UNIVERSITY OF NORTH CAROLINA Department of Statistics Chapel Hill, N. C. FOR MULTIVARIATE POPULATIONS. J. N. Srivastava.

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1 UNIVERSITY OF NORTH CAROLINA Departmet of Statistics Chapel Hill N. C. A NOTE ON THE BEST LINEAR UNBIASED ESTIMATES FOR MULTIVARIATE POPULATIONS by J. N. Srivastava November 1962 Cotract No. AF 49(638)-213 I this ote we preset a lemma o the best liear ubiased estimates for multivariate populatios. This research was supported by the Air Force Office o~ Research. Scieti~ic Istitute of Statistics Mimeo Series No. 339

2 A NOTE ON THE BEST LINEAR UNBIASED ESTIMATES FOR MULTIVARIATE POPULATIONS 1 by J. N. Srivastava Uiversity of North Carolia = == ;:::: := ::::;: = = = == = = = = = = = = = ; ; = = = == = = = = = = = Cosider the usual multivariate liear model (1) Exp(Y) = A ~ xp ma J.:h.'1l where > m ad where f:ll' Y1' Y2' J - Y12'.. YIp \ (2) Y = (Xl'!..2' Y ) =... ) f~(l)i --p = ' I say lie)j L YpJ is a matrix of p opservatiosj A is a kow matrix ad (3 ) r." f:ll S12 ~;lpl '::"... = \ = (~l'.2' 1 p ) say LSml sm2.. r \ '''mp~ is a ratrix of ukow parameters. We further assure that the vectors II( )~) (r = 1 2 p) are all ucorrelated ad that for r = 1 2 : (4) Var (X (r--) ) say where the dispersio matrix Z is also uko\v. The model (1) for the j-th variable reduces to IThis research was supported by the Air Force Office of Scietific Research.

3 (5) Exp (y. ) == AF.'J.2.j Var (Y jr ) == (J j == p r == JJ If we cosider just the j-th variable ad igore the rest we ca obtai from A (5) the best liear ubiased estimate c! S. of c! ~. where c. is ay -J -J -J -J "'J mxl vector such that ~j ~j is estimable. Let (6) P Q == L: j==l c' f:!.._j-j be a liear fuctio of all the mp ukow parameters such that for each j c I r- is estirable. Let _oj 2.j p " (7) u == L: c! 5.j j==l -J The we show that u is the best liear ubiased estimate of Q. Lemma: Let z == b l ' Y l + + b l Y - - -p-p be ay other liear ubiased estimate of Q. The provided that the space of the mpxl vector. S ) "mp cotais at least mp liearly idepedet poits we must have Var (z) > Var (u) wha.tever the populatio dispersio matrix L: may be. (Notice that o assumptio of ormality is ivolved.) Proof: Suppose (8)... + d' y -p -p Sice Exp (z) == E(u) == Q

4 we have or Exp (z - u) = = for all 1' S2' S This however implies - - -p t (b! - d -J - j ) A = 0lm' j = 1 2 P where OlI is a 1 x m matrix. servatios Y we have Also sice b. ad d. are free of the ob- -J -J (10) Var (u) = Var (~l."l + + d t Y ) -p -p P = Z (~j ~j) j=l CI' JJ + ( d! I d.) CI'j J' r -J -J ad similarly Var (z) = P I: (b j ' b. )O"j. - -J J j=l + I: (b j t t b.) CI' t j1j t - -J JJ Let A be of rak r ad let W be the vector space of rak -r which is orthogoal to the colums of A. Let ~l'.~2j1 ~-r be a orthogoal basis of 'W. The fram (9) there exist costats ~jl' ~'2' ~. J J-r (j = I 2 p) such that (11) b. = d. + ~'l Q l + IJ. _Q --J -J J IJ.. Q J- J-r --r j = 1 2 P Let vi be the vector space of rak r (orthogoal to 'W) geerated by the colums of A. The sice c! S. is estimable as a uivariate problem for the -J'-J j-th variable it follows that A Rak (A) = Rak ( t)' j = P ~j

5 4 ad hece that i j W for all j Hece we have from (11) b! b. = -J -J d j ' d. - -J + 2 IJ.j_r b! b. = d! d + lj.'l lj.'l + + IJ.. r fl.. r -J -J '-J -J J J J - J- Therefore we get Ver (z) - Var (u) + -r ~ ( ~ " " ) j~j' s=l t-"j' S t-"j" s 0" r JJ -r p 2 IJ.js 0" + ~ IJ.. IJ.j's O"jj' _7 s=l j=l JJ j~j' JS = z f ~ = ~ r IJ.' Z - -s -s- IJ. 7 where -s IJ.' = (1J.1S ' 1J.2s'. IJ. ps ) s=l But sice Z is positive defiite IJ.' Z IJ. > 0 uless IJ. = 01 (zero vector) -S -8 -s p Sice however z is differet from u we must have H s ~ 0lp' for some s. Hece Var (z) > Var (u) which proves the leimllb. The above lea opes the door for may ew lies of work i multivariate liear estimatio. It has may iterestig corrolaries of which a very obvious oe is the follo1g: Let xl' x ' x be idepedet radom variables such that 2 Exp ad Var j =

6 2 where a j are ukow ad are ot ecessarily equal. The if ai' a 2 lj a are ay real umbers the best liear ubiased estimate of ~ a. Q. is j=l J J ~ a YJ.. 1 j J= 5 It is outside the scope of the preset ote to go ito the details of applicatios of the above leimlla. However by way of illustratio two examples may be illumiatig. A detailed paper will follow later. Example 1. Cosider a 2 factorial experimet with r'repetitios of each treatmet combiatio. Also assume o blocks to be preset. Let (gl' g2' g)' g. = 0 or 1 j = 1 2 represet a treatmet combiatio J.. Q(gl' g2' g ) its 'true' effect ad y. (gl' g2' g ) the correspodig p.. ~ i-th (i = 1 2 r) observed value. Further suppose that all the 2 r observatios are idepedet ad that i = 1 2 ~ r depeds o gl' g2' g. Notice that this assumptio is cotrary to the usual oe 1-There the variaces are assumed to stay costat. However the applicatio of the leimlla (i fact merely the corrollary) shows that if i l i 2 i k are ay k (0 ~ k ~ ) factors the the estimate of this k-factor iteractio say Ai Ai Ai. 12K is of Q (gl' g2' Here is the s~e cotrast of y (gl' g2' g)' asa. A. Ai ~l ~2 l\. r ~. 1 ~= y. (gl' g2' ~ g ) I other words the best liear ubiased estimate of ay iteractio is ualtered by the assumptio of uequal variaces. Example 2. As aother area of applicatio cosider a experimet with say v treatmets repeated at differet poits of time. At a;}' fixed poit of time t the observatios are supposed to be idepedet ad have a variace

7 6 (depedet o t). Such a situatio is obtaied whe for e~~ple the treatmets are the differet ratios for pigs ad we are studyig their growth. Suppose T ad T' are ay two treatmets ad the desig is coected ad T t T t are their true effects. The if we wat to estimate somethig like where R is the total set of time poits the above lemma says that we could 1\ 1\ proceed by first obtaiig the best liear ubiased estimate (T - t T t ) for each t R ad the the best liear ubiased of Q is give by I am tha.'kful to Professor S. N. Roy for goig through this ote ad for his commets. f'!:7 REFERENCES Bose R. C. "Notes o Liear Estimatio" Upublished Class Notes UiverGity of North Carolia~ Chapel IIill N. C. (1958). f~7 Hery Scheffe: The Aalysis of Variace Joh Wiley ad Sos L"J..7 Roy S. N. "Some Aspects of Multivariate Aalysis" Joh Wiley ad Sos 1957.