Two-sample inference for normal mean vectors based on monotone missing data

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Journa of Mutivariate Anaysis 97 (006 6 76 wwweseviercom/ocate/jmva Two-sampe inference for norma mean vectors based on monotone missing data Jianqi Yu a, K Krishnamoorthy a,, Maruthy K Pannaa b a

1 Journa of Mutivariate Anaysis 97 ( wwweseviercom/ocate/jmva Two-sampe inference for norma mean vectors based on monotone missing data Jianqi Yu a, K Krishnamoorthy a,, Maruthy K Pannaa b a Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA b Actuaria Department, United Guarantee Corporation, Greensboro, NC 740, USA Received 5 January 005 Avaiabe onine 8 August 006 Abstract Inferentia procedures for the difference between two mutivariate norma mean vectors based on incompete data matrices with different monotone patterns are deveoped Assuming that the popuation covariance matrices are equa, a pivota quantity, simiar to the Hoteing T statistic, is proposed, its approximate distribution is derived Hypothesis testing confidence estimation of the difference between the mean vectors based on the approximate distribution are outined The vaidity of the approximation is investigated using Monte Caro simuation Monte Caro studies indicate that the approximate method is very satisfactory even for sma sampes A mutipe comparison procedure is outined the proposed methods are iustrated using an exampe 006 Esevier Inc A rights reserved AMS 99 subject cassification: 6H; 6H5 Keywords: Coverage probabiity; Incompete data; Maximum ikeihood estimators; Moment approximation; Powers; Sizes Introduction The probem of incompete data arises commony in many practica situations, especiay in pubic survey Missing data arises, for exampe, during data gathering recording, when the experiment is invoved a group of individuas over a period of time ike in cinica trias or in a panned experiment where the variabes that are expensive to measure are coected ony from a subset of a sampe The causes for missing data are not our concern but to ignore the process that causes missing data it is assumed that the data are missing at rom (MAR Recenty, Lu Corresponding author Fax: E-mai address: krishna@ouisianaedu (K Krishnamoorthy X/$ - see front matter 006 Esevier Inc A rights reserved doi:006/jjmva

2 J Yu et a / Journa of Mutivariate Anaysis 97 ( Copas [] pointed out that inference from the ikeihood method ignoring the missing data mechanism is vaid if ony if the missing data mechanism is MAR For forma definition exposition of MAR or missing competey at rom we refer to Litte Rubin [9] or Litte [8] There are a few missing patterns considered in the iterature, but the incompete data with monotone pattern (see dispay not ony occurs frequenty in practice but aso it is convenient for making inference In particuar, if mutivariate normaity is assumed then the monotone pattern aows the exact cacuation of the maximum ikeihood estimators (MLEs, the ikeihood ratio statistics reevant distributions Severa authors have considered the monotone missing pattern under normaity assumption, provided asymptotic as we as approximate test procedures about the norma mean vector Anderson [], one of the eariest papers in this area, gives a simpe approach to derive the MLEs present them for a specia case of monotone pattern some other patterns Ka Fujkoshi [4] studied some basic properties of the MLEs based on monotone data Many authors deveoped asymptotic inferentia procedures based on the ikeihood ratio approach for mutivariate norma distribution We note, among many other papers, Bhargava [], Morrison Bhoj [] Naik [4] Many of these papers considered primariy hypothesis testing probem, ony recenty Krishnamoorthy Pannaa [6] provided an accurate simpe approach to construct confidence region for a norma mean vector In this artice, we consider the probems of hypothesis testing confidence estimation of the difference between two norma mean vectors based on sampe data matrices that are of monotone pattern Our approach is essentiay based on the ones given in Krishnamoorthy Pannaa [5,6] Hao Krishnamoorthy [] Specificay, we deveop a pivota quantity based on the MLEs (simiar to the Hoteing T statistic for the one sampe case, derive its approximate distribution to make inferentia procedures To formuate the probem, et x foow a p-variate norma distribution with mean vector μ covariance matrix Σ We write this as x N p (μ, Σ Let y N p (β, Σ independenty of x Suppose that we have a sampe of N observations avaiabe on x, a sampe of M observations avaiabe on x Assume that the sampes have the foowing monotone pattern: x,,x Nk,,x N,,x N, y,,y Mk,,y M,,y M, x,,x Nk,,x N, y,,y Mk,,y M, x k,,x knk, y k,,y kmk, ( where x ij is a p i vector, j =,,N i, whie y ij is a q i vector, j =,,M i, i =,,k In other words, in the x-sampe, there are N observations avaiabe on the first p components, N observations avaiabe on the first p + p components, so on Notice that N N N k, M M M k, p + +p k = q + +q k = p Krishnamoorthy Pannaa [6] considered the one-sampe case, provided approximate methods for constructing confidence region hypothesis testing for the mean vector Using their idea, we deveop inferentia procedures to the present two-sampe probem However, unike the compete data case, extending the soution of the one-sampe probem to the two-sampe case is not easy Indeed, the probem is much more compex than the one-sampe probem, so methods for the two-sampe case are reay warranted for easy reference In the foowing section, we present first some preiminaries in the notations of Krishnamoorthy Pannaa [5] for the data matrices in ( with k = p i = q i, i =,, We present the MLEs of the reevant parameters in terms of these notations Using these MLEs, we propose a

3 64 J Yu et a / Journa of Mutivariate Anaysis 97 ( pivota statistic simiar to the Hoteing T statistic, derive an approximation to its distribution We outine procedures for hypothesis testing constructing confidence region for μ β based on the approximate distribution Required resuts for approximating the nu distribution of the pivota quantity are aso given for the genera case We describe a method of constructing simutaneous confidence intervas for the components of μ β We aso point out the resuts for equa monotone pattern (that is, p i = q i, i =,,k can be extended to the case of unequa monotone patterns (ie, in (, p i = q i for some i The accuracies of the approximation are appraised by Monte Caro simuation in Section Simuation studies show that the approximation is very satisfactory even for sma sampes Our imited power studies in Section 4 indicate that the proposed test has some natura power properties The methods are iustrated using an exampe in Section 5, some concuding remarks are given in Section 6 Inference on µ β To deveop inferentia procedures about μ β, we first need to obtain the MLEs of the parameters μ, β the common covariance matrix Σ In the foowing section, we present some preiminaries in the notations of Krishnamoorthy Pannaa [5], present the MLEs of the reevant parameters for the sampes of two-bock monotone pattern (that is k = in ( The MLEs can be easiy expressed for the genera case The maximum ikeihood estimators Consider the data matrices in ( with k = assume that p i = q i = r i, i =,,k, partition the data matrices as foows: x = ( x,,x N,,x N,,x N, r N ( x,,x x = N x,,x N (r +r N, x,,x N x = x,,x N ( x,,x N (r+r+r N That is, x is the submatrix of x in ( formed by the first N coumns the first p + +p rows, =,, Partition the matrix y simiary That is, y = ( y,,y M,,y M,,y M, r M ( y,,y y = M y,,y M (r +r M, y,,y M y = y,,y M ( y,,y M (r+r+r M Let x S denote, respectivey, the sampe mean vector the sums of squares products matrix based on x, =,, Simiary, et ȳ V denote, respectivey, the sampe mean

4 S = S (,, S = J Yu et a / Journa of Mutivariate Anaysis 97 ( vector the sums of squares products matrix based on y, =,, We partition these means matrices accordingy as foowing: ( x = x (, x x ( x ( = : r : r x (, x = x ( : r : r, x ( : r ( S (, : r r S (, : r r S (, : r r S (, : r r S = S (, : r r S (, : r r S (, : r r S (, : r r S (, : r r S (, S (, : r r S (, : r r S (, : r r : r r (4 Notice that x (i : r i is the mean of the ith bock of the data matrix x, i =,, =,, We aso read S (i,j : r i r j as the (i, jth submatrix of S based on the data matrix x, =,, The statistics ȳ V based on the data matrix y in ( are aso partitioned ike x S That : r i is the mean of the ith bock of data matrix y, i =,, =,,, V (i,j : r i r j is the (i, jth submatrix of V, i, j =,, =,, Finay, we partition the parameters as foows: is, ȳ (i μ : r μ = μ : r μ : r p β : r, β = β : r β : r p Σ : r r Σ : r r Σ : r r Σ = Σ : r r Σ : r r Σ : r r Σ : r r Σ : r r Σ : r r p p It shoud be noted that the way the data matrices summary statistics are partitioned is different from the one given in Krishnamoorthy Pannaa [6] for the one-sampe case We found the MLEs can be expressed in simpe forms in terms of the above partitioned sampe mean vectors, S V We now give the MLEs of the partitioned mean vectors sub-matrices of Σ Let B = (S (, + V (, (S (, + V (,, (B,B = (S (, + V (,,S (, + V (, Σ = ( S (, + V (,,S (, + V (, S (, + V (,,S (, + V (,, N + M [(S (, + V (, (S (, +V (, (S (, +V (, (S (, +V (, ]

5 66 J Yu et a / Journa of Mutivariate Anaysis 97 ( Σ = (S (, + V (, (S (, + V (,,S (, + V (, N + M ( (, S + V (, S (, + V (, S (, + V (, S (, + V (, The MLEs are given by ( (, S + V (, S (, + V (, μ = x, μ = x ( B (x ( μ, μ = x ( B (x ( μ B (x ( μ, β = y, β = y ( B (y ( β, β = y ( B (y ( β B ((y ( β, Σ = (S (, + V (, /(N + M, Σ = B Σ, Σ = Σ + B Σ, Σ =B Σ +B Σ, Σ =B Σ +B Σ Σ = Σ +B Σ +B Σ Let μ = ( μ, μ, μ β = ( β, β, β, et Ω denote the covariance matrix of ( μ β Then, the MLE of Ω is given by Σ /W Σ /W Σ /W Σ /W Σ /W W W Ω = W W B Σ Σ /W W W W W B Σ Σ /W Σ /W W W W W B Σ Σ /W W W W W (B Σ, (5 where +B Σ + W W W W B Σ B W = N + M, W = N + M W = N + M Hypothesis test confidence region for μ β The pivota quantity that we consider to construct confidence region for μ β or to test about μ β is given by Q =[( μ β (μ β] [ Ω] [( μ β (μ β] where = Q + Q + Q, Q = W ( μ β (μ β [ Σ ] ( μ β (μ β, Q = W [( μ β (μ β B (μ β ] [ Σ ] [( μ β (μ β B (μ β ], Q = W [( μ β (μ β B (μ β B (μ β ] [ Σ ] [( μ β (μ β B (μ β B (μ β ], (6

6 μ = μ B μ, β = β B β, J Yu et a / Journa of Mutivariate Anaysis 97 ( μ = μ B μ B μ β = β B β B β The expression for Q ceary suggests that it is difficut to derive the exact distribution of Q Because Q is resembing the Hoteing-T statistic, its distribution is free of any parameters (see Appendix A, it is reasonabe to approximate its distribution by the distribution of df p,ν, where d is a positive constant, F a,b denotes the F rom variabe with numerator degrees of freedom a the denominator degrees of freedom b The unknown constants d ν can be determined so that the first two moments of Q are equa to those of df p,ν In the appendix, foowing the ines of Krishnamoorthy Pannaa [6], we evauated an exact expression G for E(Q an approximation G for E(Q Using these G G (see Appendix A, we see that Q df p,ν approximatey, where ν = 4pG (p + G pg (p + G ν d = G ν (7 We again note that d ν were determined so that E(Q = E(dF p,ν E(Q = E[(dF p,ν ] Letting δ = μ β, an approximate α confidence set for μ β is the set of vaues of δ that satisfy [( μ β δ] [ Ω] [( μ β δ] df p,ν ( α, (8 where F p,ν ( α is the ( αth quantie of the F p,ν distribution An approximate α-eve test rejects the nu hypothesis H 0 : μ = β when ( μ β [ Ω] ( μ β >df p,ν ( α The resuts for a genera monotone patterns are given in Appendix B In particuar, we write Q =[( μ β (μ β] [ Ω] [( μ β (μ β] = Q + +Q k give expression G for E(Q an approximation G for E(Q Simutaneous confidence intervas Approximate simutaneous confidence intervas for δ i = μ i β i, i =,,p, can be constructed using Scheffé s S-method Towards this, we note that the inequaity in (8 hods if ony if [a (ˆδ δ] a Therefore, we have P Ωa df p,ν for a a R p (a ˆδ cα a Ωa a δ a ˆδ + cα a Ωa for a a α,

7 68 J Yu et a / Journa of Mutivariate Anaysis 97 ( where ˆδ = μ β c α = df p,ν ( α It foows from the above equation that ˆδ i c α ˆωii δ i ˆδ i + c α ˆωii for i =,,p, (9 where ˆω ii is the (i, ith eement of Ω in (5, with probabiity at east α 4 The case of unequa monotone patterns We sha now consider the data matrices in ( with different monotone pattern For convenience simpicity, et us assume that k =, without oss of generaity, p >q That is, we have the foowing data matrices (x,,x L,,x L p L, (y,,y T,,y T q T, (x,,x L p L, (y,,y T q T (0 Now, et x denote the data matrix formed by the first q rows of (x,,x L,,x L p L, x denote the data matrix (x,,x L,,x L p L ( x,,x x = L x,,x L (p +p L Notice that q = (p q + p because p + p = q + q = p Let y = (y,,y T,, y T q T, y denote the data matrix formed by (y,,y T q T as the first bock of rows (p q rows of (y,,y T q T as second bock of rows ( y,,y y = T y,,y T (q +q T Thus, we can partition the data matrices of unequa monotone patterns to make equa monotone pattern Specificay, we have x : q L, x : p L, x : (p + p L, y : q T, y : p T, y : (q +q T So, by setting (r,r,r = (q,p q,p, (N,N,N = (L,L,L (M,M,M = (T,T,T, we can appy the method for equa monotone pattern case to the present unequa monotone patterns case We aso note that any type of unequa monotone patterns data can be rearranged to form equa monotone pattern Vaidity of the approximation To appraise the accuracy of the F approximation to the distribution of the pivota quantity Q in (6, we estimated the coverage probabiities of the 95% confidence region based on (8 for various sampe size configurations using Monte Caro simuation Each simuation resut is based on 00,000 runs The mutivariate norma rom vectors were generated using IMSL subroutine RNMVN In Tabe a, we present the estimated coverage probabiities for the equa monotone pattern with (r,r,r = (,,, various vaues of (N,N,N,M,M,M The estimated coverage probabiities are given in Tabe b for -bock unequa monotone patterns with (p,p = (, (q,q = (,, in Tabe c for (p,p = (,

8 J Yu et a / Journa of Mutivariate Anaysis 97 ( Tabe Critica vaues df p,ν (095 Monte Caro estimates of the coverage probabiities of the 95% confidence region in (8 (M,M,M (N,N,N (, 6, 6 (, 5, 7 (9, 6, 6 (r,r,r = (,, (,6,6 70(950 49(95 0(948 (,,6 566(95 89(95 080(948 (9,, 57(949 48(950 09(948 (,, 75(948 57( (949 (6,,7 45(950 09(95 06(947 (8,4,7 4(95 00(95 05(948 (,5,7 49(950 95(95 048(948 (,, 65(95 0( (950 (,,8 048( ( (948 (40,0,0 75(95 5(95 98(949 (60,40,0 99(950 96( (947 (N,N (M,M (0, 5 (0, 0 (0, 0 (0, 5 (5, 0 (, 0 (p,p = (,, (q,q = (, (5,0 07(948 6(949 98(949 07(948 (949 8(948 (5,9 045(949 5( (948 06(948 4(948 08(949 (5,7 067(949 85( ( (949 9(949 84(950 (8,5 99( (948 95( ( ( (949 (8, 00(948 07( (949 00( (949 9(946 (8,9 04(948 9( (949 06(948 5(948 87(949 (5,0 95(948 98(948 96( ( ( (948 (5,5 97(949 04( ( (948 08( (947 (5,7 07(950 46( (950 0(949 50(949 5(950 (N,N (M,M (0, 5 (0, 0 (0, 7 (0, 5 (0, 0 (0, 7 (p,p = (,, (q,q = (, (0,0 76(948 (948 4(948 65(949 0(950 (949 (0,5 0(948 58(948 0(950 9(949 47(950 99(950 (0,0 49(949 49( (95 8(950 9( (95 (8,5 5(946 8(947 8(947 (947 99(948 6(949 (8, 77(947 6( (947 57(947 44(950 46(948 (8, 9 0(947 46(949 59(95 9(948 48( (950 (40,0 5(948 85(949 (948 45(948 78(949 06(950 (40,5 80(949 (949 8(949 7(947 4(949 75(949 (40,0 7(950 5(949 44(95 9(949 8(95 46(95 (q,q = (, We observe from a these three tabe vaues that the estimated coverage probabiities are very cose to the nomina eve 095 for a the cases considered Even for sma sampes, our approximate procedure is very accurate (for exampe, see (N,N,N = (M,M,M = (, 6, 6 in Tabe a, (N,N,M,M = (5, 0,, 8 in Tabe b We

9 70 J Yu et a / Journa of Mutivariate Anaysis 97 ( (a (b Fig Monte Caro estimates of the powers of the test: (a (N,N,N = (, 8, 6; (M,M,M = (5,, 7; (b (N,N,N = (0, 8,, (M,M,M = (9, 6, 0 aso estimated the coverage probabiities of the confidence region in (8 at confidence eve 099 for sampe sizes given in Tabes Because the resuts are simiar to the case of 95% confidence eve, they are not reported here 4 Power studies To underst the nature of the power function of the test in Section, we estimated the powers via Monte Caro simuation consisting of 00,000 runs The powers are estimated as a function of δ = (μ β Σ (μ β For fixed sampe sizes, the powers are estimated using Σ = I p μ β = δ, where denotes the vector of ones We observe from power pots in Fig that for fixed sampe sizes, the power is an increasing function of δ Aso, for fixed δ, the power is an increasing function of sampe sizes because the power curve for (N,N,N = (0, 8,, (M,M,M = (9, 6, 0 fa above the power curve for (N,N,N = (, 8, 6, (M,M,M = (5,, 7 The power studies for other sampe sizes parameter configurations exhibited simiar properties so they are not reported here Thus, our proposed test possesses some natura power properties 5 An iustrative exampe We sha now iustrate the methods using the Fisher s Iris Data which represent measurements of the sepa ength width, peda ength width in centimeters of fifty pants for each of three types of iris; Iris setosa, Iris versicoor Iris virginica The data sets are posted in many websites, we downoaded them from For iustration purpose, we use the data on virginica (x versicoor (y Aso, we use ony sepa ength, width peda ength as three components We appied the modified ikeihood ratio test (eg, Muirhead [, p 09] to check the equaity of covariance matrices The test produced a p-vaue of 04, so the assumption of equaity of covariance matrices is tenabe We created monotone patterns by discarding the ast 0 measurements on x (sepa ength of virginica, the ast 0 measurements on x (peda ength of virginica, the ast 8 measurements on

10 J Yu et a / Journa of Mutivariate Anaysis 97 ( y (sepa ength of versicoor, the ast 5 measurements on y (peda ength of versicoor That is, we have (N,N,N = (50, 40, 0 (M,M,M = (50, 4, 5 Let μ = (μ, μ, μ = (average sepa ength, average sepa width, average peda ength of virginica, β = (β, β, β = (average sepa ength, average sepa width, average peda ength of versicoor We want to test H 0 : μ β = d 0 vs H 0 : μ β = d 0, ( where d 0 = (04, 00, construct simutaneous confidence intervas for μ β, μ β μ β We present the resuts for three different cases: (i compete data sets containing 50 observations from each group; (ii incompete monotone pattern data, (iii partiay compete data (that is, a vector observation is discarded if any of its components are missing; in the present case, N = N = N = 0 M = M = M = 5 5 Resuts based on compete data x ȳ = (06500, 00400, 900, [ Cov( x ȳ] = Cov( x ȳ = , ( where Cov( x ȳ = (/N +/M S p S p is the pooed covariance matrix so that E(S p = Σ The Hoteing T -statistic is computed as T = ( x ȳ d 0 [ Cov( x ȳ] ( x ȳ d 0 = 080 Noticing that T statistic is distributed as (N +M p N +N p F p,n +N p, the p-vaue for testing ( is given by ( ( (N + M p 94 P N + M p F p,n +M p >T = P 96 F,96 > 080 = 0077 To get simutaneous confidence intervas, we computed c = F,96(095 = 8669 Using Scheffé s method, x i ȳ i ± (N + M p N + M p F p,n +M p ( αv ii, i =,,, where v ii is the (i, ith eement of Cov( x ȳ given in (, we computed 95% simutaneous confidence intervas for μ β, μ β μ β as respectivey 065 ± 0, 004 ± 08 9 ± 095,

11 7 J Yu et a / Journa of Mutivariate Anaysis 97 ( Resuts based on incompete data As pointed out earier in the section, here we consider the monotone data with (N,N,N = (50, 40, 0 (M,M,M = (50, 4, 5 The MLE of μ β is given by (ˆμ ˆβ = (06500, 0869, 580 the estimate ˆΩ of the covariance matrix of ˆμ ˆβ in (5 is computed as ˆΩ = ˆΩ = The vaue of the statistic Q in (6 is given by The required vaues to compute the critica vaue are G = E(Q = 876, G = E(Q = 8749, d = 4 ν = The critica vaue df p,ν (095 = 8680 The p-vaue for testing ( is given by P ( df p,ν >Q = P ( 4F,07985 > = 0045 The 95% simutaneous confidence intervas for μ β, μ β μ β based on (9 are respectivey 065 ± 07, 087 ± ± 06, 5 The resuts based on partiay compete data As we aready mentioned, here we form compete data sets by dropping vector observations with missing components In this case, N = N = N = 0 M = M = M = 5 Using these compete vector observations, we found x ȳ = (057, 057, 9, [ Cov( x ȳ] = Cov( x ȳ = ( The p-vaue other critica vaues can be computed using the formuas in Section 4 with N = 0 M = 5 The Hoteing T statistic is computed as 489 with p-vaue 0578 The 95% simutaneous confidence intervas for μ β, μ β μ β are given by 057 ± 050, 057 ± ± 0440, respectivey We observe from the above resuts that the concusions of the tests based on compete data (Section 5 on incompete data are the same Aso, as expected, the simutaneous confidence

12 J Yu et a / Journa of Mutivariate Anaysis 97 ( intervas based on incompete data are wider than the corresponding ones based on compete data, shorter than those based on partiay compete data 6 Concuding remarks In this artice, we proposed a Hotteing T type test for testing the equaity of two norma mean vectors when the covariance matrices are equa The test is simpe to use the monotone patterns of the sampes are not necessariy simiar We aso note that in many practica situations the covariance matrices need not be equa It is pausibe that we can extend the present approach for the case of unequa covariance matrices aong the ines Krishnamoorthy Yu [7] who gave a simpe test procedures when there is no missing data We are currenty working on this mutivariate Behrens Fisher probem with missing data As pointed out by a reviewer, the setup for the present probem is a specia case of the setup for a mutivariate inear regression In particuar, they are specia cases of the modes considered in Liu [0] who provides the MLEs for the parameters in mutivariate inear regression mode with missing data Even though the MLEs are readiy avaiabe, it is not straightforward to get the moment approximation for the distribution of the pivota quantity of the form in (6 We pan to investigate the appicabiity of our approach to this genera setup, pubish the resuts esewhere Acknowedgment The authors are gratefu to a reviewer for providing vauabe comments suggestions Appendix A We here evauate the first two moments of Q in (6 Define Q d = W [( x ( ȳ ( (μ β ] [S (, + V (, ] [( x ( ȳ ( (μ β ], Q d = W ([( x ( R = ȳ ( (μ β ], [( x ( ( S (, + V (, S (, + V (, S (, + V (, S (, + V (, Q ( + Q d R = Q ( + Q d ȳ ( (μ β ] (μ β ( x ( ȳ ( (μ, β ( ( x ( ȳ ( The foowing resuts can be easiy deduced from the resut in Seber [5, p 5] The variabes Q d Q d are independent with Q d Q d r N + M r F r,n +M r r + r N + M (r + r F (r +r,n +M (r +r (A

13 74 J Yu et a / Journa of Mutivariate Anaysis 97 ( Aso, Q in (6, R R are independent with Q (N + M r N + M r F r,n +M r R (N + M r N + M (r + r F r,n +M (r +r, (A R (N + M r N + M p F r,n +M p Furthermore, Q d Q d are distributed independenty of R R However, Q (Q d,q d are not independent Notice that the pivota quantity Q in (6 can be written as Q = Q + Q + Q = Q + R ( + Q d + R ( + Q d hence, it foows from (A (A that, the distribution of Q is free of any parameters We sha now evauate the first moment an approximation to the second moment of Q = Q + Q + Q Using the above distributiona resuts, we have E(Q = (N + M r N + M r, E(Q = E(R ( + Q d = E(R E( + Q d (N + M (r (N + M = (N + M r (N + M r r, E(Q = E(R ( + Q d = E(R E( + Q d (N + M r (N + M = (N + M r r (N + M p The second moments of Q i s are given by E(Q = (N + M r (r + (N + M r (N + M r 5, E(Q (N + M (N + M (N + M 5(r (r + = (N +M r (N +M r r (N +M r 5(N +M r r 5, E(Q (N + M (N + M (N + M 5r (r + = (N +M r r (N +M p (N +M r r 5(N +M p 5 Using the arguments of Krishnamoorthy Pannaa [6], it can be shown that E(Q Q E(Q E(Q E(Q Q E(Q E(Q Thus, we have E(Q = E(Q + E(Q + E(Q = G

14 J Yu et a / Journa of Mutivariate Anaysis 97 ( E(Q E(Q + E(Q + E(Q + E(Q E(Q + E(Q E(Q +E(Q E(Q = G Appendix B Generaization Assume that the sampes have the monotone pattern in ( In this genera case, the quadratic form Q in (6 can be expressed as where Q =[( μ β (μ β] [ Ω] [( μ β (μ β] = Q + Q + +Q k, Q = W ( μ β (μ β [ Σ ] ( μ β (μ β,, for =,,k, Q = W ( μ β μ β B j (μ j β j [( μ β ], j= [ Σ ] with ( (B,,B = S (, + V (,,,S (, + V (, S (, + V (, S (, + V (, S (, + V (, S (, Σ = (S (, + V (, (B,,B N + M + V (, S (, S (,, + V (,, + V (, μ = X B j X j, β = Y B j Y j j= Furthermore, for =,,k,wehave E(Q = j= (N + M p (N + M (N + M p ( (N + M p (

15 76 J Yu et a / Journa of Mutivariate Anaysis 97 ( E(Q (N + M (N + M (N + M 5(p (p + = (N +M p (N +M p ( (N + M p 5(N + M p ( 5, where p ( = i= p i p (0 = 0Aso, using the approximation that E(R Q s E(Q E(Q s for = s, we can get G = E(Q an approximation G for E(Q Thus, Q df p,ν approximatey, where ν = 4pG (p + G pg (p + G ν d = G ν References [] TW Anderson, Maximum ikeihood estimates for a mutivariate norma distribution when some observations are missing, J Amer Statist Assoc 5 ( [] RP Bhargava, Mutivariate tests of hypotheses with incompete data, PhD Dissertation, Stanford University, Stanford, CA, 96 [] J Hao, K Krishnamoorthy, Inferences on norma covariance matrix generaized variance with incompete data, J Mutivariate Ana 78 ( [4] T Ka, Y Fujikoshi, Some basic properties of the MLEs for a mutivariate norma distribution with monotone missing data, J Math Manage Sci 8 ( [5] K Krishnamoorthy, M Pannaa, Some simpe test procedures for norma mean vector with incompete data, Ann Inst Statist Math 50 ( [6] K Krishnamoorthy, M Pannaa, Confidence estimation of norma mean vector with incompete data, Canad J Statist 7 ( [7] K Krishnamoorthy, JYu, Modified Ne Van der Merwe test for the mutivariate Behrens Fisher probem, Statist Probab Lett 66 ( [8] RJA Litte, A test of missing competey at rom for mutivariate data with missing vaues, J Amer Statist Assoc 8 ( [9] RJA Litte, DB Rubin, Statistica Anaysis with Missing Data, Wiey, New York, 987 [0] C Liu, Bayesian robust mutivariate inear regression with incompete data, J Amer Statist Assoc 9 ( [] GB Lu, JB Copas, Missing at rom, ikeihood ignorabiity mode competeness, Ann Statist ( [] DF Morrison, D Bhoj, Power of the ikeihood ratio test on the mean vector of the mutivariate norma distribution with missing observations, Biometrika 60 ( [] RJ Muirhead, Aspects of Mutivariate Statistica Theory, Wiey, New York, 98 [4] UD Naik, On testing equaity of means of correated variabes with incompete data, Biometrika 6 ( [5] GAF Seber, Mutivariate Observations, Wiey, New York, 984

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(This is a sample cover image for this issue. The actual cover is not yet available at this time.) (This is a sampe cover image for this issue The actua cover is not yet avaiabe at this time) This artice appeared in a journa pubished by Esevier The attached copy is furnished to the author for interna