Ragnar Norberg University of Oslo

Transcription

1 HIERARCHICAL CREDIBILITY: ANALYSIS OF A RANDOH EFFEC'i' LINEAR ~10DEL WITH NESTED CLASSIFICATION by Ragna Nobeg Univeity of Olo Abtact A andom coefficient egeion model with multi -tage neted claification i conideed. Linea Baye etimato ae obtained fo andom effect at all tage, and etimato of the tuctual paamete ae popoed. Key od: Random effect; Regeion; Neted claification; Hieachical cedibility.

2 Intoduction In tandad cedibility analyi the unobevable chaacteitic of the individual ik ae epeented by i.i.d. (independent and identically ditibuted) andom element. Thi::o way of atto.ck i appopiate when the potfolio can be egaded a a andom ample fom a population of ik. In many ituation, hov;eve, the potfolio i compoed of ik clae that peumably diffe fom one anothe with epect to baic ik condition. In uch cae moe geneal model ae called fo that allow fo tochatic dependence between ik belonging to one and the ame cla. A an example, conide wokmen' compenation inuance witten by fim to cove employee accident: the entie potfolio can be divided into indutie (heavy induty, tanpotation, agicultue,... ), each induty can be futhe divul2d in-to banche (heavy induty into melting wok, mining, engine wok,... ), and o on until we end up with the individual fim. In thi example the ik ae claified in a neted o hieachical manne. Simila patten aie in a numbe of line ot inuance, and they have become a epaate iue of tudy unde the heading "hieachical cedibility". The topic wa launched by Jewell (1975), who- inpied by Taylo (1974) -conideed a hieachy with two tage. Taylo (1979) extended the analyi to hieachie ~vith a geneal numbe of tage. Hieachical extenion of Hachemeite' (1974) egeion model wee teated by Sundt (1979, ), fit in the cae of two tage and then in the geneal multin -tage cae.- The two-tage ege ion cae wa analyed in a Baye ia etting in an ealy wok by Lindley and Smith (1972). ' To thi mall aembly of wok on hieachical cedibility one could add an immene lit of elevant efeence fom geneal ::otatitic; jut like tandad cedibility eentially deal with andom effect model with one-way claification, hieachical cedibility fit into the famewok of andom effect model with neted

3 - 2 - claification. What i peculia to cedibility, i ::"':ldi: <-~mlf-,_.:-u~ :i.:3.i laid on etimation of andom effect and on epeenting the etimato a cedibility weighted mean. t1oeove, cedibility pimaily ha in view ituation ',vith unbalanced deign and nonpaametic familie of ditibution and, theefoe, cente on ditibution-fee method depending only on the fit and econd ode moment of the obevation. In geneal tatitic andom effect ae by tadition tudied unde nomality aumption, which incidentally - alo lead to method baed on moment up to econd ode. Bayeian liteatue, boadly epeented by Box and Tiao (1973), i concened with etimation of hidden andom effect; ampling theoetic liteatue, notably Scheffe (1959) and Swamy (1971), concentate on infeence about expected value and vaiance component (called tuctual paamete in the cedibility teminology). Thoe who hould ty to apply the exiting theoy on hieachical cedibility in pactice, may face poblem of two kind. In the fit place, thee exit no explicit algoithm fo calculation of cedibility etimato fom factual data in the geneal multi-tage egeion cae. 'In the econd place, the poblem of paamete etimation ha eceived little attention. It i the intent of thi pape to upplement the theoy at thee point. It i alo hoped that the peent teatment will open an eay way to faily geneal eult; the poof ae elementay and computationally oiented, the tating point being a model fo the obevation a they peent themelve in pactical application. The content of the pape ae oganized in ection a follow. The hieachical model i intoduced in ection 2. In ection 3 well known eult on linea Baye etimation in the nonhieachical egeion model ae eviewed and then extended to the hieachical cae. In ection 4 etimato ae popoed fo the 1t and 2nd ode moment appeaing in the linea Baye etimato, wheeby empiical linea Baye etimato ae obtained.

4 - 3 - l~e end thi intoduction by laying down a few notational convention. ~ne dimenion of a vecto o matix may be indicated by a topcipt; thu Lqxn denote a matix L with q ow and n column. n. xn. By diag(a. 1 1 )._ 1 1-1,,m i meant the (E.n. )x(l:.n.) matix with A1,...,~m placed in conecutive ode downwod the pincipal diagonal and zeo elewhee. The ymbol Cov and Va deignate covaiance and vaiance matice, epectively. Thu, if x and y ae andom column vecto 1 \ve have Cov( X 1 y 1 ) = E ( xy 1 ) - ExEy 1 1 and Va x = Cov(x,x 1 ). The conditional covaiance of x and y, given a andom element e 1 i denoted by Cov( x, y 1 I e). We emind of the elation 'Cov (X I y I ) = Cov { E (X I e) IE ( y I I e) } + E Cov (X I y I I e) ( 1 1 )

5 A hieachical egeion model Conide a ytem of quantitie anked in a hieachical ode a follow. Stage l : i 1 =l,.,m; Stage 2: i 1 =l,.,m Stage : i =1,...,m i 1 =1,...,m; Stage : e i1 i =1,, m. J.l i 1 =1,...,m; Obevable x. J.l i =1 1 o 1 ffi J.l... i i 1=1,...,m. The hieachical tuctue i viualized by Figue 1. (To be in pefect keeping with the notion of hieachy, the figue ought to be tuned upide down, the idea being that all quantitie labeled by ome elevant index e.,..., e i 1 p, with p >, ae govened by The peent olution i choen fo teminological eaon: it i convenient to peak of a~l quantitie labeled by an -tuple, e.g. i 1, a being at tage. Accodingly, we hall athe think of e.,..., e.. a "unqe lying" all "' 1 quantitie labeled i 1 i.) One intepetation of the quanti P tie i the following: x.. and e.. ae, epectively, 1 1 " " 1 the obeved claim expeience and the hidden ik chaacte~tic of a unit ik cla C.. ; thi cla i the m.. clae contituting a hype-cla l i -th of

6 - 5 - Obevable X i1 i 1 i - X m - 1 1""" 1 Stage e m " Stage ~.. I.. ~ e e m., ::J Stage 2 ~ e. 1.1f;l. 1.1 Stage 1 e 1 Figue 1. A hieachy of clae at tage

7 - 6 - hidden ik chaacteitic -cla i the i -th 1 - in it tun thi hype- hype-clae contituting a "hype-hype-cla" C.. with hidden ik chaacteitic ~, ~-2 ; and o on.... i 2 - Fomally, a cla C.. ~1 ~ i identified with the quantitie that ae pecifically belonging to it, viz. all quantitie at tage p) labeled i 1 i. In Figue 1 they ae found in the p local hieachy above and including e.. ~,... ~ ide denote by e the aembly of all latent quanti tie p intoduce eo= 0. at tage p' ; =1,..,. It i convenient alo to All quantitie intoduced o fa ae enviaged a andom element. \\Te make the following model aumption, the lat thee of which add futhe content to the notion of hiea~hy. ( i) Each i a vecto of dimenion n.. ~,... ~ exit a vecto-valued function and, fo each n.. xq ~,... ~ a nonandom, knmm matix Y. ~1 i E(x. le ) = ~,... ~ with ay. Thee uch that ). ( 2. 1 )... i.1. All the andom vecto econd ode moment.... i and b. -~ 1 have finite (ii) The tage 1 clae chaacteitic e. ~1 C. ae independent, and thei ik ~, ae identically ditibuted; i 1 =1,..,m.

8 - 7 - (iii) vh thin each cla C. 1.1 i at tage ~ aumption (ii) ha the following analogue: conditionally, given 0 the tage +1 clae C... l1... Ll+1 ik chaacteitic e ll+1 ae identically ditibuted; i + 1 =1,...,m ( i v) All the vecto (e.,e..,...,e ae independent, and thei ae identically ditibuted (implying that all hidden vaiable e l. a given tage aume thei value in the ame pace, which may be quite geneal). at He now oganize the data cla-wie a follow. Fo each cla let x.. and. '!:' vecto of.:~,ll Y.. denote, epectively, the cegeand and the matix of all egeo belonging to that cla. Moe pecifically, we tat fom the aleady intoduced elementay data and Y elated to clae at tage, and define the coeponding aggegate quantitie at lowe tage by the ecuive elation X~ = (X~ I X' I 1.1 m. 1.1 y~ = (Y~ I '. y ' i m. 1.1 ( 2. 2) ( 2. 3 ) validfoallelevant (i 1,...,i) and all =1,...,. (Inthe following we hall feel fee to dop uch lengthy and elf-evident pecification of domain of ubcipt.) Clealy, and Y. ae of ode 1.1 whee the n. ' i n.. x1 and n.. xq, epectively, ae defined ecuively a

9 - 8 - n. = I n. ~, ~, i i +l -+l Fo each (i 1,...,i) intoduce b. = E:(b. I e ) ~, ll = 1,,. ( 2. 4) Now it i een that each cla C poee~ ~.eg.eil ion tuctue imila to that of the tage by aumption (i), namely clae a pecified E(x.. le ) = ~,... ~ Y.. b '~ ~, '~ ( 2. 5 ) with element defined by (2.2)-(2.4). Thu, in a hieachy of ode thee ae embedded hieachie of all ode (. The following moment ae well defined unde the aumption ( i)-( iv): (2. 6) E Va(b.. 1e 1 ) ~,... ~ - ( 2. 7) 1 = E Va(x.. le ) ~,... ~ ( 2. 8) 'd { i 1,, i ) ; = 1,,. Thee quantitie have a taightfowad intepetation: ~ i the mean ik level in the potfolio, A meaue the ik diffeen tial between tage clae in one and the ame tage cla, and - in a ene - <I>.. meaue the vaiability in ~,... ~ claim expeience that i not explained by between cla vaiation at tage 1,...,. The baic paamete ae ~, the A, and the at tage. In ection 3 it will be hown that each at tage < i a function of and the

10 Linea Baye etimation by known paamete Fomulation of the poblem. Fo each cla C.. we eek l.,... J. an etimato ~.. of b.. given by (2.4), the pupoe being 1 1""" 1 1 1""" 1 to minimize the expected weighted quaed eo o (oveall) ik, whee qxq w = E{b.. - b.. } 'W{b b.. }, l.1"""l. i a nonandom p.d.. (poitive definite ymmetic) matix. In view of the independence aumption (ii), only X. l.1 elevant fo the pupoe. We confine ouelve to inhomogeneou linea etimato of the fom ( 3. 1 ) i v b.. = g 0 + Gx., 1 1""" (3. 2) whee qx1 go and qxn G ae allowed to depend on y and the paamete of the ditibution. The optimal etimato will be called the LB (linea Baye) etimato, and the minimum value of the ik will be called the LB ik. The cae = 1 The imple non-hieachical andom coefficient egeion model wa fit tudied in the context of cedibility theoy by Hachemeite (1975). It hall be teated in ome detail ince thee exit no ingle efeence that pell out the eult in the fom equied hee. Thu, conide the model (i)-(iv) in the pecial cae = 1, wheeby item (iii) and (iv) become void. The poblem i to etimate b. = b( 9. ) (ay) by mean of x., the paamete (2. 6)- ( 2. 8) )_1 being aumed to be known. To ave notation, dop all indice and conide the imple model ~elation nxl nxq qx1 E(x 19) = Y b (9) ( 3. 3) and the paamete

11 b( e ), A qxq = Va b( e) I nxn ~ = E Va(xle). ( 3 4) Lemma 3.1 (tandad). Aume ~ to be p.d.. The LB etimato of b(e) i - b = Ac + (I -AM)~, ( 3 5) whee qx1 c and Hqxq ae defined a c = Y' (YAY' + ~) X (I MA)Y'~ = - X M = Y' (YAY'+ ~) y Y(AY'~ = Y'~ y + I ). (3.6a) (3.6b) (3.7a) (3.7b) Lemma 3.1 i a claical eult in cedibility and Bayeian egeion. It i tated hee in a fom that i paticulaly well uited fo ou pupoe. Notice that neithe A no Y need to be of full ank. In fact, fomula (3.5) emain valid if A= 0 and even if n = 0 (no obevation) if c and.ivi ae then intepeted a 0. In both cae b educe to ~, of coue. The o-called cedibility matix i defined a Z = AM (3.8.a) = AY' (YAY' + ~) Y (3.8.b) = I - (AY'~ Y + I). (3.8.c) In cae n ) q and Y ha full ank q, b can be cat in the fom of a cedibility weighted mean, b = z~ + ( I-z ) ~, ( 3. 9) with /\ b = (Y'~ Y) Y'~ x. (3.1 0)

12 The identitie in (3.6) and (3.7) tun out to be intumental in the analyi of the geneal hieachical model. They ae alo computationally convenient; if n ( q, ue fomula (3.6a) and (3.7a), which involve inveion of an nxn matix; othewie povided that ~ i eaily inveted- ue (3.6b) and (3.7b), which equie inveion of a qxq matix. The meit of fomula (3.9) i on the intepetative athe than the computational ide, ee e.g. Nobeg (1980). Fo the ake of completene, and in the abence of a uitable efeence, we pove the lemma. Then the poof of (3.9) will be an eay execie. Poof of Lemma 3.1. The LB etimato i (ee e.g. Nobeg, 1980) b = E b(9) + Cov{b(e),x' }(Va x)- 1 (x-ex). (3.11) In the peence of aumption (3.3) the moment appeaing on the ight of (3.11) can be expeed by the paamete in (3.4) a E b( e) = ~, co v { b ( e ), x ' } = tty ', Va x = YAY' + ~. Ex= Y~. (3.12) (3. 1 3) (3.14) (3. 15) The expeion (3.13) and (3.14) ae eay conequence of (1.1). On ineting (3.12)-(3.15) in (3.11), we obtain b = ~ + AQ(x-Y~), (3.16) whee Q = Y'(YAY' + ~). (3.17) The elation (3.16) and (3.17) ae equivalent to (3.5), (3.6a), and (3.7a). It emain to demontate the identitie aeted in (3.6) and (3. 7).

13 Potmultiplication by YAY' + ~ in (3.17) yield the equivalent elation QYA y I + Q~ = y I (3.18) Futhe, potmultiplying by ~ y in (3.18) give QYAY'~ Y + QY = Y'~ Y, fom which we olve QY = Y'~ Y(AY'~ Y + I). (3.19) A M = QY (compae (3.7a) and (3.17)), (3.19) i jut the identity aeted in (3.7). Finally, ubtitute (3.19) back into (3.18) to obtain Q = (I - MA)Y'~, which implie that (3.6a) and (3.6b) ae identical. To complete the poof, it mut be etablihed that i alway invetible when ~ i. Thi follow by ue of the identity x and B one get AY I~ y + I I I x lab+ II = BA +I, valid fo any pai of matice A (Zellne, 1971, p. 231 ). Putting A= AY' and B = ~ Y, IAY'~y + II = ~~- 1 YAY' + II = I~. IIYAY' + ~I. ( 3. 20) Since ~ i p.d.., o i both ~ and YAY' + ~. hence both deteminant in (3.20) ae tictly poitive. It can be concluded that AY'~ Y + I ha a non-zeo deteminant and, theefoe, i invetible. D The hieachical cae, > 1. We now tun to the genuine hieachical model and et out by demontating ome ueful ecuive elation. On ineting (2.2) in the definition of the dipeion

14 matice in (2.8) and ecalling the independence aumption (iii), it i een that = diag{e Va(x. le }. _ 1 1 i ,...,m (3.21) Applying (1.1) to the conditional vaiance in (3.21 ), noticing that e C8 1 andthenuing (2.5), (2.7),and (2.8), yield +1 E Va(x.. le ) ~1 ~+1 = E[Va{E(x.. le +1 ) le } 1 1 ~+ 1 + E { va ( x. I e ) I e } ] ~ = E Va (Y.. b.. I e ) ~1 ~+1 ~, ~+1 + E Va(x.. 1e+1 ) ~, ~+1 (3.22) Combination of (3.21) and (3.22) give the ecuive fomula = diag(y.. A + 1 Y~ ~1 " 1 +1 ~1 i+1 (3.23) Fo each i 1 define, by vitue of (3.6) and (3.7), ( 3 24 a) (3.24b) M.. ~,... ~ (3.25a). +I). ' l. t" (3.25b) (The poitive definitene of each eli.., which i neceay fo ~1 ~ the identitie in (3.24) and (3.25) to hold tue, i a tivi~l conequence of (3.23) and the poitive definitene of each eli. ~1 at

15 at tage.) By ue of (2.2), (2.3), (3.23), and (3.24a), it follow that Y'.. q;.. X ' ' 1 ~1.L = I c.... l1... l l ( ) whence, by ubtitution in (3.24b), = (I-M.. A )I c... l1... l ll+1 l+1 ( 3. 27) Similaly, (3.25) give y~. <li.. Y.. l1oool l,... l l,... l = I M ].]. +1 J.+1 (3.28) and = I M... (A I 1 1 ' M... +I) 1 1 ' ( 3 29 ) We ae now in a poition to contuct LB etimato of all andom effect b.. given by (2.4). Cqnide fit the poblem 1 1 ' 1 of etimating effect at tage 1. Application of Lemma 3.1 to the egeion (2.5) with = 1, give: Lemma 3.2. The LB etimato of b. i ].1 (3.30) whee c. l1 and M. ].1 ae defined by (3.24) and (3.25).

16 To contuct the LB etimato b.. of a geneal b.., 1.1 ".l. 1.1.l. we call on a beautiful eult poved in the univaiate cae by Jewell (1975) and genealized to the egeion cae by Sundt (1980). Lemma 3.3(Jewell/Sundt). LB ~timato at neighbouing tage ae elated by the fomula (3.31) By compaion of fomula (3.30) and (3.31 ), it i een that i obtained by fomally teating b.. l.1... l. - a a tage effect in the local hieachy of with b l. the place of ~. Sundt pove (3.31) by veifying that the expeion on the ight hand ide atifie the nomal equation detemining b. -. l.1... l. He hall give an altenative, diect agument, which may peent an inteet of it own. in Poof of Lemma 3.3. Since LB etimato depend only on the 1t and 2nd ode moment of the ditibution, it uffice to pove the -eult fo one paticula choice of ditibution with the equied moment tuctue. 'Ide pick the following. Suppoe that all the latent quantitie ae mutually independent, each having a q-vaiate nomal ditibution with mean and vaiance A, and that, conditional on e, each... ~. -vaiate nomal dit.ibqti.on '"ith mean Y.. b ]_1 and vaiance ~.., whee b. = ~ + e. + +e..ll. ' i (and, conequently, b i = ~ + e. + + e.. ) the In thi model the joint ditibution of the... i ' i multivaiate nomal, hence the conditional mean ' and

17 of b.., given the x. ', i a linea function (Andeon, ~, ~ ~, ~ 1958, p.29). Now it i well known that thi conditional mean i the uneticted Baye etimato of b.., and- being of the fom ~,... l (3.2) - it i alo LB. Theefoe, in the peent cae l,... ~ = E(b.. lx. ) ~, = E{E(b.. le 1,x. ) lx. } ll... ~ - ~, ~, = E{E(b.. le 1,x.. ) lx. }, ~,.. ~ - l,... ~ ll (3.32a) (3.32b) the lat equation being a conequence of aumption (iii). Unde the peent aumption the inne expectation in (3.32b) aume the fom E(b.. 1e 1,x. ll... ~ - ~,... i ( 3. 33) Thi aetion follow by noting that, conditional on e-l, the local hieachy within C.. ha the ame ditibutional tuc- 1i ~ tue a the global hieachy, with b.. I A,,A I ~,... ~ ~- playing the ole of ~, A1,...,A, ~- ~, l. It follow ~, that the conditional mean on the left of (3.33) i - fomally - the LB etimato of b il in the conditional model, which, accoding to Lemma 3.2, i the expeion on the ight of (3.33). Now, upon ubtituting (3.33) in (3.32b) and then uing (3.32a), thi time at tage, we aive at (3.31). 0 The eult above ae now ummaized a a complete algoithm fo computation of LB etimato. Theoem 3.4. LB etimato ae found fo all andom effect by fit calculating all the quantitie c ~ and M. by the ecuive elation (3.27) and (3.29), ll... l tating fom tage, and then calculating all the b.' ~,...]_ by the ecuive elation (3.31 ), tating fom (3.30) at tage 1.

18 It i notewothy that the LB etimato do not depend on the weighting w. Comment. Sundt' (1980) etting i diffeent fom the peent one; in tem of ou definition, he focue attention at one tage cla C.. and wok with tatitic x 1,..., x, whee ~1 ~ x = x.. and each x ; =1,..., ; conit of the tage ~ 1 ~ tatitic x.. h ; h +l * i +1. The tating point of ou ~1 ~ +1 model i the aembly of all obevable x. in unit ik ~1... i clae, and Theoem 3.4 explain how to actually calculate LB etimato fom thee data. The poof given hee of Lemma 3.3 i elementay and contuctive. The technique can be caied ove to a numbe of complex etimation poblem that othewie would equie eithe heavy algeba o efined Hilbet pace method (de Vylde, 1976) fo thei olution. It eve, inte alia, to jutify the witch fom conditional to unconditional moment in Taylo' poof of Lemma 3.3 fo the univaiate cae (Taylo, 1979, Theoem 4, item 1). If Y.. i of ank q, then ( ) can be ewitten in ~1 ~ the fom of a cedibility weighted mean, b.. ~1 ~ whee and = z.. is.. ~1 ~ ~1 ~ + (I-Z.. )b.. ~ 1 ~ ~ 1 ~ = (Y ~. ~.. Y.. ) Y~. ~.. X.. ~1. ~ ~1 ~ ~1 ~ ~1 ~ ~1. ~ ~1 ~ = o: M. ) ~1 i i I c.... ~,.. ~ ~ +1 ~+l LB +l ( 3. 34) (3.35) z. = A M. ~, ~, (3.36) The demontation of thee fomula i taightfowad by vitue of (3.8)-(3.10). Fom a computational point of view (3.31) i moe convenient than (3.34). Beide, the cae whee Y.. i not of ~1 ~ full ank q i of geat pactical elevance: inuance companie ae contantly facing the poblem of fixing pemium fo new witten

19 buine. If a new ik o cla of ik can be claified with othe ik, poibly in a hieachical ytem a viualized in Figue 1, then it pemium can be detemined in a ational manne within the famewok of a andom effect model, the point being that the model pecifie a elationhip between the ik. An inuance company woking with a fixed effect model, a advocated by Gebe (1982), would be at a lo when confonted with the poblem of aeing a ik with no claim expeience of it own. Fo a futhe dicuion of thee matte, ee Nobeg (1985). Linea Baye ik. The LB ik fo the poblem of etimating i eaily hown to be (Nobeg, 1980) (3.37) whee 6.. i the LB ik matix given by 11 ol = V a b.. - Cov ( b ol i, X~ ) (Va X. ) Cov (X., b ~. ). 11 ll ll ll. 0 ol (3.38) Notice that the LB ik depend on the weighting matix H, ~vheea the LB ik matice - jut like the LB etimato - do not. The following theoem tate that the LB ik.in (3.37) can be calculated by a ecuive pocedue once the matice have been found by ue of the ecuion (3.29). Theoem 3.5. The by the ecuive elation LB ik matice in (3.38) can be calculated... ~ t:' ) +A}, t:' ( ) tating fom tage with ( ) the matice ae given by (3.29) and (3.36) 0

20 Poof: To demontate (3.40), ubtitute (3.13) and (3.14) with b.,x.,y.,a 1,~ =1. Thi give in the ole of b(e),x,y,a,~ into (3.38) fo which, by (3.25a) and (3.36), i jut fomula (3.40). To pove (3.39), apply the ame tick a in the poof of Lemma 3.3. In the nomal model pecified thee the LB ik matix coincide with the uneticted Baye ik ~tix, that i, t:... = E Va(b.. lx. ) (3.41) Applying pinciple (1.1) to the conditional vaiance and aguing a in the poof of Lemma 3.3, ewite (3.41) a t:... = E [Va { E (b.. I e 1 'X. ) I X. } E { Va (b.. I e 1 'X. ) I X. } J = E v a { E ( b.. I e 1, x.. ) I x. } (3.42) + EE { va ( b.. I e 1, x.. ) I e 1 } Now ubtitute (3.33) in the fit tem in (3.42) and ob~eve that the inne expectation in the econd tem i - fomally - the Baye ik fo the poblem of etimating b.. a a tage the local hieachy of C.., conditional on pinciple (3.40) applie. It follow that effect in e 1 SO that in = E V a { ( I-A ~.. ) b.. I x. } + ( I-Z.. ) A, 11.. olli ~ which by (3.36) and (3.41) i the ame a (3.39). 0

21 Etimation of paamete; empiical linea Baye etimato Fo the pupoe of paamete etimation only clae with obevation ae elevant, and o all n. 1.,... i be geate than 0. ' ae now taken to Etimation of the econd ode moment in (2.7) and (2.8) i in geneal not feaible unle futhe tuctue i impoed on the matice ~... Theefoe, in the peent ection it will be 1., ; 1. aumed - a i tandad in egeion theoy - that thee exit a poitive function v and nonandom, kno\'m p.d.. matice P.. uch that l. Va(x. 1.1 Then I e ) = < e e. ) P-:-. 1 v.,..., l. '1.1.. ol. whee... ]. ~ = Ev ( e.,..., e i and the elevant et of paamete become ~I A,,,A I~. ( 4. 1 ) The tating point fo contucting etimato ae th~ expeion... ~ ' ~ I ( 4. 2) (~~ +I&.... A )Y~. = ' J1 J J1 J... ~p. (4.3).. ol.,jl.. J l..l whee equal if and 0 othewie. Relation (4.3) i obtained by epeated ue of pinciple ( 1. 1 )

22 A cla of unbiaed etimato of the paamete in (4.1) i contucted a follow. Fo each i 1,...,i, denote by the ank of the egeo matix Y.., and let L (Y.. ) be the. -dimenional linea pace panned by the column of i Y.. Let A. and B.. be matice of ode t 1.1 i n.. x q and n.. 11 "1 11 "1 uch that the column of A. column of L(Y. - l.l... i ). x ( n ), epectively, i pan L(Y.. ) and the - l., l. i a bai of the othocomplement of By (4.2), a taightfowad etimato of ~ i (4. 4) Fo each i 1,...,i 8, intoduce the tatitic 1 ( 4. 5 ) = t ( B.. B ~. X. X ~ 1 1"" ' 1 1 1"' ' 1 1.1,,_ 1.1 ). By (4.3) and fact that the column of y ]. thoe of ae othogonal to = <!> t ( B... i 1.. B ~. P.. ) 1... ] ,_ ]. Theefoe, an unbiaed etimato of <1> i * <!> = {I t (B.. B ~. P.. ) },...,i 1 1 ' ' ' ' 1 i 1 with defined by (4.5). ~. 1 1,, ]. (4.6)

23 Etimato of A 1,...,A ae baed on the quantitie I j + 1 ' 'j : :i_ } 5 j +1 * i+1 (I ( ) =O,..., (4.7) i 1,...,i whee the um E ( ) extend ove all i 1,..., i fo,,,hich the indicated inveion ae.valid. It i obviou how to intepet the ummation in the cae =O and =: fo = the um ove j+1,...,j include only i 1,...,i. It i. een that involve all poduct of obevation in diffeent tage clae within the ame tage G +l cla. Thu, all poduct x.. x'.. ae utilized in the etimation. Uing (4.3), it J1 J i eaily checked that EG = ~~ with ~~ + A + + A 1 ~~ + Al + + A + <PH =O: =1,..., : =: H = cp{l() (A'.. Y ,...,15 )... i A~. P.. A } <I < > i1 '...,i (4. 8) It follow that a et of y~metic i given by unbiaed etimato of the A (4. 9) whee the G and H ae defined by (4.7) and (4.8), epectively. Note that the matice in (4.9) need not be poitive definite. Thu the old poblem of negative etimate of vaiance component may aie.

24 The etimato in (4.4) i well defined if and only if the egeo matix of all the obevation unit, Y = (Y~,...,Y~)', i of full ank q. The etimato in (4.6) i well defined if at leat one n.. -. i geate than 0. Both thee ~,.. 1 1, i ~equiement ae vey weak. The etimato A * in (4.9) i well defined if G in ( 4. 7) i, which equie that thee exit at leat one tage cla C. with a tage +1 ubcla uch that the matix obtained by deleting the block fom Y i of ank q. If any one of thee equiement i not fulfilled, the coeponding paamete i not identifiable fom the obevation. The quetion of how to chooe the matice A.. and hall not be dicued in any detail hee. If the deign i balanced, it i, of coue, optimal to,take all (A..,B.. ) equal (Nobeg, 1977), but thi cae i of 11 " 1 11 ' 1 little inteet in inuance and othe non-expeimental field. In geneal no unifomly optimal choice can be made. A "natual" choice, paying egad to the amount of infomation contained in the individual ik clae, i A.. = Y'. P. 11 ' ' if. = q.., B.. B~. = P Y.. (Y~ P.. Y.. ) Y!. P. ",_,_1.. ' ' 1 11 ' ' ' i and, Then the expeion in (4.4) become y~. p,. y, 1 1 ' ' I ' ' ' I ~... ~. ) I i1,..,i which would be the bet linea unbiaed etimato if A= =A=O, 1 and (4.5) educe to the odinay ample etimate of

25 v ( 9. ].1 9 I I 11 multiplied by the degee of feedom, n.. - q.. If the econd ode moment wee known, J.1.. J. J.1 J. the bet linea unbiaed etimato of { Y 1 ( V a x ) Y} Y 1 ( V a x ) x, would be then (4.10) whee x = ( xl,..., x~) 1 and Y = (Y l,..., Y~) 1 By independence between clae at tage 1, the expeion in (4.10) i equal to m L Y! cf?. x. i =1 ].1 ] o, by ineting (3.26) and (3.28), (4.11) which can be calculated ecuively by (3.27) and (3.29). A genuine etimato (4.11) by thei etimato. i obtained by eplacing ail econd ode moment in The etimato defined by (4.4), (4.6), and (4.9) ae conitent a m - the numbe of tage clae - inceae, povided that the ank.. and the peciion P.. do not tend J.1 J. J.1 1 to too mall value, oughly peaking. Thi eult i, howeve, mainly of theoetical inteet to inuance people becaue in thei application thee vlill uually be a limited numbe of clae at tage 1. Finally, empiical LB etimato of andom effect ae obtained upon eplacing the paamete occuing in the fomula in ection 3 by thei etimato in (4.4), (4.6), and (4.9).

26 Refeence Andeon, T.W. (1958). An Intoduction to Multivaiate Statitical Analyi. Wiley, New Yok. Box, G.E. and Tiao. G.C. (1973). Baveian Infeence in Statitical Analyi. Addion-Weley. De Vylde (1976). Geometical cedibility. Scand. Actuaial J., 1979, Gebe, H.U. (1982). An unbayeed appoach to cedibility. Inuance Math. Econ., 1, Hachemeite, C.A. (1975). Cedibility fo egeion model with application to tend. In Cedibility: Theoy and application (ed. P.M. Kahn), pp , Hiley, New Yok. Jewell, w.. (1975). The ue of collateal data in cedibility: a hieachical model. Reeach Memoandum of the Intenational Intitute fo Applied Sytem Analyi, Laxenbug, Autia. Lindley, D.V. and Smith, A.F.M. (1972). Baye etimate fo the linea model. (Hith dicuion.) J.R. Statit. Soc. B, 34, Nobeg, R. (1977). Infeence in andom coefficient egeion model with one-way and neted claification. Scand. J. Statit. 4, Nobeg, R. (1980). Empiical Baye cedibility. Scand. Actuial J. 1980, Nobeg, R. (1985). Unbayeed cedibility eviited. ASTIN Bulletin 15, Scheffe, H. (1959). The Analvi of Vaiance. Wiley, New Yok. Sundt, B. (1979). A hieachical cedibility egeion model. Scand. Actuaial J., 1979, Sundt, B. (1980). A multi-level hieachical cedibility egeion model. Scand. Actuaial J., 1980, Swamy, P.A.V.B. (1971). Statitical infeence in andom coefficient egeion model. Lectue Note in Oneation Reeach and Mathematical Sytem, 55, Spinge-Velag, Belin. Taylo, G.C. (1974). Expeience ating vdth cedibility adjutment of the manual pemium. ASTIN Bulletin, 7, Taylo, G.C. (1979). Cedibility analyi of a geneal hieachical model. Scand. Actuaial J., 1979, 12. Zellne, A. (1971). An Intoduction to Bayeian Infeence in Econometic. Hiley, New Yok.