NK? D T IC ELECTE. Prepared Under Contract ESTIMATION OF VARIANCE COMPONENTS ,IROVED IN MIXED MODELS. ALAN E. GELFAND and DIPAK K.

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1 NK?,IROVED ESIMAION OF VARIANCE COMPONENS IN MIXED MODELS BY ALAN E. GELFAND and DIPAK K. DEY ECHNICAL REPOR NO. 434 AUGUS 30, 1990 D IC ELECE Ppad Und Contact N J-167 (NR-04-67) Fo th Offic of Naval Rsach Hbt Solomon, Pojct Dicto Rpoduction in Whol o in Pat is Pmittd fo any pupos of th Unitd Stats Govnmnt Appovd fo public las; distibution unlimitd. DEPARMEN OF SAISICS SANFORD UNIVERSIY SANFORD, CALIFORNIA

2 ~V ABSRAC aking Albt's (1976) fomulation of a mixd modl MA,, w consid impovd stimation of th vaianc componnts fo balancd dsigns und squad o loss. wo appoachs a psntd. On xtnds th idas of Stin (1964). h oth is dvlopd fom th fact that vaianc componnts can b xpssd as lina combinations of chi-squa scal paamts. Encouaging simulation sults a psntd. '. 1. INRODUCION Albt (1976) xhibits ncssay and sufficint conditions fo a sum of squas dcompoition und a mixd modl to b an ANOVA, i.., fo th tms of ti dcomposition to b indpndnt and to b distibutd as multipl.s of chi-squa. W consid impovd stimation of th vaianc componnts und squad o loss in such a st-up. W mak no attmpt to discuss th nomous litatu on this poblm. S Havill (1977) fo such a viw. Rath, w spcializ to th "balancd" cas considing dsigns consisting of cossd and nstd classifications and combinations thof. Ruls of thumb fo fomalizing th associatd ANOVA tabl a thus wll known (s,.g., Sal (1971, Chap. 9)). Customay stimatos of th vaianc componnts a th unbiasd ons obtaind as dscibd in Sal, pp Und nomality ths stimatos a UMVU (Gaybill (1954), Gaybill and Wotham (1956)) and, in fact, stictd maximum liklihood (REKL) (hompson (196)). Howv, positiv pat coctions a usually takn yilding impovd man squa o ssion Fo but sacificing ths "optimalitis." Baysian appoachs to GRA&I vaianc componnt stimation in this stting a discussd in, AB.g., Hill (1965) and ' Box and iao (1973). zound :-fication El Sinc th positiv pat stimatos a not smooth and, thus, not admissibl und squad o loss (SEL), it is natual to.ibutioz - [Availability Cods Avail and/o 1ti D, s t Spoial Li,

3 3 sk dominating stimatos. h alist wok of this typ is du to Klotz, Milton and Zacks (1969) fo th on-way layout. hy show, fo xampl, that th MLE of th "btwn" vaianc componnt (s Hbach (1959)), which is itslf a positiv pat stimato, dominats th UJMVU and, using idas of Stin (1964), iht it in tun can bc dd~mna.zd. h objctiv of this pap is to dscib two gnal appoachs fo cating impovd stimatos of th vaianc componnts und SEL. In Sction w dvlop a mthod by xtnding th afomntlond Stin ida. W hav discussd a spcial cas of this appoach in lina gssion modls in Glfand and Dy (1987b). Sinc th vaianc componnts a lina combinations of chi-squa scal paamts, w can daw upon som litatu fo impovd stimation of lina combinations of scal paamts. his scond 0 ppoach is offd in Sction 3. Wok of Dy and Glfand (1987) fo abitay scal paamt distibutions and of Kloncki and Zontk (1985, 1987) fo th Gamma family of distibutions is ptinnt h. Finally, in Sction 4 w psnt som simulation sults. In th maind of this sction w dvlop notation fo and fatus of th modl w will b woking with. Consid th gnal balancd mixd modl of th fom p Y = -I + * H + X: + (1.i) nxl =l wh Y is an n x 1 vcto of obsvations, w is an ovall man ffct o intcpt, H a known n x m incidnc matics wh H 1 mxl 1 n nxl and H H = I (i.., \ is th numb of nonzo ntis in a typical column of H ), a indpndnt distibutd as N(O,o I ), X is a known n x s dsign matix involving m

4 4 possibly fixd ffcts and covaiats, is th associatd s x 1 vcto of cofficints and E is an n x I vcto of os distibutd N(0,o 1 ) indpndnt of th n wh hus, Y 'I N(m,W) m I + X6 nxl and W is th pattnd covaianc matix P W= I + - HH n =1 ) Lt = ( ":, "'' c p ) Ou pimay intst is in stimating th individually (as it has bn don histoically) although w shall say somthing in Sction 3 about simultanous stimation. As in Albt (1976) w consid a complt st of othogonal p pojctions, PI, P Pp' P, PL, P, P + P + P + P- = I 1'' ' p fu~ n In paticula, P is associatd with th o, i.., Y P Y is th full modl o sum of squas. Pu is associatd with th intcpt (P 1 n ), i.., Y P Y = ny. wh Y. is th nxn avag of th Y's. SS is th modl sum of squas fo th H )- P -- ducd ANOVA modl, i.., SS = Y H(H H) H Y H Y P Y + i ny. i=li wh H - (H 1... H ). Not that w hav a sum of squas fo ach andom ffct. Finally, lt SS,. b th sum of squas fo th fixd ffcts and covaiats adjustd fo th ANOVA, i..,

5 5 SSBI H - Y P Y. ypically, PB is itslf xpssd as a sum of BIH othogonal pics. Accoding to Albt (1976) (s also Bown (1984) and Havill (1984) in this gad), w hav an ANOVA if and only if fo =,,..., p, H H P = k P, k = 1,...,, HH P k u Pu HH P X P and HH P =B P B wh X ov accoding to whth o not H P = 0. hn th Q = y P Y a - indpndnt and distibutd as wh p +, = ank(p ), k k 0. 0, 0, =l m P m/', = m P m/ y. W not that sinc fo ach, I S >(H ), and, nxl. thus, = + 0 " Sinc H P =0 and H P = 0, = 0 and =, i.., =7 and =. h can not b k dtmind... icitl', without spcifying: th dsi:n Howv, fo two andom ffcts, with spctiv sums of squas y Y P Y k and Y P k Y, if th latt is any nstd o c-ossd ffct involving all th factos in th fom, thn ) > '. k -- k' his is in fact, Rul 1 of Sal (1971, p. 393). Obviously, <k -- k < - and typically th is a patial oding amongst th k Finally, again as in Sal (1971, p. 405), if w dfin S= (0o I."' y ) with Y = w hav = AO wh p 0 A with { = k k whnc

6 6-1 -A Y. (1.) Expssion (1.) vals a ky point. h vaianc componnts a xpssibl as lina combinations of chi-squa scal paamts. In fact, this xpssion is usually mployd to -1 cat th familia unbiasd stimatos of th o using f kq k k th unbiasd stimato of -y. IMPROVED ESIMAORS USING SEIN'S MEHOD Consid stimating ay k + by k. Fo appopiat choics of a, b, k, k' (in fact ab < 0), this paamt will b a vaianc componnt. o pocd w utiliz th following lmntay lmma whos poof is immdiat. Lmma.1. Lt S b an stimato of and lt dominat S I und 1 SEL. Lt S b an stimato of wh S is 1 + s indpndnt of S 1 and C hn in stimating a1 a I+ bs domi-ais as + bs und SEL if 1 DbE.,S - I) E, (S - v) > 0 (.1) In ou applications w will mt (.1) by having ab < 0, I < SI E (S ) < K-. W also qui th following sult which is a mino gnalization of a thom statd and povd in Glfand and Dy (1987b). hom.1. Lt S 0 + S (y + 0 O n 0 i 0 n i 1,..., t all indpndnt wh > 0, > > 0. Din R - Sj

7 7 hn in stimating 0 und SEL 6 << 6 <<... << 6 wh 6 << 6 mans 6 dominats 6 i j j i Lastly w nd a lmma which appas, fo xampl, in Klotz, Milton and Zacks (1969, p. 1394). Lmma.. If < S and, in stimating > 0, S << und + + SEL thn S <<, und SEL wh + dnots positiv pat. hs sults will b synthsizd in th following way. Assum ay k + byk, > 0 and w.l.o.g. that a > 0, b < 0. his will b th cas if ayk + byk, dfins a vaianc componnt. st of all Y > Y k (xcluding Y k' gadlss). his st is Find th nonmpty sinc at th vy last Y, > 'k" Qk and th associatd st of Q fom th S and S i, spctivly, fo hom.1 and nabl th cation of a acasing squnc of stimatos which -l dominat (fk + ) Qk, th bst invaiant stimato of k' h sultant suitably dfind play th ol of S and in Lmma -l and will b indpndnt of S = (f + ) Q ' > 0 whnc k' k') i+ Lmma.1 holds. Finally, using Lmma., [as + bs ] << 4 1 [a I + bs Rmak.1. hum.1 allows fo a vaity of impovd stimatos fo -. Lt S 0 = Q with S i bing th Q = i,..., p as wll as Q and Q!. hn t = p + and w may adily cat. In fact, cosponding to any spcifid t pmutation, 7, of th S th will b a sultant S t i.., i th will b t! such stimatos. How might w combin thm to poduc a pmutation invaiant stimato? It can b agud that th minimum of ths will b "too small" and that th avag is a btt pactical choic. S Glfand and Dy (1987b) fo dtails. W illustat using th on-way ANOVA, Yij + a, + ij I ij'. I, J =,..., J, ( i N(O, G), C ij N(O, c ), all indpndnt. In this cas p = 1 with QI (7 + Jl )x, 1 I1-1

8 8 u (c + Jo7)x, IJ and Q ~ -t(j-1) (o + Jo ) In stimating 0 w may dominat th bst invaiant -1 stimato R 0 = (l(j-l) + ) Q3 using 6 = min(r 0, R I ) which in tun is dominatd by, = min(r0, Rip R) (.) o using 6' = min(r O, RI') which in tun is dominatd by, =min(ro, R', R) (.3) -l -l H R, = (I + i) (Q + Q1), R I 1 = (I(3 -i) + 3) (n + Q ) and R= (1 +,)(Q + Q + Q ) " Estimatos and appa in I,l, Klotz, Milton and Zacks (1969). In pactic, if w suspct small w would us if w suspct. small w would us,, and if w hav no pio suspicions w would commnd, + ' (.4) - uning to w may wit - = J ) whnc th usual -1 -l -I unbiasd stimato is givn by J [(I - 1) Q = (1(0-1)) Q I. *1 1 -i -l By Lmma.1 this is immdiatly dominatd by J ((I + 1) Q - (1( - I))- IQ ] which in tun is dominatd by =, -l -I -l -I J [min{(i + I) Q 19 (I + ) (QI + Q )1 - (1(0-1)) Q ]. U + Using Lmma. w aiv at th positiv pat vsion,

9 9 Altnativly, again by Lmma.1, th usual unbiasd stimato is dominatd by J [(I-I) Q - (l(j - 1) + ) Q ] which is dominatd by J [(I + 1) Q1 - (I(J - 1) + ) Q ] which is dominatd by, J [min{(i + 1) Q (I + ) (Q + Q'} -1Q+ - (I(J - 1) + ) Q]. Again by Lmma., w aiv at. + 1, h stimato - appas in Klotz, Milton and Zacks. Ncta that whil << sinc 8 > w cannot conclud gading ",i, 1, 1, and. In fact, th simulation sults in Sction 4 show that nith dominats th oth. In concluding this sction w mak that utilizing th idis in Glfand and Dv (1987a) and in Glfana (1987), along with th,ifomntiond sults, w can impov in th stimation of thatio.k' his allows fo Impovd stimation of,.g.,. S Loh (1986) in this gad. W omit th dtails. Unfotunatly, w cannot xtnd this to,.g., th intaclass colation cofficint sinc it is a non-lina function of such atios. 3. IMPROVED ESIMAES USING A GEOMERIC MEANS APPROACH In this sction w dvlop a mthod fo obtaining impovd stimats which aiss fom xpssion (1.), th fact that th vaianc componnts a xpssibl as lina combinations of chisqua scal paamts. Consid a singl C which w wit as P p -k ckik and lt Q b a candidat stimato. H k=0 k=ok w wdoq dnot Q by Q 0 " Whn can 7 k~k Q b dominatd and what is th

10 fom of th dominating stimato? Dy and Glfand (1987) discuss this poblm whn th ) k a scal paamts fom abitay distibutions. Kloncki and Zontk (1985, 1987), assuming th k a scal paamts fom Gamma distibutions, obtain conditions which nabl assssmnt of lina admissibility fo ZlkQk, i.., k k admissibility within th class of lina stimatos of. hy also off a slightly boad class of dominating stimatos than in Dy and Glfand (1%7) Mo pcisly, th following sult appas in Dy and lfand ki,<> hom 3.1. Lt Y f i = 1,,..., t, t >, ind pndn t and 'uch that EY < Consii th os ti mata lot I+t t - i i ii A wh i I E(Y I /V(Y i i I) n d d m in d d = Mlla i hn (3.1) dominats Y (1) 1' (t) i i und SEL in stimating c i if ith if (i) d > C and 0) < b < td (1) (1) (i) d (t) < 0 and td ( t ) < b < 0 t -1-1 t t wh E(Y 1 If w dnot by G(- ii ) th gamma dnsity f (y) = i

11 'I ~-i -y/d i i-it ] / -,thna 'A+t and:= F i- + t ( +t ). i 7( a i ) i Now lt D b a diagonal matix whos diagonal ntis a th and lt G b of th fom - wh 7 is a nonngativ d dfinit matix such that (7)ij O,( > 0 and E d is a diagonal matix whos diagonal ntis a (7)ii" hn Kloncki and Zontk (1985) show: hom 3.. If Y G(t 7 ) = I 1... t indpndnt, thn v is linaly admissibl fo c if and only if th xists a matix C of th abov fom such that (I + GD) = Gc wh - (,.** ), c (c,..., c t 1 t hom 3.1 is oftn too stictiv. If instad w allow a q. Mo g LlA' poduct Y w can choos q to achiv suitably dfind "d all having th sam sign. In fact, fo a spcifid i st of q > 0 such that q = I Kloncki and Zontk (1987), again I - fo gamma distibutions, povid ncssay and Sufficint conditions fo th xistnc of an stimato of this fom which impovs upon Y. W stat a vsion of thi Lmma I which is ii in a fom paalll to hom 3.1. xists b hom 3.3. :f Yi I C(, ; ' I ) i = I... t indpndnt th 0 such that th stimato Y + by, ii (3.).ij wh q. 0,.q 1, dominats y und SEL, in stimating c i if and only if ith (i) o (ii) blow holds. Dfin d * = ii i

12 1 c +q = mind* andd* =msxd*. i i II'(1) I i (t) 1 i (i) d * > 0, qi = 0 if d* = 0 and 0 < b < b* (1) - i (ii) d * < 0, qi = 0 if d* = 0 and -b* < b < 0 (t) -- i + q" d qj. wh b* =. (a + q. ) {j:d.#0} j +q q. Rmak 3.1. j hom 3.3 holds mo gnally than fo th gamma family. Its poof only quis spcification of th l+q q in-asing functions w (q) E(Y, i = l)/e(y i = ). (In th gamma cas w (q) = + q). Givn ii w w can chaactiz i th sts of q's which mak th cosponding d*'s J all hav th i sam sign, thus nabling domination by (3.). Rmak 3.. h bounding of th isk diffnc in hom 3.1 is not as shap as is possibl Lind th Gamma assumption in hom 3.3; hnc, th sulting bounds on b in hom 3.3 whn J all q a qual a mo liual than thos in hom 3.1. Rtuning to th stimation of a vaianc componnt -= p c k consid th stimato Q wh k k=o k=o -l ck' (fk + -k), 0 < k < 1. h tms kq k ang fom th unbiasd to th bst invaiant stimato of c k as kk k angs fom 0 to 1. Sinc is G(. f' k)' kf k d* = ck(k - q )/(f + 1 ) (3.3) k k k k k k If c k = 0 w must st qk = 0. kkk hus, if Y k dos not appa in using hom 3.3, Qk dos not hlp in stimating -. his d claly diffs fom th appoach In Sction wh, fo xampl,

13 13 in stimating all th Qk can b usd to impov upon th bst invaiant stimato. Not that with k as dfind abov, fom k (3.3), th sign of d* dpnds only upon sgn(c (F k - q )); fo k kk k spcifid E k and qk th magnitud of c k dos not play a ol with spct to whth an stimato of th fom (3.) can dominat. Fom (3.3) if all k = 0 o all E = I this appoach will k k povid a dominating stimato if and only if at last two c k diff fom 0 and all nonzo ck hav th sam sign. Fo a vaianc componnt som pai of c will hav opposit signs. k hfo, a dominating stimato will not b obtaind if fo any such pai both 's a 0 o both 's a 1. If w can choos qk to mak d * a "shink." < 0 thn th dominating stimato in (3.) will b Hnc, using hom.3, th positiv pat of p + (3.) will dominat [ k- Q k=o In this spiit it is natual to ask whth th appoach of this sction can b combind with that of th pvious sction. Can w impov upon th stimatos dvlopd though Lmma.1 and hom.1 using a mo gnal vsion of hom 3.3 as suggstd in Rmak 3.1? notation of hom.1 h answ appas to b no sinc in th i0 is not a scal paam fo th distibution of -.. h ad might suggst that 10 could b J0 viwd as a scal paamt fo th distibution of F und J suitabl conditioning. Following th agumnt lading to hom 3.3, whil b must b chosn unconditionally, it would hav to povid impovmnt at ach conditional lvl. show that vn in th simplst cas, t = achiv this. W can adily, no b unqual to 0 can As an xampl, w tun again to th on-way ANOVA using th j-l notation in Sction. Rcalling j I - ) w consid dominating th stimato

14 14-1 -l -1Q J [(I - I + E-I) Q - (I(J -i) + 6 ) ] (3.4) hus, d I > 0 as q < tp d *> 0 As notd abov, 1 1 > < as < this appoach unfotunatly dos not povid a dominating stimato in th two impotant cass wh E = E = 0 and wh 1. E = C I - 1. Instad, w tak (M) = E =0 fo which any ql, q' > 0, q + q = I wok i ' anq 1 q,1 with (I - 1) + q I(J - 1) + q 0<Kb<-x J (I - 1) + 4 q I(J - 1) ql iq q (l + 1) I(J - ) and (3.) bcoms -I -1 -(35 J [(I + 1) QI - (I(J - 1) Q ] + b O Q (3.5) (ii) 1 = 0, = I fo which any q, q > 0, q + q 1 1 I wok with (I - 1) + J (I - 1) + q I(J - 1) + q 4 q (I(J - 1) + I 1 I q - 1) q[(. - b) + ] and (3.) bcoms

15 15 J-l[( I-I1 (J I)+-i~ O ql 1 Q q J M - 1) Q - (I(J - 1) + ) Q + b Q Q (3.6) W conclud this sction with a mak. Rmak 3.3. Rsults applicabl to th simultanous stimation of vaianc componnts und unwightd SEL a givn in Kloncki and Zontk (1987). In paticula, xtnsions of homs 3. and 3.3 a givn fo th stimation of a vcto C 6 using L Y. h spcial cas C = I (not of intst h) has bn xtnsivly discussd. S,.g., Bg (1980), Das Gupta (1986), Dy and Glfand (1987), and Das Gupta, Dy and Glfand (1987). 4. SIMULAION RESULS In th on-way ANOVA w studid impovd stimation of both -. and 7 by undtaking a substantial simulation study ov vaious valus of I, J,, and -. Each cas civd 10,000 plications. Evn with so many plications, simulation of paticula cass suggsts that th statd pcnt impovmnts (P1's) will only b accuat within %. In stimating : som slctd cass a psntd in abl 1. In this tabl PI is lativ to th bst invaiant stimato givn abov (.). Not supisingly (.) outpfoms (.3) whn is small, and vic vsa whn is small. h stimato (.4) 1 sms lik a good compomis. Fo fixd 7 a ), P1's incas in I, dcas in J. Although th P1's a small th fact that (.)-(.4) a so simpl to calculat ncouags thi us. In th stimation of G th fnc stimato is th positiv pat of th unbiasd stimato. QI Q + [ ] /j (4.1) I - 1Il(J - 1)

16 16 ABLE 1 PERCEN IMPROVEMENS IN ESIMAING a ) PI fo (G, C, (.) (.3) (.4) at I =, J = 5 (0, 1, 10) (1,.1, 10) (1, 1, 1) I = 5, J = 5 (0, 1, 10) (1,.1, 10) (1, 1, 1) I = 10, J = 5 (0, 1, 10) (1,.1, 10) (1, 1, 1) As shown in Sction, (4.1) is dominatd by and by q Q (4.) I + 1 l(j - 1) QI Q l(j - 1) + Nith of (4.) and (4.3) dominats th oth. Howv, fom Sction,. + dominats (4.),s dominats (4.3). Fo b a'l a, sufficintly small (3.5) dominats (4.) ignoing th positiv pats. With positiv pats applid to both stimatos this is no long tu. Sinc th is no obvious optimal choic w took b

17 17 at th middl of th allowabl ang. + + In abl w compa 6 and 6 with (4.1). W s ci a, nomous impovmnt fo both, that th Pl's a ssntially indistinguishabl and that nith of th 's dominats th oth. W would daw th sam conclusions in th compaison of (4.) and (4.3) with (4.1). of cous, if a > c thn 6 will tnd to C al b nonngativ + whnc th domination sult in Sction agus + fo 6 * uning to a compaison of 6 with (4.3) w s 4 a, that if P is small th gain may b substantial. A compaison of + 6 with (4.) would yild ssntially th sam magnituds of impovmnt. Again, sinc ths stimatos a so simpl to calculat, thi us is ncouagd. Finally, th compaison of th positiv pat of (3.5) with (4.) is discouaging whn -, is small than. Modst impovmnt will usually occu whn > :. his is asonabl sinc thn th positiv pat modification is aly applid and th dominanc sult coms into play. + W conclud by commnding (.4) fo and- fo, ACKNO'hLED C) N h authos acknowldg Bad Calin fo pfoming th computations.

18 18 ABLE PERCEN IMPROVEMENS IN ESIMAING Go Pl's vs a Vs vs (3.5) vs (., ca, c') (4.1) (4.1) (4.3) (4.) I =, J = 5 (0, 1, 1) (0, 1,.1) (0, 1, 1) (1,.1, 1) (1, 1,.1) (1, 1, 1) , J = 5 (0,.1, 1) (0, 1,.1) (0, 1, 1) (1, 1,.1) (1, 1, 1) = 10, J = 5 (0,.1, 1) (0, 1,.1) (0, 1, 1) (1,.1, 1) (1, 1,.1) (1, 1, 1)

19 19 REFERENCES Albt, A. (1976). Whn is a sum of squas an analysis of vaianc? Ann. Statist. 4, Bg, J. 0. (1980). Impoving on inadmissibl stimatos in continuous xponntial familis with applications to simultanous stimation of gamma scal paamts. Ann. Statist. 8, Box, G. E. P. and iao, G. C. (1973). Baysian infnc in statistical Analysis. Addison Wsly, Rading, Mass. Bown, K. G. (1984). Ann. Statist. 1, On analysis of vaianc in th mixd modl. Das Gupta, A., Dy, D. K. and Glfand, A. E. (1987). A nw inadmissibility thom with applications to stimation of suvival and hazad ats and mans in th scal paamt family. o appa in Sankhya. Dy, D. K. and Glfand, A. E. (1987). Impovd stimatos in simultanous stimation of scal paamts. ch. Rpt. ':399. Stanfod Univ. Glfand, A. E. (1987). Estimation of a stictd vaianc atio. o appa in Pocdings Scond Intl. amp Confnc in Statistics. Glfand, A. E. and Dy, D. K. (1987a). On th stimation of a vaianc atio. o appa in J. Statist. Planning and Infnc.

20 0 Glfand, A. E. and Dy, D. K. (1987b). Impovd stimation of th distubanc vaianc in a lina gssion modl. o appa in Jounal of Economtics. Giybill, F. A. (1954). On quadatic stimats of vaianc componnts. Annals of Math. Statist. 5, Gaybill, F. A. and Wotham, A. W. (1956). A not on unifomly bst unbiasd stimatos fo vaianc componnts. Jounal Am. Statist. Assn. 51, H~vill, D. A. (1977). Maximum liklihood appoachs to vaianc componnt stimation and latd poblms. Jounal Am. Statist. Assn. 7, Havill, D. A. (1984). A gnalizd vsion of Albt's thom, with applications to th mixd lina modl. Expimntal Dsign Statistical Modls and Gntic Statistics, Essays in Hono of Osca Kmpthon (K. Hinklmann, d.), 31-38, Dkk, NY. In Hbach, L. H. (195). Poptis of modl ll-typ analysis of vaianc tsts, A: Optimum natu of th F-tst fo modl II in th balancd cas. Annals of Math. Statist. 30, Hill, B. M. (1965). Infnc about vaianc componnts in th on-way modl. Jounal Am. Statist. Assoc. 60, Kloncki, W. and Zontk, S. (1985). On admissibl stimation of paamtic functions fo mixd lina modls. Poc. Bkly Confnc in Hono of Jzy Nyman and Jack Kif, Vol., , Wadswoth, Monty, Calif.

21 1 Kloncki, W. and Zontk, S. (1987). Inadmissibility Rsults fo Lina Simultanous Estimation. in th Multipaamt Gamma Distibution. Institut of Mathmatics, polish Acadmy of Scincs, Wasaw. Klotz, J. H., Milton, R. C. and Zacks, S. (1969). Man squa fficincy of stimatos of vaianc componnts. Jounal Am. Statist. Assn. 64, Loh, W. (1986). Imp~G;d stimatos fo atios of vaianc componnts. Jounal Am. Statist. Assn. 81, Sal, S. R. (1971). Lina Modls. J. Wily and Sons, Nw Yok. Stin, C. (1964). Inadmissibility of th usual stimato fo th vaianc of a nomal distibutton with unknown man. Ann. Inst. Statist. Math. 41, hompson, J., W. A. (196). h poblm of ngativ stimats of vaianc componnts. Annals of Math. Statist. 33,

22 UMCASSIFIED SECURIY CLASSIFICAION OoP 1141 PAGE (Wloom D& Eatoos REPOR DOCUMENAION PAGE READ [NSRUCIONS BEFORE COMPLEING FORM Rpl NUNOR L OV ACCCSSION NO. LACCIPE01IIa CAALOG NUMBER AILE (sood 8uIftl) Impovd Estimation Of Vaianc Componnts In Mixd Modls SL ype o nepoa a Pamoo covekes ECHNICAL REPOR 6. PERFORMING 0RGL REPOR HNBMEE 7. AUYHOR(O) S. CONRAC OR ORAm Nuigan). Alan E. Glfand and Dipak K. Dy N J-167 I. PCRPORMING ORGANIZAION WNM AND AOORESE 1. _PROGRAI&6LZME9N. PROJ9C. AM ARCA 6 WORK UNI NUMSERS, Dpatumnt of Statistics Stanfod Univsity Stanfod, CA $I. CONROLLING OPPICE HNAMC AND ADORES8 NR IL REPOR DAE Offic of Naval Rsach August 30, 1990 Statistics & Pobability Pogam Cod 11111&HN^OFP" 14 MONIORIMO0 AGENCY NAME 6 AOORES5(i diffot., bas ConlliMU Offi) 15. SECURIY CLASS. (of this nowns IS. OISRfUI0 SAEMEN (at ahap t) AiPP~)VED FO9R PUBLIC RELEASE: DI Sf1Il BUION U'NLIMI'1ED UNCLASSIFPIED IS. OXCLASSIFICAIONiO01110GRAOING SCmEOUL OiSRI UION SAEMEN (of th obact satmdo &lac "1.It.difttm osav Rpot) IS. SUPPLEMENARY NOES IS. KEY 9ORDS (Calfao an s ole. to mosooy OW IUaio 6F b90*5 nminbt) vaianc componnts; ANOVA dcomposition; balancd dsigns; squad o loss. IS. ASS 0 AC (Csflm so pw* old nsoim ldma sit 6Vlk oupnlw aking Albt's (1976) fomulation of a mixd modl ANOVA, w consid in- Ljovd stimation of th vaianc componnts fo balancd dsigns und squad o loss. wo appoachs a psntd. On xtnds th idas of Stin (1964). h oth is dvlopd fom th fact that vaianc componnts can b xpssd as lina combinations of chi-squa scal paamts. Encouaging simulation sults a psntd. DD 1473 E~tIoo o, UCAS l 1 NmOV5 St iss oso 5/N4 *IOC0I4 SAS I F I ED SICuIDI C1 ASSIPCASOU op os PaE (6%a Data Sao~

Extinction Ratio and Power Penalty

Extinction Ratio and Power Penalty Application Not: HFAN-.. Rv.; 4/8 Extinction Ratio and ow nalty AVALABLE Backgound Extinction atio is an impotant paamt includd in th spcifications of most fib-optic tanscivs. h pupos of this application