m ij, Σ = σ ij and Γ = γ ij as follows: E Γ 2 C 22Γ 2 C Γ 2 Γ ij = p 2 p 2 p p i = i 2 = i 3 = i 4 = γ ii4 C i 4i 3 m i 3 i 2 C i 2i { m ii C i i 2 22 σ i 2 jc i 3i 4 γ i 4 i 3 22 σ i j γ ii3 C i 3i 4 } m ii C i i 2 22 m i 2 i 3 C i 3i 4 m i 4 j m i 4 jc i i 2 22 σ i 2 i Therefore by noting Σ and Γ are symmeic, E Γ 2 C 22Γ 2 C Γ 2 Γ = C Γ Γ Σ Σ 2C 22 Σ 22 Γ C Γ Σ Σ 2C 22Σ 22 Γ C Γ Σ Σ 2 C 22 Σ 22 Γ Σ Σ 2C 22 Σ 2 Σ Γ C Γ Σ Σ 2 A5 9

20 By A3, multiplying Γ on the left-hand of A5 and putting C = Γ C Γ into A5 lead the result to prove Next we complete the proof of A2 A2 is also expressed as E C 2Γ 2 D 2Γ 2 Γ, A6 where C 2 = C 2 Γ and D 2 = D 2 Γ Therefore A6 is expressed by Γ 2 = Γij 2, Γ 2 = Γij 2, C 2 = C ij 2 and D 2 = D ij 2 as follows: E C 2Γ 2 D 2Γ 2 By using Lemma 3 with putting Γ = = p 2 p p 2 p i = i 2 = i 3 = i 4 = p 2 p p 2 p i = i 2 = i 3 = i 4 = E Γ i i 2 2 Γi 3i 4 2 Γ E C i i 2 2 Γi 2i 2 Di 3i 4 2 Γi 4i 3 2 Γ C i i 2 2 Di 3i 4 2 A7 again, A7 is also expressed by X = Γ 2, M = Σ 2Σ Γ, Σ = Σ 22 and Γ = Γ M = m ij, Σ = σ ij, Γ = γ ij, M = m ij and Σ = σ ij as follows: E C 2Γ 2 D 2Γ 2 Γ = p 2 p p 2 p C i i 2 i = i 2 = i 3 = i 4 = C i i 2 2 m i 2 i D i 3i 4 2 m i 4 i 3 2 γ i 2 i 4 D i 4i 3 2 σ i 3 i Therefore by noting that Σ is symmeic, E C 2Γ 2 D 2Γ 2 Γ = C 2Γ D 2Σ 22 D 2Γ Σ Σ 2 C 2Γ Σ Σ 2 A8 By A6, putting C 2 = C 2 Γ and D 2 = D 2 Γ into A8 leads the result to prove The other expectations can be obtained in the same way as the above Thus Lemma 32 has been completed 20

21 Appendix B Proof of Lemma 33 By Lemma 24, Γ Γ2 Σ Γ A = β I,p W W Σ 2 A Σ 2 W W 2 2, B where β I,p W W 2 2 W W W 2 2 has B I p n /2, n 2 2/2 Since β I,p is independent of W W 2, we can calculate conditional expectation of B given W W 2 Besides Konno 988 gave the expectation of β I,p as follows: E β I,p = n p n p I p Thus the conditional expectation to obtain can be calculated as n p n p W W 2 Σ 2 A Σ 2 W W 2 B2 Next we also calculate the expectation of B2 concerning W W 2 Since W W 2 is disibuted as W p n, I p, the expectation can be obtained by Marx 98: E Γ Γ2 Σ Γ A = c Σ 2 A Σ 2, B3 where c = n /{n p n p 3n p } By putting A = Σ and A = µ µ 2 µ µ 2 into B3, Lemma 33 has been completed Appendix C Proof of Theorem 34 Since µ g = Ip 0 Ψ 2 I p2 η g, Ip 0 Σ = Ψ 2 I p2 Ψ 0 0 Ψ 22 I p Ψ 2 0 I p2 and Ip 0 Ψ 0 Ip Ψ 2, Σ = Ψ 2 I p2 0 Ψ 22 0 I p2 U m is also expressed as U m = η η 2 Ψ η µ η 2 η Ψ 22 η 2 Ψ 2 µ µ 2

22 2 η η 2 Ψ η η 2 2 η = x T x2 T Ψ x T µ x 2F x2 2F x F x2 F Ψ 2 2 x 2F x2 2F x 2 η 2 2 2F x2 2F Ψ 22 η 2 η 2 2 Ψ 22 x 2F µ 2 Ψ 22 Ψ 2 x F µ x F x2 F Ψ 2 Ψ 22 Ψ 2 x F µ 2 x Ψ 22 x 2F x2 2F x 2 x F x2 F Ψ 2 Ψ Ψ 22 2 x F x2 F 2F x2 2F T x2 T Ψ 22 x 2F µ 2 Ψ x T x2 T Ψ 22 Ψ 2 x F x2 F Here all sample mean vectors x g T, xg F and xg 2F are independent of Γ, Γ 2 respectively Then the expectation of U m given Γ and Γ 2 is EU m Γ, Γ 2 = n N N 2 2N N 2 n N N 2 2N N 2 nn N 2 2N N 2 n N N 2 N N 2 n N N 2 2N N 2 Γ 22 Σ 22 Γ 22 Σ 2Σ Σ 2 Γ Γ2 Σ Γ Γ 2 Γ 22 Σ 2 Γ Γ 2 Γ 22 Γ 2 Γ Σ n 2 µ µ 2 Γ Γ2 µ µ 2 n µ µ 2 Γ Γ 2 Γ 22 µ 2 µ 2 2 n 2 µ 2 µ 2 2 Γ 22 µ 2 µ 2 2 n 2 µ µ 2 Γ Γ 2 Γ 22 Γ 2 Γ µ µ 2 Now we express EU m Γ, Γ 2 by using EU m Γ, Γ 2 and EU m2 Γ, Γ 2 which are separated this expectation given Γ and Γ 2 into the parts containing Γ Γ 2 and not respectively Therefore where EU m Γ, Γ 2 = EU m Γ, Γ 2 EU m2 Γ, Γ 2, EU m Γ, Γ 2 = n N N 2 N N 2 Γ Γ 2 Γ 22 Σ 2 22

23 n N N 2 2N N 2 Γ Γ 2 Γ 22 Γ 2 Γ Σ n µ µ 2 Γ Γ 2 Γ 22 µ 2 µ 2 2 n 2 µ µ 2 Γ Γ 2 Γ 22 Γ 2 Γ µ µ 2 and EU m2 Γ, Γ 2 = n N N 2 2N N 2 n N N 2 2N N 2 nn N 2 2N N 2 Γ 22 Σ 22 Γ 22 Σ 2Σ Σ 2 Γ Γ2 Σ n 2 µ µ 2 Γ Γ2 µ µ 2 n 2 µ 2 µ 2 2 Γ 22 µ 2 µ 2 2 Then we consider EU m at first Since U m does not include Γ 2 and Γ 22 is independent of Γ, Γ 2, we can obtain EU m Γ, Γ 2 By Lemma 22, EU m Γ, Γ 2 is as follows: EU m Γ, Γ 2 = n n p µ µ 2 Γ Γ 2 Σ 22 µ 2 µ 2 2 n 2n p µ µ 2 n N N 2 N N 2 n p n N N 2 2N N 2 n p Γ Γ 2 Σ 22 Γ 2 Γ µ Σ 22 Σ 2Γ Γ 2 Γ Γ 2 Σ 22 Γ 2 Γ Σ Moreover EU m Γ can be obtained by Lemma 22 and Lemma 32 EU m Γ = p 2 n 2n p µ µ 2 Γ µ µ 2 n n p µ µ 2 n 2n p µ µ 2 p 2n N N 2 2N N 2 n p n N N 2 2N N 2 n p Σ Σ 2Σ 22 µ 23 2 µ 2 2 Σ Σ 2Σ 22 Σ 2Σ µ Γ Σ Σ 22 Σ 2Σ Σ 2 µ 2 µ 2

24 Finally by Lemma 22, p 2 n EU m = 2n p n p µ µ 2 n n p µ µ 2 n 2n p µ µ 2 Σ Σ 2Σ 22 µ p p 2 n N N 2 2N N 2 n p n p n N N 2 2N N 2 n p Σ µ 2 µ 2 µ 2 2 Σ Σ 2Σ 22 Σ 2Σ µ On the other hand, we can calculate EU m2 with leaving Γ Γ 2 making use of Lemma 22, n EU m2 = 2n p µ µ 2 n 2n p µ 2 µ 2 n N N 2 2N N 2 n p p nn N 2 2N N 2 n p µ 2 Σ 22 Σ 2Σ Σ 2 C Σ µ 2 Σ 22 µ out of consideration By µ 2 2 µ 2 2 Σ 22 Σ 2Σ Σ 2 p 2n N N 2 2N N 2 n p C2 Thus EU m = EU m EU m2 is derived by C and C2 On the other hand, V m is expressed as V m = η η 2 Ψ Ψ Ψ η η 2 2 η η 2 Ψ Ψ Ψ 2 Ψ 22 η 2 η η η 2 Ψ Ψ Ψ 2 Ψ 22 η 2 η 2 2 η 2 η 2 2 Ψ Ψ 22 2 Ψ Ψ 2 Ψ 22 η 2 η η 2 η 2 2 Ψ 22 Ψ 2 Ψ Ψ 2 Ψ 22 η 2 η 2 2 η 2 η 2 2 Ψ 22 Ψ 2 Ψ Ψ 2 Ψ 22 η 2 η 2 2 η 2 η 2 2 Ψ 22 Ψ 22 Ψ 22 η 2 η 2 2 = x T x2 T Ψ Ψ Ψ x T x2 T 2x T x2 T Ψ Ψ Ψ 2 24 Ψ 22 x 2F x2 2F

25 2x 2x 2x T x2 T T x2 T T x2 T x 2F x2 2F 2x F x2 F Ψ 2 x F x2 F Ψ 2 2x 2F x2 2F 2x F x2 F Ψ 2 2x F x2 F Ψ 2 2x F x2 F Ψ 2 x 2F x2 2F 2x F x2 F Ψ 2 x F x2 F Ψ 2 x 2F x2 2F Ψ Ψ Ψ 2 Ψ Ψ 22 2 x F x2 Ψ Ψ Ψ 2 Ψ 22 x 2F x2 2F F Ψ Ψ Ψ 2 Ψ Ψ 22 2 x F x2 Ψ 22 Ψ 2 Ψ Ψ 2 Ψ 22 Ψ 2 Ψ Ψ 2 F Ψ 22 x 2F x2 2F Ψ 22 x 2F x2 2F Ψ Ψ 22 2 Ψ Ψ 2 Ψ Ψ 22 2 x F x2 Ψ 22 Ψ 2 Ψ Ψ 2 Ψ 22 x 2F x2 2F Ψ Ψ 22 2 Ψ Ψ 2 Ψ 22 x 2F x2 Ψ 22 Ψ 2 Ψ Ψ 2 Ψ 22 x 2F x2 2F 2F F Ψ 22 Ψ 2 Ψ Ψ 2 Ψ Ψ 22 2 x F x2 Ψ 22 Ψ 2 Ψ Ψ 2 Ψ 22 Ψ 22 2x F x2 F Ψ 2 x F x2 F Ψ 2 Ψ 22 Ψ 2 Ψ Ψ 2 Ψ 22 x 2F x2 2F Ψ 22 x 2F x2 2F Ψ 22 Ψ 2 Ψ Ψ 2 Ψ Ψ 22 2 x F x2 Ψ 22 x 2F x2 Ψ 22 Ψ 22 2F Ψ 22 x 2F x2 2F Ψ 22 Ψ 22 Ψ Ψ 22 2 x F x2 F Also in this calculation, we separate EV m Γ, Γ 2 into the terms containing Γ Γ 2 in the same way as calculating EU m Thus we express EV m Γ, Γ 2 as follows: EV m Γ, Γ 2 = EV m Γ, Γ 2 EV m2 Γ, Γ 2, where EV m Γ, Γ 2 and EV m2 Γ, Γ 2 are the terms which contain Γ Γ 2 and do not respectively First EV m Γ, Γ 2 is calculated as follows: EV m Γ, Γ 2 = 4nn N N 2 N N 2 Γ Γ2 Σ Γ Γ 2 Γ 22 Σ 2 2nn µ µ 2 Γ Γ2 Σ Γ Γ 2 Γ 22 µ 2 µ 2 2n2 N N 2 N N 2 F F Γ 22 Σ 22 Γ 22 Γ 2 Γ Σ 2 2n 2 µ µ 2 Γ Γ 2 Γ 22 Σ 22 Γ 22 µ 2 µ 2 2 2nn µ µ 2 Γ Γ2 Σ 2 Γ 22 Γ 2 Γ µ µ 2 2n2 N N 2 N N 2 Γ 22 Σ 2Γ Γ 2 Γ 22 Σ

26 2n2 N N 2 N N 2 Γ 22 Σ 2Γ Γ 2 Γ 22 Σ 2Σ Σ 2 2n 2 µ 2 µ 2 2 Γ 22 Σ 2Γ Γ 2 Γ 22 µ 2 µ 2 2n2 N N 2 N N 2 2 Γ 22 Σ 2Σ Σ 2Γ 22 Γ 2 Γ Σ 2 2n 2 µ µ 2 Γ Γ 2 Γ 22 Σ 2Σ Σ 2Γ 22 µ 2 µ 2 2 2nn N N 2 N N 2 Γ Γ2 Σ Γ Γ 2 Γ 22 Γ 2 Γ Σ 2nn µ µ 2 Γ Γ2 Σ Γ Γ 2 Γ 22 Γ 2 Γ µ µ 2 n N N 2 N N 2 n N N 2 N N 2 Γ 22 Γ 2 Γ Σ Γ Γ 2 Γ 22 Σ 22 Γ 22 Γ 2 Γ Σ Γ Γ 2 Γ 22 Σ 2Σ Σ 2 n 2 µ 2 µ 2 2 Γ 22 Γ 2 Γ Σ Γ Γ 2 Γ 22 µ 2 µ 2 n2 N N 2 N N 2 Γ Γ 2 Γ 22 Σ 22 Γ 22 Γ 2 Γ Σ n 2 µ µ 2 Γ Γ 2 Γ 22 Σ 22 Γ 22 Γ 2 Γ µ µ 2 2n2 N N 2 N N 2 Γ 22 Σ 2Γ Γ 2 Γ 22 Γ 2 Γ Σ 2 2n 2 µ µ 2 Γ Γ 2 Γ 22 Γ 2 Γ Σ 2 Γ 22 µ 2 µ 2 2n2 N N 2 N N 2 Γ Γ 2 Γ 22 Σ 2Γ Γ 2 Γ 22 Σ n 2 µ µ 2 Γ Γ 2 Γ 22 Σ 2Γ Γ 2 Γ 22 µ 2 µ 2 n2 N N 2 N N 2 2 Γ Γ 2 Γ 22 Σ 2Σ Σ 2Γ 22 Γ 2 Γ Σ n 2 µ µ 2 Γ Γ 2 Γ 22 Σ 2Σ Σ 2 Γ 22 Γ 2 Γ µ µ 2 4n2 N N 2 N N 2 Γ Γ 2 Γ 22 Γ 2 Γ Σ Γ Γ 2 Γ 22 Σ 2 2n 2 µ µ 2 Γ Γ 2 Γ 22 Γ 2 Γ Σ 26

27 Γ Γ 2 Γ 22 µ µ 2 2n 2 µ µ 2 Γ Γ 2 Γ 22 Σ 2 Γ Γ 2 Γ 22 Γ 2 Γ µ µ 2 n2 N N N N 2 Γ Γ 2 Γ 22 Γ 2 Γ Σ Γ Γ 2 Γ 22 Γ 2 Γ Σ n 2 µ µ 2 Γ Γ 2 Γ 22 Γ 2 Γ Σ Γ Γ 2 Γ 22 Γ 2 Γ µ µ 2 Now we consider separating EV m Γ, Γ 2 again by the number of Ψ 2, ie, EV m Γ, Γ 2 EV m Γ, Γ 2 EV m2 Γ, Γ 2 EV m3 Γ, Γ 2 EV m4 Γ, Γ 2, where V mr r =,, 4 denotes the terms including r Ψ 2 s in V m Since we find that Γ 22 is independent of Γ, Γ 2, Γ2 and we have the conditional expectation of Ψ 2 given Γ in Lemma 32, we can calculate conditional expectation of V mr given Γ, Γ2 for each r By making use of Lemma 22, we can obtain EV m Γ, Γ 2, Γ2 as follows: EV m Γ, Γ 2, Γ2 = 4nn N N 2 N N 2 n p Γ Γ2 Σ Γ Γ 2 Σ 22 Σ 2 2nn n p µ µ 2 Γ Γ2 Σ Γ Γ 2 Σ 22 µ 2 µ 2 2 2nn n p µ µ 2 Γ Γ2 Σ 2 Σ 22 Γ 2 Γ µ µ 2 4n 2 N N 2 N N 2 n pn p 3 4n 2 N N 2 N N 2 n pn p 3 Σ 22 Σ 2Γ Γ 2 Σ 22 Σ 2Σ Σ 2 N N 2 4p 2 n 2 N N 2 n pn p n p 3 Σ 22 Γ 2 Γ Σ 2 27 Σ 22 Σ 2Γ Γ 2

28 4n 2 N N 2 N N 2 n pn p n p 3 Σ 22 Γ 2 Γ Σ 2 Σ 22 Σ 2Σ Σ 2 2n 2 n pn p 3 µ 2 µ 2 2 Σ 22 Σ 2Γ Γ 2 Σ 22 µ 2 µ 2 2n 2 2 n pn p n p 3 Σ 22 Γ 2 Γ Σ 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 2n 2 n pn p 3 µ µ 2 Γ Γ 2 Σ 22 Σ 2Σ Σ 2Σ 22 µ 2 µ 2 2n 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Γ Γ 2 Σ 22 µ 2 µ 2 2 2n 2 n p n pn p n p 3 µ µ 2 Γ Γ 2 Σ 22 µ 2 µ 2 2 By making use of Lemma 32, we can obtain EV m Γ, Γ2 as follows: EV m Γ, Γ2 = 4nn N N 2 N N 2 n p Γ Γ2 Σ 2 Σ 22 Σ 2 2 2nn n p µ µ 2 Γ Γ2 Σ 2 Σ 22 µ 2 µ 2 2 2nn n p µ µ 2 Γ Γ2 Σ 2 Σ 22 Σ 2Σ µ µ 2 4n 2 N N 2 n p N N 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 4n 2 N N 2 N N 2 n pn p 3 Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ Σ 2 N N 2 { 4n 2 N N 2 n pn p n p 3 } Σ 22 Σ 2Σ 2 Σ 2 28

29 2n 2 n pn p 3 µ 2 µ 2 2 Σ 22 Σ 2Σ Σ 2Σ 22 µ 2 µ 2 2n 2 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 2n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ Σ 2Σ 22 µ 2 µ 2 2n 2 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 2 2n 2 n p n pn p n p 3 µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 2 By making use of Lemma 22, EV m2 Γ, Γ 2, Γ2 can be calculated as follows: EV m2 Γ, Γ 2, Γ2 = 2nn N N 2 N N 2 n p Γ Γ2 Σ Γ Γ 2 Σ 22 Γ 2 Γ Σ 2nn n p µ µ 2 Γ Γ2 Σ Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 N N 2 2n 2 N N 2 n p n pn p n p 3 Σ 22 Γ 2 Γ Σ Γ Γ 2 n 2 N N 2 N N 2 n pn p 3 Σ 22 Γ 2 Γ Σ Γ Γ 2 Σ 22 Σ 2Σ Σ 2 N N 2 Σ 22 Σ 2Σ Σ 2 2n 2 N N 2 n pn p n p 3 n 2 n pn p 3 Σ 22 Γ 2 Γ Σ Γ Γ 2 µ 2 µ 2 2 Σ 22 Γ 2 Γ Σ Γ Γ 2 Σ 22 µ 2 µ

30 n 2 n pn p n p 3 Σ 22 Γ 2 Γ Σ Γ Γ 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 N N 2 2n 2 N N 2 n pn p 3 Σ 22 Σ 2Γ Γ 2 Σ 22 Γ 2 Γ Σ 2 N N 2 4n 2 N N 2 n pn p n p 3 Σ 22 Σ 2Γ Γ 2 2n 2 n p 2 n pn p n p 3 Σ 22 Σ 2Γ Γ 2 µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ Σ 2 Σ 22 µ 2 µ 2 2n 2 n pn p n p 3 µ µ 2 Γ Γ 2 Σ 22 Σ 2Γ Γ 2 Σ 22 µ 2 µ 2 4n 2 n pn p n p 3 Σ 22 Σ 2Γ Γ 2 µ µ 2 Γ Γ 2 Σ 22 µ 2 µ 2 2 2n 2 N N 2 N N 2 n pn p 3 Γ Γ 2 Σ 22 Σ 2Γ Γ 2 Σ 22 Σ 2 2n 2 n p 2 n pn p n p µ µ 2 Γ Γ 2 Σ 22 Σ 2Γ Γ 2 Σ 22 µ 2 µ 2 2n 2 n pn p n p 3 2 µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ Σ 2 Σ 22 µ 2 µ 2 2 n 2 N N 2 N N 2 n pn p 3 Γ Γ 2 Σ 22 Σ 2Σ Σ 2Σ 22 Γ 2 Γ Σ n 2 n pn p 3 µ µ 2 Γ Γ 2 Σ 22 Σ 2 30

31 Σ Σ 2Σ 22 Γ 2 Γ µ µ 2 n 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 n 2 n p n pn p n p 3 µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 By making use of Lemma 32, EV m2 Γ, Γ2 can be calculated as follows: EV m2 Γ, Γ2 = 2p 2nn N N 2 N N 2 n p Γ Γ2 Σ Γ Σ 2nn N N 2 N N 2 n p Γ Γ2 Σ 2 Σ 22 Σ 2 2p 2nn n p µ µ 2 Γ Γ2 Σ Γ µ µ 2 2nn n p µ µ 2 Γ Γ2 Σ 2 Σ 22 Σ 2Σ µ µ 2 2p 2 n 2 N N 2 n p N N 2 n pn p n p 3 Γ Σ N N 2 2n 2 N N 2 n p n pn p n p 3 Σ 22 Σ 2Σ Σ 2 6n 2 N N 2 N N 2 n pn p 3 Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ Σ 2 N N 2 2n 2 N N 2 n p n pn p n p 3 Σ 22 Σ 2Σ Σ 2 Γ Σ 3

32 N N 2 { 6n 2 N N 2 n pn p n p 3 } Σ 22 Σ 2Σ 2 Σ 2 n 2 n pn p 3 µ 2 µ 2 2 Σ 22 Σ 2Σ Σ 2Σ 22 µ 2 µ 2 n 2 n p n pn p n p 3 Γ Σ µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 n 2 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 N N 2 2p 2 n 2 N N 2 n pn p 3 Σ 22 Σ 2Γ Σ 2 N N 2 4n 2 N N 2 n pn p n p 3 Σ 22 Σ 2Γ Σ 2 2p 2 n 2 n p 2 n pn p n p 3 µ µ 2 Γ Σ 2 Σ 22 µ 2 µ 2 4n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2 Σ Σ 2Σ 22 µ 2 µ 2 6n 2 2 n pn p n p 3 2 µ µ 2 Γ Σ 2 Σ 22 µ 2 µ 2 4n 2 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 2 N N 2 2n 2 N N 2 n pn p 3 Σ 22 Σ 2Γ Σ 2 32

33 2n 2 n p 2 n pn p n p 3 µ µ 2 Γ Σ 2 Σ 22 µ 2 µ 2 n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ Σ 2 Σ 22 Σ 2Σ µ µ 2 n 2 n p n pn p n p 3 Σ 22 Σ 2Σ Σ 2 n 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 2 µ µ 2 Γ µ µ 2 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2 p 2 n 2 n p n pn p n p 3 µ µ 2 Γ µ µ 2 n 2 n p n pn p n p 3 Since V m3 does not include Γ 2, EV m3 Γ Lemma 22 as follows: µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2, Γ 2 can be calculated by making use of EV m3 Γ, Γ 2 = 2n 2 N N 2 N N 2 n pn p 3 Γ Γ 2 Σ 22 Γ 2 Γ Σ Γ Γ 2 Σ 22 Σ 2 N N 2 Γ 2 Γ 2n 2 N N 2 n pn p n p 3 Σ Γ Γ 2 Σ 22 Σ 22 Σ 2Γ Γ 2 2n 2 n pn p 3 µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ Σ Γ Γ 2 Σ 22 µ 2 µ 2 2n 2 2 n pn p n p 3 33

34 Γ 2 Γ Σ Γ Γ 2 Σ 22 µ µ 2 Γ Γ 2 Σ 22 µ 2 µ 2 2 2n 2 N N 2 N N 2 n pn p 3 Γ Γ 2 Σ 22 Σ 2Γ Γ 2 Σ 22 Γ 2 Γ Σ 2n 2 N N 2 N N 2 Σ 22 Σ 2Γ Γ 2 n pn p n p 3 Γ Γ 2 Σ 22 Γ 2 Γ Σ 2n 2 n pn p 3 µ µ 2 Γ Γ 2 Σ 22 Σ 2Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 2n 2 n pn p n p 3 Σ 22 Σ 2Γ Γ 2 µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 By making use of Lemma 32, EV m3 Γ can be calculated as follows: N N N 2 EV m3 Γ = 4n 2 N 2 n pn p 3 Σ 22 Σ 2Γ Σ 2 N N 2 4p 2 n 2 N N 2 n pn p 3 Σ 22 Σ 2Γ Σ 2 4n 2 N N 2 N N 2 n pn p 3 Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ Σ 2 2n 2 n pn p 3 µ µ 2 Γ Σ 2 Σ 22 µ 2 µ 2 2p 2 n 2 n pn p 3 2 µ µ 2 Γ Σ 2 Σ 22 µ 2 µ

35 2n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ Σ 2Σ 22 µ 2 µ 2 4n 2 n pn p n p 3 µ µ 2 Γ Σ 2 Σ 22 µ 2 µ 2 2 2n 2 n p n pn p n p 3 Γ Σ µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 2 2n 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 2 N N 2 8n 2 N N 2 n pn p n p 3 Σ 22 Σ 2Γ Σ 2 N N 2 4n 2 N N 2 n p n pn p n p 3 Σ 22 Σ 2Σ Σ 2 Γ Σ N N 2 { 4n 2 N N 2 n pn p n p 3 } Σ 22 Σ 2Σ 2 Σ 2 2p 2 n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Γ µ µ 2 2n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Γ µ µ 2 2n 2 n pn p 3 2 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ µ µ 2 4n 2 n pn p n p 3 µ µ 2 Γ Σ 2 Σ 22 Σ 2Σ µ µ 2 35

36 2n 2 n p n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Γ µ µ 2 2n 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2 Since V m4 also does not include Γ 2, EV m4 Γ Lemma 22 as follows: N N N 2 EV m4 Γ, Γ 2 = n 2 N 2, Γ 2 can be calculated by making use of n pn p 3 Γ Γ 2 Σ 22 Γ 2 Γ Σ Γ Γ 2 Σ 22 Γ 2 Γ Σ N N 2 n 2 N N 2 n pn p n p 3 Σ 22 Γ 2 Γ Σ Γ Γ 2 n 2 n pn p 3 µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ Σ Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 n 2 n pn p n p 3 Σ 22 Γ 2 Γ Σ Γ Γ 2 Γ Γ 2 Σ 22 Γ 2 Γ Σ µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 By making use of Lemma 32, EV m4 Γ can be calculated as follows: EV m4 Γ = p 2 2 n2 N N 2 N N 2 n pn p 3 Γ Σ Γ Σ N N 2 p 2 n 2 N N 2 n pn p 3 Γ Σ Γ Σ 2p 2 n 2 N N 2 N N 2 n pn p 3 36

37 Σ 22 Σ 2Γ Σ 2 N N 2 2n 2 N N 2 n pn p 3 Σ 22 Σ 2Γ Σ 2 N N 2 2n 2 N N 2 n pn p 3 Γ Σ Σ 22 Σ 2Σ Σ 2 n 2 N N 2 N N 2 n pn p 3 Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ Σ 2 N N 2 2p 2 n 2 N N 2 n pn p n p 3 Γ Σ Γ Σ N N 2 p 2 n 2 N N 2 n p n pn p n p 3 Γ Σ Γ Σ N N 2 Γ Σ N N 2 2p 2 n 2 N N 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 4n 2 N N 2 n pn p n p 3 Σ 22 Σ 2Γ Σ 2 N N 2 { n 2 N N 2 n pn p n p 3 } Σ 22 Σ 2Σ 2 Σ 2 p 2 2 n2 n pn p 3 µ µ 2 Γ Σ Γ µ µ 2 p 2 n 2 n pn p 3 µ µ 2 Γ Σ Γ µ µ 2 2p 2 n 2 n pn p 3 37

38 µ µ 2 Σ Σ 2Σ 22 Σ 2Γ µ µ 2 2n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Γ µ µ 2 n 2 n pn p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ µ µ 2 2p 2 n 2 n pn p n p 3 µ µ 2 Γ Σ Γ µ µ 2 p 2 n 2 n p n pn p n p 3 Γ Σ µ µ 2 Γ µ µ 2 n 2 n p n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Γ µ µ 2 4n 2 n pn p n p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Γ µ µ 2 n 2 n p n pn p n p 3 Γ Σ µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2 n 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2 Now that EV m Γ, Γ2 = EV m Γ, Γ2 EV m2 Γ, Γ2 EV m3 Γ EV m4 Γ, EV m Γ, Γ2 = p 2 n 2 n p n pn p n p 3 µ µ 2 Γ µ µ 2 2p 2nn n p µ µ 2 Γ Γ2 Σ Γ µ µ 2 p 2 2 n2 n pn p 3 µ µ 2 Γ Σ Γ µ µ 2 p 2 n 2 n pn p 3 µ µ 2 38 Γ Σ Γ µ µ 2

39 2p 2 n 2 n pn p n p 3 µ µ 2 Γ Σ Γ µ µ 2 p 2 n 2 n p n pn p n p 3 Γ Σ µ µ 2 Γ Σ Γ µ µ 2 n 2 n p n pn p n p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2 n 2 n p n pn p n p 3 Γ Σ µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2 2n 2 n p n pn p n p 3 µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 2 2nn n p µ µ 2 Γ Γ2 Σ 2 Γ 22 µ 2 µ 2 2 2n 2 n p n pn p n p 3 Γ Σ µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 2 n 2 n p n pn p n p 3 Γ Σ µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 n 2 n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 n 2 n pn p 3 µ 2 µ 2 2 Σ 22 Σ 2Σ Σ 2Σ 22 µ 2 µ 2 2p 2 n 2 N N 2 n p N N 2 n pn p n p 3 p 2 2 n2 N N 2 N N 2 n pn p 3 p 2 n 2 N N 2 N N 2 n pn p 3 2 Γ Σ Γ Σ Γ Σ Γ Σ Γ Σ 39

40 2p 2 n 2 N N 2 N N 2 n pn p n p 3 2p 2nn N N 2 N N 2 n p Γ Γ2 Σ Γ Σ N N 2 p 2 n 2 N N 2 n p n pn p n p 3 Γ Σ Γ Σ 2n 2 N N 2 n p N N 2 n pn p n p 3 Γ Σ Γ Σ Σ 22 Σ 2Σ Σ 2 2nn N N 2 N N 2 n p Σ 22 Σ 2Γ Γ2 Σ 2 n 2 N N 2 N N 2 n pn p n p 3 n 2 N N 2 N N 2 n pn p 3 { } Σ 22 Σ 2Σ 2 Σ 2 Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ Σ 2 Finally EV m can be derived by making use of Lemma 22 and Lemma 33 as follows: EV m = p 2 n 2 n pn p n p 3 µ µ 2 Σ µ µ 2 2p 2 nn n n p n p 3n p n p µ µ 2 p 2 p 2 n 2 n n pn p 3n p n p n p 3 Σ µ µ µ 2 Σ µ µ 2 2p 2 n 2 n n pn p n p 3n p n p n p 3 µ µ 2 Σ µ µ 2 p p 2 n 2 n p 2 n pn p n p 3n p n p 3 µ µ 2 Σ µ µ 2 2p 2 n 2 n pn p n p 3n p n p 3 µ µ 2 Σ µ µ 2 n 2 n n pn p n p 3 µ µ 2 Σ Σ 2Σ 22 Σ 2Σ µ µ 2 2n 2 n n pn p n p 3 µ µ 2 Σ Σ 2Σ 22 µ µ 2 2 µ

41 2nn n p n p µ µ 2 p n 2 n pn p n p 3 µ 2 µ 2 n 2 Σ Σ 2Σ 22 µ 2 Σ 22 µ n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 n 2 n pn p 3 µ 2 µ 2 2p p 2 n 2 N N 2 N N 2 n pn p n p 3 2 µ 2 2 µ 2 2 Σ 22 Σ 2Σ Σ 2Σ 22 µ µ 2 p p 2 p 2 n 2 n N N 2 N N 2 n pn p 3n p n p n p 3 2p p 2 n 2 n N N 2 N N 2 n pn p n p 3n p n p n p 3 2p p 2 nn n N N 2 N N 2 n p n p 3n p n p p 2 p 2n 2 N N 2 n p 2 N N 2 n pn p n p 3n p n p 3 2p p 2 n 2 N N 2 N N 2 n pn p n p 3n p n p 3 2n 2 N N 2 n p N N 2 n pn p n p 3 2nn N N 2 N N 2 n p n p n 2 N N 2 N N 2 n pn p n p 3 n 2 N N 2 N N 2 n pn p 3 On the other hand, EV m2 Γ, Γ 2 is calculated as follows: Σ 22 Σ 2Σ Σ 2 Σ 22 Σ 2Σ Σ 2 2 { } Σ 22 Σ 2Σ 2 Σ 2 Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ Σ 2 EV m2 Γ, Γ 2 = n2 N N 2 N N 2 Γ Γ2 Σ Γ Γ2 Σ n 2 µ µ 2 Γ Γ2 Σ Γ Γ2 µ µ 2 2nn N N 2 N N 2 4 Γ 22 Σ 2Γ Γ2 Σ 2 C3

42 2nn µ µ 2 Γ Γ2 Σ 2 Γ 22 µ 2 µ 2 n2 N N 2 N N 2 n2 N N 2 N N 2 2 Γ 22 Σ 22 Γ 22 Σ 22 Γ 22 Σ 22 Γ 22 Σ 2Σ Σ 2 n 2 µ 2 µ 2 2 Γ 22 Σ 22 Γ 22 µ 2 µ 2 n2 N N 2 N N 2 n2 N N 2 N N 2 2 Γ 22 Σ 2Σ Σ 2Γ 22 Σ 22 Γ 22 Σ 2Σ Σ 2Γ 22 Σ 2Σ Σ 2 n 2 µ 2 µ 2 2 Γ 22 Σ 2Σ Σ 2Γ 22 µ 2 µ 2 2 By making use of Lemma 22 and noting that Γ Γ2 and Γ 22 are independent, EV m2 can be calculated easily as follows: EV m2 = n 2 n n p n p n p 3 µ µ 2 Σ µ µ 2 2nn n p n p µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 n 2 n p n pn p n p 3 µ 2 µ 2 n 2 2 Σ 22 µ n pn p n p 3 Σ 22 Σ 2Σ Σ 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 n 2 n pn p 3 µ 2 µ 2 2 Σ 22 Σ 2Σ Σ 2Σ 22 µ 2 µ 2 2n 2 N N 2 n p N N 2 n pn p n p 3 2nn N N 2 N N 2 n p n p n 2 N N 2 N N 2 n pn p n p 3 n 2 N N 2 N N 2 n pn p 3 p n 2 n N N 2 N N 2 n p n p n p µ 2 2 Σ 22 Σ 2Σ Σ 2 Σ 22 Σ 2Σ Σ 2 { } Σ 22 Σ 2Σ 2 Σ 2 Σ 22 Σ 2Σ Σ 2Σ 22 Σ 2Σ Σ 2

43 p 2 n 2 n p N N 2 N N 2 n pn p n p 3 C4 Now we have obtained EV m = EV m EV m2 by C3 and C4 Thus Theorem 34 has been completed Appendix D Proof of Theorem 36 First we derive the unbiased estimator of δ 2 Since d 2 m = µ µ 2 Σ µ µ 2 we find the expectation of it given Γ, Γ 2 : = nx T x2 T Γ Γ2 x T x2 T, E d 2 m Γ, Γ 2 = nn N 2 N N 2 Γ Γ2 Σ nµ µ 2 Γ Γ2 µ µ 2 Thus we can obtain the following expectation as follows by Lemma 22: E d 2 m = n δ 2 p N N 2 n p N N 2 Therefore we derive the following statistic as the unbiased estimator of δ 2 : δ 2 = n p d 2 m p N N 2 n N N 2 D Next we derive the unbiased estimator of 2 Since Dm 2 = µ µ 2 Σ µ µ 2 µ µ 2 Σ Σ 2 Σ Σ 22 2 Σ µ µ 2 2 µ µ 2 Σ Σ 2 Σ 22 µ 2 µ 2 2 µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 = nx T x2 T Γ Γ2 x T x2 T n x 2F x2 2F Γ 22 x 2F x2 2F 2n x F x2 F Γ Γ 2 Γ 22 x 2F x2 2F n x F x2 F Γ Γ 2 Γ 22 Γ 2 Γ x F x2 F, 43

44 the conditional expectation of D 2 m given Γ, Γ 2 can be obtained: E D m Γ 2, Γ 2 = nn N 2 N N 2 Γ Γ2 Σ nµ µ 2 Γ Γ2 µ µ 2 n N N 2 N N 2 n N N 2 N N 2 Γ 22 Σ 22 Γ 22 Σ 2Σ Σ 2 n µ 2 µ 2 2 Γ 22 µ 2 µ 2 2n N N 2 N N 2 2 Γ 22 Σ 2Γ Γ 2 2n µ µ 2 Γ Γ 2 Γ 22 µ 2 µ 2 n N N 2 N N 2 2 Γ 22 Γ 2 Γ Σ Γ Γ 2 n µ µ 2 Γ Γ 2 Γ 22 Γ 2 Γ µ µ 2 Therefore we find the conditional expectation of D 2 m given Γ, Γ 2, Γ2 by Lemma 22 as follows: E Dm Γ 2, Γ 2, Γ2 = nn N 2 N N 2 Γ Γ2 Σ nµ µ 2 Γ Γ2 µ µ 2 p 2n N N 2 N N 2 n p n N N 2 N N 2 n p n n p µ 2 µ 2 2n N N 2 N N 2 2n n p µ µ 2 n p Σ 22 Σ 2Σ Σ 2 2 Σ 22 µ 2 µ 2 2 Σ 22 Σ 2Γ Γ 2 Γ Γ 2 Σ 22 µ 2 µ 2 2 n N N 2 N N 2 n p Σ 22 Γ 2 Γ Σ Γ Γ 2 n n p µ µ 2 Γ Γ 2 Σ 22 Γ 2 Γ µ µ 2 44

45 Moreover we can calculate the conditional expectation of D 2 m given Γ, Γ2 by Lemma 32 as follows: E Dm Γ 2, Γ2 = nn N 2 N N 2 Γ Γ2 Σ nµ µ 2 Γ Γ2 µ µ 2 p 2n N N 2 N N 2 n p n n p µ 2 µ 2 2 Σ 22 µ 2 µ 2 2 2n n p µ µ 2 Σ Σ 2Σ 22 µ 2 µ 2 p 2n N N 2 N N 2 n p p 2n n p µ µ 2 n n p µ µ 2 Γ Σ Γ µ µ 2 2 Σ Σ 2Σ 22 Σ 2Σ µ By using the result of Lemma 22 again, we can obtain ED 2 m as follows: ED 2 m = n n p 2 n n p n n p δ 2 µ 2 p nn N 2 N N 2 n p p 2n N N 2 N N 2 n p p p 2 n N N 2 N N 2 n p n p D2 By the results of D and D2, Theorem 36 has been completed 45

46 Table The Comparisons of the Accuracy of Asymptotic Approximations for EPMC where p = 2 p = 2 p =, p 2 = N, N 2, N 2, N 2 2 Lachenbruch/Shutoh Fujikoshi and Seo Simulation 0, 0, 0, , 0, 20, , 0, 20, , 0, 40, , 20, 0, , 20, 20, , 20, 20, , 20, 40,

47 Table Continued p = 2 p =, p 2 = N, N 2, N 2, N 2 2 Lachenbruch/Shutoh Fujikoshi and Seo Simulation 0, 30, 0, , 30, 20, , 30, 20, , 30, 40, , 0, 0, , 0, 20, , 0, 20, , 0, 40,

48 Table Continued p = 2 p =, p 2 = N, N 2, N 2, N 2 2 Lachenbruch/Shutoh Fujikoshi and Seo Simulation 20, 20, 0, , 20, 20, , 20, 20, , 20, 40, , 40, 0, , 40, 20, , 40, 20, , 40, 40,