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1 This ticle ws downloded by: [Ntionl Chio Tung Univesity 國立交通大學 ] On: 8 Apil 014, At: 01:16 Publishe: Tylo & Fncis Infom Ltd egisteed in Englnd nd Wles egisteed Numbe: egisteed office: Motime House, Motime Steet, London W1T 3JH, UK Jounl of Applied Sttistics Publiction detils, including instuctions fo uthos nd subsciption infomtion: Byesin nlysis of genel gowth cuve model with pedictions using powe tnsfomtions nd A(1) utoegessive dependence Jck C. Lee & Kuo-Ching Liu Published online: 0 Aug 010. To cite this ticle: Jck C. Lee & Kuo-Ching Liu (000) Byesin nlysis of genel gowth cuve model with pedictions using powe tnsfomtions nd A(1) utoegessive dependence, Jounl of Applied Sttistics, 7:3, , DOI: / To link to this ticle: PLEASE SCOLL DOWN FO ATICLE Tylo & Fncis mkes evey effot to ensue the ccucy of ll the infomtion (the Content ) contined in the publictions on ou pltfom. Howeve, Tylo & Fncis, ou gents, nd ou licensos mke no epesenttions o wnties whtsoeve s to the ccucy, completeness, o suitbility fo ny pupose of the Content. Any opinions nd views expessed in this publiction e the opinions nd views of the uthos, nd e not the views of o endosed by Tylo & Fncis. The ccucy of the Content should not be elied upon nd should be independently veified with pimy souces of infomtion. Tylo nd Fncis shll not be lible fo ny losses, ctions, clims, poceedings, demnds, costs, expenses, dmges, nd othe libilities whtsoeve o howsoeve cused ising diectly o indiectly in connection with, in eltion to o ising out of the use of the Content.

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3 Jounl of Applied Sttistics, Vol. 7, No. 3, 000, 31± 336 Byesin nlysis of genel gowth cuve m odel with pedictions using powe tnsfomtions nd A(1) utoegessive dependence JACK C. LEE & KUO-CHING LIU, Institute of Sttistics, Ntionl Chio Tung Univesity, Hsinchu, Tiwn ABSTACT In this ppe, we conside B yesin nlysis of the unblnced ( genel) gowth cu ve model with A(1) utoeg essive dependence, while pplying the B ox± Cox powe tnsfomtions. We popose exct, simple nd Mkov chin M onte Clo ppoximte pmete estimtion nd pediction of futue vlues. Numeicl esults e illustted with el nd simulted dt. 1 Intoduction The min pupose of this ppe is to ddess the poblem of nlyzing gowth cuve dt fom Byesin point of view, using n unblnced gowth cuve model with A(1) utoegessive dependence nd with the Box± Cox powe tnsfomtions pplied to the obsevtions. It hs been obseved in the litetue, such s Lee (1988), Lee nd Lu (1987) nd Kemids nd Lee (1990), tht the dt of biologicl gowth nd technologicl substitutions tend to exhibit stong coeltion coss diþ eent time peiods. In Lee nd Lu (1987), temendous impovement ws found in pedictive ccucy using the dt-bsed tnsfomtion models fo technology substitutions. This is pimily becuse of the fct tht the lineity ssumption fo the gowth function cn be enhnced signi cntly with the Box± Cox powe tnsfomtion, long with incopoting into the model the pope dependence stuctue m ong the obsevtions. Kemids nd Lee (1990) combined the concepts of powe tnsfomtion nd genelized gowth cuve model with seil stuctue fo foecsting technologicl substitutions bsed on the mximumlikelihood (ML) method when epeted mesuements of shot time seies e vilble. Insted of the blnced cse, we will ddess the poblem of gowth Coespondence: J. C. Lee, Institute of Sttistics, Ntionl Chio Tung Univesity, Hsincho, Tiwn. ISSN pint; ISSN online/00/ Tylo & Fncis Ltd

4 3 J. C. Lee & K.-C. Liu cuve when unblnced epeted-mesues dt e vilble. This is ppently moe genel nd of moe pcticl impotnce. The unblnced epeted-m esues gowth cuve model with the Box± Cox powe tnsfomtions is de ned s Y (k i 5 X is + e i, i 5 1,,..., ; 5 N N ; 5 1,..., (1) whee s is n unknown m 3 1 vecto of egession coeý cients of goup, nd X i is known design mtix of nk m, whee < m < p i, nd N 5 N N. The distubnce tems e i e independent p i -vite noml with men vecto 0 nd covince mtix i. In genel, p i is the numbe of time (o sptil) points obseved on mesuement i; m nd, which specify the degee of polynomil in time (o spce) nd the numbe of distinct goups, espectively, e ssumed known. The powe tnsfomtions of Box nd Cox (1964) e de ned s Y (k i j 5 {(Y k ij + y ij) 1, if k ¹ 0 k log(y ij + y i j), if k 5 0 whee y ij is known constnt such tht Y i j + y i j > 0 fo ll i, j nd Y i 5 (Y i1,..., Y ipi )Â. In pctice, y i j 5 0 if ll Y ij tems e positive. Without loss of genelity, we will ssume y ij 5 0 fo the est of the ppe, becuse y ij is ssumed known. The convention Y (k ) i 5 (Y (k ) i1,..., Y (k ) ip )Â is dopted thoughout the ppe. Also, k is unknown. If ll the p i e equl, i.e. blnced epeted mesuements, then eqution (1) cn be witten s the genelized gowth cuve model with Box± Cox powe tnsfomtions. The genelized gowth cuve model, when no powe tnsfomtions e pplied, is de ned s Y 5 X p 3 N p3 m s m 3 e 3 A + 3 N p N whee Y 5 (Y 1,..., Y N ), e 5 (e 1,..., e N ) nd s 5 (s 1,..., s ) is unknown, X nd A e known design mtices of nks m < p nd < N espectively. Futhemoe, the columns of e e independent p-vite noml with men vecto 0 nd common covince mtix. The model ws st poposed by Potthoþ nd oy (1964), nd subsequently consideed by mny uthos, including o (1967, 1987), Khti (1966), Gizzle nd Allen (1969), Geisse (1970, 1980), Lee nd Geisse (197, 1975), Fen (1975), Lee (1988), Jennich nd Schluchte (1986) nd Von osen (1991). Lee (1988) studied eqution (3) when hs A(1) dependence, while Kemids nd Lee (1990) incopoted into the model the Box± Cox powe tnsfomtions in the foecst of technology substitutions. Lee (1988), Lee nd Lu (1987) nd Kemids nd Lee (1990) demonstted epetedly the impotnce of A(1) dependence, o seil covince stuctue, fo the covince mtix fo eqution (3). Applying the A(1) dependence to the covince mtix i fo eqution (1), we hve () (3) i 5 C i (4) ½ d dâ ½ whee C i 5 (q ); d, dâ 5 1,,..., p i; nd > 0 nd 1 < q < 1 e unknown. The estimtion of pmetes nd pediction of futue vlues fo equtions (3) nd (4) hve been studied using the M L method by Lee (1988). The pupose of this ppe is to conside eqution (1) fom Byesin point of view. The esults using the ML method e lso included in the ppendices.

5 B yesin nlysis of genel gowth cu ve model 33 We will conside two types of pediction poblem fo unblnced gowth cuve models with powe tnsfomtions s speci ed by equtions (1), () nd (4). Fo the st type of pediction, let V be set of p 3 1 futue mesuements of goup, ssumed to hve been dwn fom eqution (1); i.e., the set of futue vlues of goup is such tht, given the pmetes s nd, we hve E(V (k ) 5 Xs (5) whee E( ) denotes the expected vlue, X is known p 3 m mtix nd V (k is multivite noml with the covince mtix. We ddess the pediction of V given Y 5 (Y 1,..., Y N ), the unblnced epeted-mesues dt, s the smple. It is noted tht Y is no longe p 3 N, becuse Y i is p i 3 1. The second type of pediction poblem is concened with pedicting the futue vlues of the obseved smple. Let y j be futue q-dimensionl vectol mesuement of goup. We e inteested in pedicting y j given Y. This is time seies pediction, so is impotnt in pctice. This type of pediction is clled extended pediction of y j, becuse the pediction is mde beyond the obseved time nge of the smple. It is noted tht extended pediction is identicl to conditionl pediction s studied in Lee nd Geisse (197, 1975), Fen (1975), nd Lee (1988) when using the model speci ed by equtions (1) nd (4). In this ppe, Byesin infeence by mens of numeicl integtion, the M kov chin M onte Clo (MCM C) method, nd some simple ppoximtions e studied fo eqution (1) with A(1) dependence. In ecent yes, sttisticins hve been incesingly dwn to M CM C methods, especilly the Metopolis± Hstings (M H) lgoithm (M etopolis et l., 1953; Hstings, 1970) nd the Gibbs smple (Gemn & G emn, 1984; G elfnd & Smith, 1990). They hve emeged s extemely popul tools fo the nlysis of complex sttisticl models. W hile they hve been most widely used in Byesin nlysis, they hve lso been employed by fequentists in missing nd dependent dt settings, whee the likelihood itself involves complicted high-dimensionl integls (see, fo exmple, Gelfnd & Clin, 1993). Excellent tutoils on the methodology hve been povided ecently by Csell nd Geoge (199) nd Gilks et l. (1996). In Section, Byesin estimtion of pmetes is consideed fo the model. In Section 3, two types of pediction poblem e pesented. In Section 4, Byesin infeence by mens of M CM C methods is studied with non-infomtive joint pio fo the pmetes. The esults developed in this ppe e illustted in Section 5, with el nd simulted dt. Finlly, some concluding emks e pesented in Section 6. Byesin estimtion of pmetes Combining the likelihood function of, s 1,..., s, k 1,..., k nd q with the noninfomtive pio (Zellne & Tio, 1964), i.e. P (s,, q, k )µ (6) nd integting with espect to P (q, k ½ Y) µ (1 q nd s 1,..., s, we hve ) (n N)/ ½ J ½ B (n m) / P 5 1 XÂ i C 1 j / p i X i (7)

6 P P 34 J. C. Lee & K.-C. Liu whee B i (Y (k i X isã ) C 1 i (Y k i X isã ), k 5 (k 1, k,..., k ) Y 5 (Y 1,..., Y N1, Y N1 + 1,..., Y N * ), N * J P 5 N i p i Y k 1 j 5 1 i j, n 5 p i, N 5 i 5 1 N 5 1 Also, simple ppoximtion is P (s ½ Y) 5 Ç P (s ½ q Ã, kã, Y) (8) whee (q Ã, kã ) is the mode of P (q, k ½ Y), if P (q, k ½ Y) is concentted nd nely symmetic, s pointed out by Ljung nd Box (1980). Thus, we hve the following posteio distibution of s, fo 5 1,..., : whee N * s ½ Y ~ Ç T m (sã,[b à /(n m)] 1 X i C à 1 i X i, n m) (9) i sã 5 ( N * ) 1 X i C à 1 i X i i C à i 5 (q à i X i C à 1 i ½ d dâ ½ ), d,dâ 5 1,..., p i Y (k à i k 5 k à q 5 q à q à k ½ 3 ~ m Bà is the vlue of B evluted t nd, with, kã mxim ize P (q, Y), s given in eqution (7). It is noted tht, fo p 1 vecto y, we sy tht y T p (l,, ), the multivite T-distibution, if its density is whee f(y) 5 K(m, p) ½ ½ 1/ [1 + m 1 (y l ) K(m, p) 5 C Simil to eqution (8), we hve (p + )/ m C ½ Y ~ Ç IG(n m 1 (p + m ) / (y l )] ) (m (m p ) p /, B à ) whee IG(m 1, m ) is the invese gmm distibution with pmetes m 1 nd m. To compute the posteio egion fo s of goup, fo 5 1,...,, let F 1 (m 1, m ) be the uppe 100 pe cent point of the F distibution. Then, we hve tht ( N * B à 1 (s sã ) X ic ) à 1 m i X i (s sã ) < i n m F 1 (m, n m) (10) which will povide 1 posteio egion fo s.

7 P B yesin nlysis of genel gowth cu ve model 35 3 Pediction Two types of pediction fo the model speci ed by equtions (1), () nd (4) will be consideed in this section. 3.1 Pediction of the futue vlues V of goup The pediction of the futue vectol mesuement V (V 1,..., V p ) given the smple Y is consideed hee. The density function of V (k given,, nd is f(v (k ½ s,, q, k )µ ½ C p ½ 1/ exp )] [ 1 (V (k Xs ) C 1 p (V (k Xs (11) U pon combining eqution (11) with the joint posteio density of s,, q nd k given Y nd integting out, s 1,..., s nd V (k, we obtin P 1 (q, k ½ Y) µ (1 q N k ¹ * k X i C 1 i 5 N * k ) [n + p (N + 1)]/ ½ J ½ ½ Q 1/ ½ M ½ 1/ i X j (n m)/ B whee J nd B e de ned in eqution (7), nd 1/ Q 5 Q 1 + Q, Q 1 5 X i C 1 i X i, Q 5 X C 1 p X i M 5 C 1 p X(X C 1 p X ) 1 Q 1 Q 1 Q (X C 1 p X ) 1 X C 1 p + Z(Z C pz) 1 Z (1) W ith guments simil to eqution (9), we obtin the following ppoximtion fo the pedictive distibution of V (k : whee sã 1 5 ( V (k ½ Y ~ Ç T p (Xsà 1,[B à /(n m)] 1 M Ã, n m) (13) ) 1 X j C à 1 j X j i i X j C à 1 j Y (k à j BÃ, M Ã, Q à nd C à i e B, M, Q nd C i evluted t q 5 q à nd k 5 kã, with q Ã, kã mxim ize P 1 (q, k ½ Y), s given in eqution (1). Next, fom eqution (13) nd fom the de nition of the powe tnsfomtion in eqution (), we cn obtin the following ppoximte pedictive density of V, when kã ¹ 0: P (V ½ Y) 5 Ç C [B à v + (V kã 1 ) Xsà 1 kã C à 1 p (V kã (n + p m)/ 1 k à Xsà 1 ½ J v ½ (14) )] whee C v 5 {[ò [B à + (V (k Xsà 1 ) C à 1 p (V (k Xsà 1 )] (n + p m)/ dv (k } 1 ½ J v ½ 5 P p k 5 1 V k à 1 k

8 5      36 J. C. Lee & K.-C. Liu Theefoe, when kã ¹ 0, we cn pedict V by the ppoximte pedicto V à 5 (1 + kã Xsà 1 ) 1/k à (15) ¹ Simil to eqution (10), when kã 0, we hve 1 pediction egion fo V by the inequlity B à Fo the cse when kã 1(V kã 1 ) Xsà 1 kã 5 C à 1 p (V kã 1 k à ) Xsà 1 < p n m F1 ( p, n m) (16) 0, the esults e simil, so e omitted hee. 3. Pediction of futue obse vtions when thei pst is vilble We next conside pediction of y j, futue q-dimensionl vlue of Y j, given the obseved unblnced epeted mesuements Y 5 (Y j, Y ( j )), whee Y ( j) 5 (Y 1,..., Y j 1, Y j +1,..., Y N ) nd j 5 1,..., N* ; 5 1,...,. This is time seies pediction which is of pcticl inteest fo mny types of gowth cuve dt. To mke this type of pediction, the covince stuctue genelly hs to be extendble to the futue vlues of the individuls obseved. The A(1) dependence stis es this equiement. Let x j, q 3 m, be design mtix coesponding to y j ; Y j is in goup. Also, E(Y (k j Â, y (k j ) 5 (X j, xâ j ) s nd 5 Cov(Y (k j, y (k j ) 5 C 5 (C i j ), i, j 5 1,, ½ d dâ ½ whee C 5 (q ), d, dâ 5 1,..., ( p j + q), C 11 is p j 3 p j, C 1 is p j 3 q, C is q 3 q, nd C 1 5 C 1. Let Y (k * j 5 (Y (k j, y (k j )Â, X* 5 (X j, xâ j )Â. Aguments simil to those in Section 3.1 led to the following ppoximte pedictive density of y j: whee P (y j ½ Y) 5 Ç C [B à y 1 + (y kã C y 5 {ò [Bà 1 + (y (k j l j 1 l kã (kã (kã y ) G à (y kã j 1 k à l y ) G à (k ) (y j l y )] (n + q m)/ dy ½ J y ½ 5 P q k 5 1 y k à 1 jk (kã (n + q m) / (kã y ½ J y ½ (17) )] (k ) j } 1 G à 5 C à 1 X*(X* C à 1 X*) 1 Q à * 1 Q à * 1 Q à * (X* C à 1 X*) 1 X* C à 1 + Z*(Z* C à Z*) 1 Z* (G à 11 G à ) 1 G à 1 G à Q* 1 5 X kc 1 k X k, Q* 5 X* C 1 X*, Q* 5 Q* 1 + Q* k k ¹ j G à G à 11 G à 1 G à 1 G à 1, l à (kã y 5 xsã * 1 G à 1 Bà 1 5 Bà + (Y (k à j X jsã * 1 ) G à 11.(Y (k à l ) j X jsã * 1 ) G à 1 (Y (k à j X jsã * 1 )

9 l P P B yesin nlysis of genel gowth cu ve model 37 sã * 1 5 ( N * i i ¹ j X i C à 1 i X i ) 1 i i ¹ j X i C à 1 i Y (k à i Z*, ( p j + q) 3 ((p j + q) m), such tht X*Z* 5 0, nd q Ã, k à mxim ize P (q, k ½ Y)µ (1 q ) (n + q N) / ½ J ½ ½ Q * ½ 1/ ½ G ½ 1/ i ¹ N i X j C 1 j 5 1 1/ p j X j (n m)/ B 1 (18) Theefoe, we cn pedict y j by the ppoximte pedicto yã j 5 (1 + kã l (kã y nd 1 pedictive egion fo y j fom the inequlity B à 1 1 (y kã k à j 1 l (kã y ) G à (y kã j 1 l kã ) 1/k à (19) ) (kã < q y n m F1 (q, n m) (0) Anothe ppoximte posteio density is the o± Blckwelliztion ppoximtion P (y j ½ Y)> 1 L L s 5 1 P (y j ½ Y, q (s), k (s) ) (1) whee q (s) (s), k e dwn fom P (q, k ½ Y). Using this method, we cn obtin the ppoximte pedicto nd pedictive egion of y j s well. We cn lso obtin the following joint density of q, k nd y j: whee P (y j, q, k ½ Y) 5 C * y (1 q N k ¹ * k X j C 1 j 5 N * k / j X j B 5 B + (Y (k j X js * 1 ) G 11. (Y (k j X js * 1 ) + ) (n + q N)/ ½ J ½ ½ Q* ½ (n + q m)/ B ½ J y ½ (y k j 1 l k (k y ) 1/ G (y k j 1 l k () ) (k y (k (kã y is l ) y with k à eplced by k, nd s * 1 is sã * 1 with k à nd q à eplced by k nd q, espectively, nd C * y is nomlized constnt. Fom eqution (), we cn obtin the following exct pedictive density of y j : P (y j ½ Y) 5 ò ò P (y j, q, k ½ Y) dq dk (3) 4 Byesin infeence vi M CM C sm pling The M H lgoithm is n extemely poweful method tht cn be used in conjunction with the Gibbs smple to extct mginl distibutions of inteest. Some detils cn be found in Chib nd Geenbeg (1995).

10 q s q k 38 J. C. Lee & K.-C. Liu 4.1 Model nd lgoithm In this section, we will outline n M CM C smpling pocedue with eqution (1). Fom the joint posteio density of,,, nd given the smple Y, the M CM C smpling poceeds s follows (1) Genete s given, q, k nd Y fom N m(sã, ( j X j C 1 1) ) j X j (4) () G enete given s, q, k nd Y fom the invese gmm distibution whee S(s, q, k, Y) 5 (3) Genete q given s, f(q ) µ (1 (4) Genete k given s, q, with f(k µ ½ J ½ exp [ IG 5 1 i ) (n S(s, q, k, Y), (5) (Y (k i X is ) C 1 i (Y (k i X is ), k nd Y using the MH lgoithm fom ) (n N) / exp [ ] S(s, q, k, Y) (6) nd Y using the M H lgoithm, whee 1 (7) )] (Y (k j X js ) C 1 j (Y (k j X js j J 5 P P j p j k 5 1 Y k 1 jk (8) To elbote on the MH lgoithm in step (3), let us ssume tht the pio on q unifom ove ( 1, 1). We cn tnsfom q to q  Π( `,` ) by q  5 log 1 + q 1 q is Then, we pply the MH lgoithm to the function g(q  )µ exp[q  (n + N) /] (1 + exp(q  )) n + N exp [ ] S (s, q Â, k, Y) (9) whee S (s, q Â, k, Y) is obtined fom S(s, q, k, Y) with q eplced by [exp(q  ) 1] / [exp(q  ) + 1]. We lso need to specify q  in the tnsition kenel. The quntity q  is usully chosen to e ect the conditionl stndd devition of q  given s,, k, Y. We use the following method to estimte the vince q Â. Fom eqution (9), let l(q  ; s,, k, Y) 5 log[ g(q  ; s,, k, Y)]. Then, fom this,

11 h B yesin nlysis of genel gowth cu ve model 39 invet the smple infomtion given the q  vlue in the M CM C smpling to obtin the pelimin y vince estimtes of q Â, nd put it into the M CM C pocedue. H ving obtined q  fom the M H lgoithm, we tnsfom q  bck to q by [exp(q  ) 1] /[exp(q  ) + 1]. The sme opetion cn lso be pplied to the vite k with the pio on k being unifom ove ( 4, 4). 4. Foecst H ving obtined the posteio distibution of the unknown pmetes fom the bove M CM C smples, we cn use it to pedict the futue vlues V of goup nd extended futue q-dimensionl vlues y j of goup. We will illustte the pocess of nding the functionl, such s pedictive density, estimto nd intevls nd quntiles of the futue vlues y j. Fom Section 3., we obtin f(y (k [ 1 (y (k j 1 ) (y (k j ½ h, Y j ) 5 Const exp l.1 C 1 j )] 1.1 l.1 (30) k k whee h 5 (s,, q, k ), l.1 5 xs + C 1C 1 11 (Y (k j X js ), C.1 5 C C 1C 1 11 C 1, q ½ J yj ½ 5 P k 5 1 y k 1 jk. Pediction fo the functionl y j follows fom the pedictive density f(y j ½ Y j ) 5 ò f(y j ½ h, Y j )p (h ½ Y) dh (31) This density cn be ppoximted by M onte Clo integtion fom the M CM C sm ples fã (y j ½ Y j ) 5 Ç 1 s 5 1 f(y j ½ (k,s), Y j ) (3) whee h (k,s) is the vite of h dwn in the kth itetion nd sth epliction of the M CM C smple. The men of this pedictive distibution is computed fom E(y j ½ Y) 5 E(E(y j ½ h, Y j ) ½ Y) (33) To evlute the inne expecttion in eqution (33), let us conside two cses. Cse 1 is k ¹ 0. It follows fom equtions () nd (9) tht whee y (k j 5 l.1 + n, n ~ N q (0, y j 5 (y (k j k + 1) 1/k (34) C.1 ). Theefoe, we hve y j 5 [(l.1 + n )k + 1] 1/k (35) Cse is k 5 0. We cn deive the following eqution by simil method: y j 5 exp(l.1 + n ) (36) whee l.1 5 xs + C 1 C 1 11 [log(y j) X js l] nd exp() 5 (exp( 1 ),..., exp( q))â fo 5 ( 1,..., q )Â. Let y (k,s) j denote the functionl in eqution (35) evluted t the kth itetion (k,s) nd sth epliction of the smples, noting tht n is geneted fom the N q (0, (k,s) C (k,s).1) distibution in ech of the MCM C smples. Theefoe,

12 330 J. C. Lee & K.-C. Liu when k ¹ 0, we pedict y j by yã j 5 1 s 5 1 y (k,s) j (37) Altentively, we cn lso pedict y j using the medin of the M CMC smples yä j 5 medin({y (k,s) j } s 5 1 ) (38) 5 k 5 k 5 5 k ¹ Pediction intevls nd quntiles of the functionl y j cn be computed similly fom the smples y (k,s) j, s 1,...,. Likewise, when 0, we cn use the sme method to pedict y j. Theoeticlly, the posteio pobbility of the cse 0 is 0. Howeve, we hd to pogm the two cses (k 0 nd 0) of the foecst ule septely to void n ove ow poblem. The foecst of the futue vlues V cn be done quite similly, so is omitted. 5 Num eicl illusttion 5.1 Applictions of technologicl substitutions with concu ent shot time seies In this section, we will pply eqution (1), s speci ed in Section 1, to set (egion B) of telephone switching dt studied by K emids nd Lee (1990). These dt e obtined fom egion in the USA consisting of ve sttes. Since, in the e of technologicl substitutions, the logistic gowth cuve is most popul, we will estict ou ttention to this pticul model in which the vible Y jt is de ned s Y jt 5 F j (t) /[1 F j (t)] (39) whee F j (t) denotes the new technology penettion t time t of mesuement j. The technology penettion is the fction of the totl numbe of new technology uses divided by the totl numbe of new nd old technology uses. Theefoe, the genel gowth cuve model with powe tnsfomtion s descibed in equtions (1), () nd (4) cn be pplied fo technologicl substitutions. It is noted tht this dt set hs been studied cefully by Kemids nd Lee (1990). Fo this ppe, we will estict ou ttention to the specil sitution in which 5 1, i.e. thee is only one goup, nd pply the tnsfomtion in eqution () to Y jt de ned in eqution (39). Next, s q 5 1, we show in Fig. 1 compison of ppoximte nd exct pedictive densities of y, the lst ye in stte 5, given the entie dt except the lst ye in tht stte s ou smple Y. Note tht the exct pedictive density of y is obtined by integting out w..t. q nd k vi numeicl integtion s given in eqution (3). Figue exhibits tht the joint posteio density of q nd k given Y is well concentted nd nely symmetic. H ence, the ppoximtion given by eqution (17) fo the pedictive distibution y should be quite dequte. Fo the esults of Byesin infeence vi M CM C smpling, Tble 1 lists the estimtes of the stndd devition, nd the pecentiles fo ech of the pmetes fo the entie dt with Y jt given in eqution (39) except ye 1 of ll sttes. The Byes estimtes e computed fom the M CM C smples with 50 itetions nd 5 eplictions. M oeove, 50 loops e cied out in ech M H lgoithm. The stting points fo the eplictions fo ech pmete e chosen fom ndom petubtions ound the mximum likelihood estimtes s developed in Appendix

13 B yesin nlysis of genel gowth cu ve model Appoximte pedictive density of Y Exct pedictive density of Y MCMC ppoximtion FIG. 1. Compison of exct nd ppoximte pedictive densities of y given Y. Hee, Y is the ye 1 in stte 5 given the entie dt set except the lst ye in tht stte s the smple. P(ho, lmd/y) lmd ho FIG.. Posteio of q nd k given Y. The smple is the sme s in Fig. 1. TABLE 1. MCMC ppoximtions fo switching dt Men SD.5% 5% 5% 50% 75% 95% 97.5% s q k.146, 0.143,.419,.370,.4,.14,.054, 1.96, 1.863,

14 33 J. C. Lee & K.-C. Liu TABLE. MCMC ppoximte pedictive mens, the stndd devitions, nd the pecentiles fo ve sttes when foecsting the penettion of ye 1 Stte Men SD.5% 50% 97.5% A. The convegence of the M CMC smples is monitoed by exmining thei empiicl quntiles nd the mesue, Î Ã, poposed by Gelmn nd ubin (199). It is noted tht the choice of stting points cn e ect the speed of convegence fo the pmetes. Theefoe, dequte stting vlues will ccelete the te of convegence fo the estimtion of pmetes. The M CM C posteio intevls cn esily be obtined fom Tble 1. Fo exmple, the 95% intevls e ed fom the.5% nd 97.5% columns. By the M CM C smples mentioned bove, we cn lso obtin the ppoximte conditionl pedictive density of y s developed in Section 4. Figue 1 shows compison of exct density, M CM C ppoximtion nd the simple Byesin ppoximtion given by eqution (17) fo the pedictive density of y. It is cle tht the ppoximtion vi M CM C smpling is bette thn the simple Byesin ppoximtion given by eqution (17), lthough eqution (17) pefoms quite well. The o± Blckwelliztion ppoximte pedictive density, which is given by eqution (11), is expected to give excellent ppoximtion s well. Tble lists the MCM C ppoximte pedictive mens, the stndd devition nd the pecentiles of the penettion fo ech of the ve sttes when ye 1 is being foecst. Figue 3 shows compison of 95% pedictive intevls fo the penettion of ye 1 by thee Byesin methods: simple ppoximte, M CM C nd exct, when the dt consist of the entie dt set except ye 1 of ll sttes. It is cle tht the intevls vi the simple ppoximte Byesin method hve slight upwd-shift fom those poduced by M CM C smpling. M enwhile, the intevls vi MCM C smpling e lmost the sme s the exct. 5. Simultions In this subsection, we will pesent some simultion esults using coss-vlidtion ( pedictive smple euse o `leve-one-out ), vi the simple ppoximte Byesin nd M L methods, fo the compison of 95% pedictive intevls of y given Y, the simulted dt except the mesuement of the lst vlue being foecst. Hee, we set p 5 6, q 5 1, , s 5 ( 1., 0.)Â, 5 0.0, q , k 5 0.7, N 5 5, 10, 0, nd the numbe of epliction g This gives N pedicted intevls fo the lst N obseved vlues in ech dt set. Ovell, thee e 50 3 N pedicted intevls to be comped with 50 3 N ctul obsevtions fo ech method. Tble 3 lists compison of covege pobbility fo N 5 5, 10 nd 0. It is cle tht the simple ppoximte Byesin method is much bette thn the M L method, becuse the pecentge of the Byesin intevls coveing the tue vlues is close to 0.95 thn ML intevls fo ech of the thee situtions. It is noted tht the covege pobbility by the M L method will be close to 0.95 s N inceses. Theefoe, the simple ppoximte Byesin method tends to povide moe elible

15 B yesin nlysis of genel gowth cu ve model el dt Appoximtion Exct MCMC Mesuement FIG. 3. Compison of 95% pedictive intevls fo the penettion of ye 1. The dt consist of the entie dt set except ye 1 of ll sttes. TABLE 3. Compison of covege pobbilities fo the pediction of y N Covege pobbility Byesin ML N N N Hee, p 5 6, , s 5 ( 1,,0.)Â, 5 0.0, q , k 5 0.7, q 5 1 nd no. of eplictions pedictive intevls thn the M L method when the smple size is modete o smll. Bette esults cn be obtined by using M CM C smples. 6 C oncluding em ks The Byesin methods pesented in this ppe, including simple ppoximte Byesin nd Byesin vi M CM C smpling, povides ltentive wys of deling with the genel (blnced o unblnced) gowth cuve dt when the seil covince stuctue holds, while pplying the Box± Cox powe tnsfomtion on the obsevtions. The model s speci ed in equtions (1), () nd (4) cn be vey useful in foecsting technologicl substitutions fo concuent shot time seies.

16 334 J. C. Lee & K.-C. Liu The seil covince stuctue (A(1) dependence) is de nitely n impotnt dependence stuctue fo the unblnced epeted mesues. It is noted tht the Byesin methods pesented in this ppe povide supeio wys of constucting moe elible pedictive intevls nd egions fo the futue vlues. M enwhile, the foecst ccucy fo the futue vlues vi the simple ppoximte Byesin method is bette thn the M L method. M oe ccute ppoximtion cn be obtined fom M CM C smples. Futhemoe, the computtions involved e not diý cult nd do not tke much unning time. It is noted tht ll computing esults in this ppe e conducted in the S-plus envionment. Finlly, it is fi to sy tht the poposed ppoximte Byesin methods with powe tnsfomtion nd A(1) dependence covince stuctue should be quite useful fo pctitiones in deling with foecsting technologicl substitutions with concuent shot time seies, s well s with othe dt in which the tnsfomtion is helpful. The esults e useful fo the sitution in which no tnsfomtion is needed s well, becuse it will simply be specil cse. EFEENCES BOX, G. E. P. & COX, D.. (1964 ) An nlysis of tnsfomtion (with discussion), Jounl of the oyl Sttisticl Society, Seies A, 6, pp. 11 ± 5. CASELLA, G. & GEOGE, E. I. (199) Explining the Gibbs Smple, Ameicn Sttisticin, 46, pp. 167± 174. CHIB, S. & GEENBEG, E. (1995) Undestnding the Metopolis± Hstings lgoithm, The Ameicn Sttisticin, 49, pp. 37± 335. FEAN, T. (1975 ) A Byesin ppoch to gowth cuves, B iometik, 6, pp. 89± 100. GEISSE, S. (1970) Byesin nlysis of gowth cuves, Snkhy, Seies A, 3, pp. 53± 64. GEISSE, S. (1980) Gowth cuve nlysis. In: P.. KISHNAIAH (Ed.), Hndbook of Sttistics, Vol. 1, pp. 89 ± 115 (Amstedm, Noth-Hollnd). GELFAND, A. E. & CALIN, B. P. (1993) Mximum likelihood estimtion fo constined o missing dt models, Cndin Jounl of Sttistics, 1, pp. 303± 31. GELFAND, A. E. & SMITH, A. F. M. (1990) Smpling bsed ppoches to clculting mginl densities, Jounl of the Ameicn Sttisticl Assocition, 85, pp. 398± 409. GEMAN, S. & GEMAN, D. (1984) Stochstic elxtion, Gibbs distibutions nd the Byesin estotion of imges, IEEE Tnsctions on Ptten Anlysis nd Mchine Intelligence, 6, pp. 71± 741. GILKS, W.., ICHADSON, S. & SPIEGELHALTE, D. T. (Eds) (1996 ) Mkov Chin Monte Clo in Pctice (London, Chpmn nd Hll). G IZZLE, J. E. & A LLEN, D. M. (1969 ) Anlysis of gowth nd dose esponse cuves, B iometics, 5, pp. 357± 381. HASTINGS, W. K. (1970) Monte Clo smpling methods using Mkov chins nd thei pplictions, B iometik, 57, pp. 97 ± 109. JENNICH,. I. & SCHLUCHTE, M. D. (1986) Unblnced epeted mesues models with stuctued covince mtices, B iometics, 4, pp. 805 ± 80. KEAMIDAS, E. M. & LEE, J. C. (1990) Foecsting technologicl substitutions with concuent shot time seies, Jounl of the Ameicn Sttisticl Assocition, 85, pp. 65± 63. KHATI, C. G. (1966) A note on MANOVA model pplied to poblems in gowth cuves, Annls of the Institute of Sttisticl Mthemtics, 18, pp. 75 ± 86. LEE, J. C. (1988) Pediction nd estimtion of gowth cuve with specil covince stuctues, Jounl of the Ameicn Sttisticl Assocition, 83, pp. 43 ± 440. LEE, J. C. & GEISSE, S. (197) Gowth cuve pediction, Snkhy, Seies A, 34, pp. 393 ± 41. LEE, J. C. & GEISSE, S. (1975 ) Applictions of gowth cuve pediction, Snkhy, Seies A, 37, pp. 39± 56. LEE, J. C. & LU, K. W. (1987) On fmily of dt-bsed tnsfomed models useful in foecsting technologicl substitutions, Technologicl Foecsting nd Socil Chnge, 31, pp. 61± 78. LJUNG, G. M. & BOX, G. E. P. (1980) Anlysis of vince with utocoelted obsevtions, Scndinvin Jounl of Sttistics, 7, pp. 17± 180. M ETOPOLIS, N., OSENBLUTH, A. W., OSENBLUTH, M. N., TELLE, A. H. & TELLE, E. (1953)

17 B yesin nlysis of genel gowth cu ve model 335 Equtions of stte clcultions by fst computing mchines, Jounl of Chemicl Physics, 1, pp. 1087± 109. POTTHOFF,. F. & OY, S. N. (1964) A genelized multivite nlysis of vince model useful especilly fo gowth cuve poblems, B iometik, 51, pp. 313± 36. AO, C.. (1967) Lest sques theoy using n estimted dispesion mtix nd its ppliction to mesuement of signls. In: L. M. LECAM & J. NEYMAN (Eds), Poceedings of the Fifth B ekeley Symposium on Mthemticl Sttistics nd Pobbility (Vol. 1), pp. 355± 37 (Bekeley, Univesity of Clifoni Pess). AO, C.. (1987) Pediction of futue obsevtions in gowth cuve models, Sttisticl Science,, pp. 434± 471. VON OSEN, D. (1991 ) The gowth cuve model: eview, Comm. Stt., 0, pp. 791 ± 8. ZELLNE, A. & TIAO, G. C. (1964 ) Byesin nlysis of egession model with utocoelted eos, Jounl of the Ameicn Sttisticl Assocition, 59, pp. 763± 778. Appendix A: The M L estim tes of the pmetes The M L estimtes of pmetes s, nd sã 5 ( 1 à 5 [ n N * ) 1 X j C à 1 j X j j j ½ b ½ ),, b 5 1,..., p j nd q Ã, k à 1,..., k Ã, mxi- with n de ned in eqution (7), C à j 5 mize the po le likelihood function L mx (q, k 1,..., k ) 5, q nd k denoted s sã, Ã, q à nd kã e (Y (k à j j X jsã  Cà j (q à X j C à 1 j Y (k à j (A1) )] (Y (k à j X jsã (A) [ à (q, k 1,..., k )] n/ (1 q ) (n N) / ½ J ½ (A3) à k k à q à k à q k k whee J, the Jcobin of the powe tnsfomtion, is de ned in eqution (7), nd (q, 1,..., ) is the given in eqution (A) with, 1,..., kã eplced by, 1,..., espectively. Appendix B: Pediction of y j given Y with the M L m ethod W ith the sme ssumption s descibed in Section 3., we hve s ou pedicto fo y j, given Y yã j 5 {1 + k à [xsã + C à 1 C à 1 11 (Y (k à j X jsã )]} 1/k Ã, k à ¹ 0 5 exp{xsã + C à 1 C à 1 11 [log(y j) X jsã ]}, k à 5 0 (B1) whee sã is given in eqution (A1), 1 5 (1,..., 1)Â, q 3 1. In eqution (B1), we use the convention tht b d 5 (b d 1,..., b d p )Â. In ddition to the point pediction, we cn obtin intevl pediction fo y j given Y. Fo q 5 1, we hve the ppoximte pedictive intevl yã j 6 z /à y (B)

18 l 336 J. C. Lee & K.-C. Liu whee z / is the 100 / pe cent point of the stndd noml distibution y 5 [hâ (l.1)] (C.1 + AW 1 A C 1 C 1 11 X jw 1 A ) A 5 x C 1 C 1 l.1 5 xsã + C 1 C 1 11 X j, W 5 j (Y (k à X j C 1 j X j j X jsã ), hâ (X ) 5 (1 + k X ) (1 k /k It is noted tht y is the vince of the foecst eo fo y j when the pmetes q, k nd e ssumed known, nd à y is its estimte obtined by substituting the M L estimtes fo the unknown pmetes. Appendix C: Pediction of futue penettion with the M L m ethod Fom equtions (39) nd (B1), we hve ou pedicto fo F j (t) the futue penettions, given by Fo q 5 Fà j(t) 5 yã jt/(1 + yã jt) (C1) 1, we hve the ppoximte pedictive intevl fo F j (t) Fà j(t) 6 z / à f (C) whee z / is the 100 / pe cent point of the stndd noml distibution It is noted tht f 5 [ gâ (l (k.1 )] A 5 x C 1 C 1 (k.1 5 xsã + C 1 C 1 (C.1 + AW 1 A C 1C 1 11 X jw 1 A ) 11 X j, W 5 j (Y (k à j X jsã ) X j C 1 j X j gâ (X ) 5 (1 + k X ) (1/k +1) [1 + (1 + k X ) 1/k ] f is obtined in mnne simil to y discussed in Appendix B.