TR/17/85 May Properties of statistical inference procedures for a gamma regression model. by A. M. Al-Abood and D. H. Young

Transcription

1 TR/7/85 May 985 Propertes of statstcal nference procedures for a gamma regresson model. by A. M. Al-Abood and D. H. Young

2 95938

3 Propertes of Statstcal Inference Procedures For A Gamma Regresson Model by D.H. Young and A.. Al-Abood SUMMARY A gamma regresson model wth an exponental ln functon for the means s consdered. Approxmatons to the percentles of the dstrbutons of the maxmum lelhood and weghted least squares estmators of the regresson coeffcents are presented and evaluated for the case of a sngle explanatory varable. These are used to develop approxmate confdence nterval and hypothess testng prcedures for the regresson coeffcents whch are assessed by smulaton. Fnally, the null dstrbuton propertes of goodness of ft tests for the exponental ln functon are nvestgated. CONTENTS. Introducton. Approxmatons to the dstrbuton of the ML estmator 3. Approxmatons to the dstrbuton of the LS estmator 4. Monte Carlo results 5. Confdence nterval estmaton 6. Tests of hypotheses concernng the regresson coeffcents 7. Goodness of ft tests for the exponental ln functon.

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5 . Introducton Let Y, Y,..., Y g represent g ndependent gamma random varables where Y has densty r r y r exp( r y /μ f (y, y > μ μ (r! (. where the shape parameter r s a nown postve nteger. The mean μ. s assumed to depend on the values x., x.,..., x for explanatory varables through the model μ exp (x β,,...g where x (, x,..., x and β ' ( β, β,..., regresson coeffcents wth unnown values. β Ths model has a number of mportant applcatons. s a vector of For example, consder accelerated lfe testng (Lawless (976 n whch there are (. g groups of tems, the th group contanng n tems and havng values x.,...,x for test varables. Suppose that there s type II censor- ng wthn groups and let Y (. < Y (, <... <Y (r represent the observed order statstcs n the th group, the remanng n - r observ- atons beng rght censored at the value Y (r.,... If the underlyng ds- trbutons are exponental and the means are gven by (., then the statstcs Y r r Y (n r Y (r, j (j +,..., are dstrbuted ndependently wth p.d.f's gven by (.. The {Y } are suffcent unbased and mnmum varance estmators of {μ.} and provde the basc observatons for a lelhood analyss. The model also arses n connecton wth the analyss of ntervals between events n g Posson processes, where the rates λ - depend on explanatory varables. If the observaton Y. for the th process denotes the tme from the orgn to the r th followng event, then Y has p.d.f. gven by (. wth μ λ r. The model wth a sngle explanatory varable s consdered by Cox and Lews (966 n the analyss of trend n a sngle Posson process. g (.3

6 3 Two well-nown methods of estmaton of the regresson coeffcents are maxmum lelhood and weghted least squares. The ML estmates are gven by the soluton of the + equatons r x r y exp( x ' βˆ r x r, r,,..., (.4 and an teratve soluton s requred. The LS method (Kahn (979 yelds the non-teratve soluton where βˆ w (x' D w x X' D w z D dag {ψ (r,, ψ (r },z log y +log r ψ(r, w X s (.5 the desgn matrx and where ψ(, ψ'( are the dgamma and trgamma functons, respectvely. Asymptotcally, the ML and LS have the same effcency as r.,,...g, but for small values of the {r }, some loss of effcency does occur n usng LS. Abood and Young (985 contrast the moment propertes of the estmators for small to moderate values of the {r } and propose modfcatons to the ML estmators leadng to bas reducton and mproved mean square error effcency. In ths report, we consder statstcal nference procedures based on the ML and LS estmates of. β The man objectve s to assess the performance of the procedures by means of a large scale smulaton nvestgaton for the case of a sngle explanatory varable. In Sectons and 3, approxmatons to the percentles of the dstrbutons of the ML and LS estmators of β for the case when β are presented. Monte Carlo results assessng the percentle approxmatons are gven n secton 4. Confdence nterval estmaton procedures are dscussed n secton 5 and test procedures for hypotheses concernng the regresson coeffcents are evaluated n secton 6. Fnally, null dstrbuton propertes of goodness of ft statstcs for testng the assumpton of the ln functon gven by (. are presented n secton 7.

7 . Approxmatons To The Dstrbuton Of The MLEstmator Under the gamma regresson model, we have 4 E L β r β s r x r x s I rs say, (. where L denotes the log-lelhood. Ths leads to the well-nown frst order expresson for the covarance matrx of B cova ^ β ( X ' D X I where D dag (r, r,., r g and where n the usual notaton, ((Irs I denotes the nformaton matrx. For 'large' values of the {r }, we have the ordnary normal approxmaton ^ β rr r approx N(βr, I, r,,..., (. (.3 where I rs denotes the element n the(r+st row and (s+st column of I. The ML estmators are asymptotcally unbased but wll show some bas for small values of {r }. Abood and Young (985 show that the ^ - bases br E(βr βr are to order R, where R r, gven by b r where K rst correcton we tae ^ β r r x approx I rs tu stu, r s t u r x s x t N( β r,,...,. Usng a normal approxmaton wth bas + b rr r, I, r,,..., ^ Statstcal nferences for the ß r based on the ML estmatorsβ r are made usng the dstrbuton propertes of the random varables ^ βr βr r,,..,,. These random varables are dstrbuted ndependently of β " and we have the exact dstrbuton result (.5 (.4 ^ ( ^ β β d r β r, r,,..., (.6 where ^ ( β r denotes the ML estmator of βr when β Approxmatons to the percentles of the dstrbuton of ^ βr ( are

8 5 needed for both hypotheses testng and confdence nterval estmaton. ^ β ( If we let b r (αdenote the α percentle of the dstrbuton of r the ordnary normal approxmaton s br (α rr / uα (I, r,,.., (.7 where u α denotes the α percentle of the N(, dstrbuton. The normal approxmaton wth bas correcton s ( α b u (I / b r r +, r,,.., (.8 α rr 3. Approxmatons To The Dstrbuton Of The LS Estmator The LS estmators are lnear functons of log-gamma random varables and hence have the advantage over the ML estmators that ther moments are nown exactly for all values of the {r }. In partcular, the covarance matrx s gven by ^ cov β w ( D L X V say x' (3. where Vrs x /ψ '(r r x s (3. Usng an ordnary normal approxmaton, we tae ^ βwr ^ β N( V rr wr βr,. d (3.3 rtng a Z, then from ahn (979 the sewness and r ^ urtoss coeffcents of βwr are 3 ( (3 ( ± (r a 4 r ψ (r ar ψ γ, γ r 3/ r (3.4 a ψ ( (r a r ψ((r r where Ψ(s ( denotes the sth dervatve of the dgamma functon and (± s the sgn of a r. These coeffcents may be used n an Edgeworth seres representaton to provde sewness and urtoss correctons to the normal ^ approxmaton to the c.d.f. of β wr.

9 6 ^ For the random varables β wr βr' r,,..., we have the exact dstrbuton result ^ ^ β d β ( wr β r wr (3.5 here ^ βwr ( denotes the LS estmator of Percentle of the dstrbuton of approxmaton s βr when β. Denotng the α ^ β ( wr by b wr ( α,the ordnary normal b wr ( α u α (V rr /, r,,...,. (3.6 The normal approxmaton wth sewness correcton s bwr rr / uα (uα 6,r r,, (3.7 ( α (V + γ, and the normal approxmaton wth sewness and urtoss correcton s ( / (u 3 b wr α (Vrr uα + (uα γ r + α 3uαγ r r 6 4,,... ( Monte Carlo Results In order to assess the varous approxmatons to the percentles ^ ^ of the dstrbutons of, β ( ( r and βwr we consder the partcular case when there s a sngle explanatory varable x, the means beng gven by μ exp(ß + ß x,,...,g. (4. thout loss of generalty we suppose that the x values are centred such that r x In ths case the results for the estmators tae on partcularly smple forms. For the ML estmators we have βˆ R Vara R bo,, b βˆ vara r rx 3 / x r x (4. (4.3 The LS estmators are

10 7 ˆβ wo { z /ψψ(r } { /ψ/ (r }, ˆβ w { x z /ψψ(r } { x /ψψ(r } (4.4 wth where var ( r, var(β ˆ D { /ψ/ (r } ˆ (βwo D x /ψ' w (4.5 D { / ψ/ (r } x /ψψ (r x/ ψ (r (4.6 The sewness and urtoss coeffcents of the LS estmators are then γ γ 3 ( ( (3 { } 4 ψ (r / ψ (r} ψ (r / ψ((r, γ 3/ ( / ψ (r / ψ/ψ( 3 ( ( 3 4 (3 ( 4 x ψ ( r / ( x ( / ( ψ r ψ r ψ r, γ 3 / ( ( x / ψ ( r x / ( r ψ (4.7 (4.8 In order to assess the accuracy of the percentle approxmatons for ˆβ and ˆ r βwr ' a large scale smulaton nvestgaton was made for the case when the explanatory varable x has equally spaced values wth x (g+,,...,g. Equal alues for the shape parameter were taen wth r r (((55 for,...,g wth g 5, and r r 6( for g 6(9. A smulaton run-sze of 4 was used. 3 For the gven values of x, we have x and hence the approx matng bas b and the sewness coeffcent γ are both zero. Hence the bas correcton and sewness correcton approxmatons wll only apply to b (α and b wo (α, respectvely. The broad conclusons from the nvestgaton are For the ML estmator βˆ (, the use of the bas correcton gves a mared mprovement n the approxmaton to the percentles.

11 8 The ordnary normal approxmaton tends to underestmate the upper percentles and to overestmate the lower percentles of the dstrbuton of βˆ ( for r, but gves satsfactory results for larger values of r. For the LS estmator ( βˆwo the use of the sewness correcton to the normal approxmaton appears to be worthwhle for small values of r and small values of the tal probabltes. v The ordnary normal approxmaton to the percentles of the dstrbuton of βˆ ( w wor satsfactorly for all values of r. These fndngs are llustrated n tables and whch gves the upper and lower percentles respectvely for βˆ ( and βˆ ( and n tables 3 and 4 whch gve the upper and lower percentles respectvely for βˆ ( wo and βˆ ( w, for the cases r (6(, g 5 and α.,.5,..

12 Table Upper percentles of th ML estmatorsβˆ ( ( andβˆ when g 5 and x. (g+, -,...,g. Actual tal probabltes assocated wth the percentles are shown n parentheses. r b ( App( (.4 (.49 (.6 (.65 (.64 (.68 (.76 (.76 App( (. (. (.3 (.98 (.93 (.95 (. (.98 b ( App( (.4 (. (.3 (.7 (.6 (.9 (.37 (.36 App( (.47 (.43 (.46 (.48 (.44 (.47 (.54 (.49 b o ( App( , (.3 (.4 (.5 (.43 (.43 (.48 (.6 (.7 App( (.75 (.85 (.63 (.75 (.83 (.83 (.5 (.8 b ( App( (.4 (.4 (. (.7 (.97 (.94 (.9 (.5 b( App( (.74 (.6 (.5 (.53 (.5 (.5 (.56 (.48 b ( App( (.53 (.78 (.48 (.5 (.95 (.95.8 (.3

13 Table Lower percentles for the ML estmators βˆ ( and βˆ ( when g 5 and x (g+,,...,g. r b ( App( , (.9 (.78 (.59 (.5 (.48 (.4 (.39 (.34 App( (.3 (.66 (. (.6 (.7 (.6 (.7 (.6 b ( App( (.45 (.6 (.93 (.87 (.85 (.79 (.78 (.75 App( (.78 (.66 (.59 (.6 (.58 (.58 (.53 (.57 b ( App( (.54 (.34 (.58 (.6 (.3 (.8 (.5 (.85 App( (.5 (.9 (.6 (.7 (.6 (.6 (.4 (.4 b ( App( (.5 (.7 (.6 (.7 (.7 (.9 (.98 (. b ( App( (.75 (.65 (.63 (.5 (.59 (.57 (.53 (.53 b ( App( (.3 (.9 (.6 (.4 (.5 (.5 (. (.3

14 Table 3 Upper percentles of the LS estmators βˆ ( w andβˆ ( w when g 5 and x (g +,,...,g. r b w ( App( ,86 (.8 (.89 (.94 (.93 (.87 (.94 (.95 (.98 App( b w (.95 App(3.6 App(3.7 (.9 (.97 (.4 (.96 (.9 (.96 (.4 ( ( ( ( ( ( ( ( ( (.49 (.5 (.49 (.54 (.48 (.5 (.53 (.5 b w ( , App( (.8 (.55 (.43 (.65 (.65 (.6 (.8 (.83 App( (.75 (. (.95 (.5 (. (.8 (.5 (.3 b w ( App( (.94 (.93 (.86 (.98 (.9 (.9 (.3 (. b w ( App( , (.53 (.47 (.44 (.47 (.46 (.45 (.53 (.47 b w ( App( , (.6 (.5 (.98 (.95 (.93 (.78 (.5 (.

15 Table 4 Lower percentles of the LS estmators βˆ ( w and βˆ ( w when g 5 and x (g+,,...,g, r b wo ( App( (.9 (.7 (.6 (. (.6 (.4 (.9 (O.5 App( (.96 (. (.97 (.99 (. (.97 (.4 (.3 b wo ( , App( (.6 (.58 (.54 (.55 (.58 (.57 (.53 (.55 App(3, (.46 (.46 (.47 (.47 (.5 (.5 (.48 (.5 b wo ( App( (.68 (.73 (.38 (.35 (.8 (.43 (.3 (.38 App( (.73 (. (.88 (. (.88 (.5 (. (.5 b w ( App( (.87 (.93 (.97 (.97 (. (. (.93 (,99 b w ( App( , (.46 (.5 (.56 (.47 (.49 (.58 (.5 (.5 b w ( App( (.9 (.8 (. (. (.8 (.5 (.3 (.3

16 3 5. Confdence Interval Estmaton In ths secton, we consder confdence nterval estmaton for the regresson coeffcents {β j }, based on the use of the ML and LS estmates. If the exact percentles of the dstrbutons ofβˆ j andwere βˆ wj were nown, confdence ntervals wth confdence coeffcents equal to the nomnal confdence coeffcents -α could be found. The two-sded central (-α% confdence ntervals for β j. based on the ML and LS estmators would be gven by ˆ ˆ ^ ˆ β j b j ( α, β j b j ( α, β wj b wj ( α, β wj b wj ( α ( 5. respectvely. For one-sded (l-α% ntervals, the lower confdence bounds for β j - would be ˆ β b ( α, ˆβ b ( α (5. j j wj j respectvely. Upper confdence bounds could be found smlarly. Snce the exact dstrbutons ofβˆ j andβˆ w are unnown, approxmate j confdence ntervals may be found usng the percentle approxmatons developed n sectons and 3. Based on the ML estmate, the ordnary normal approxmatng confdence nterval for B J s gven by ˆ β u (I jj /, ˆβ + u (I jj / (5.3 j α j α th bas correcton, the approxmate nterval s ˆ β / (I jj / j b j u (I jj, ˆ β j b α j + u α (5.4 Smlarly, usng the LS estmate the ordnary normal approxmatng confdence nterval for β j. s ˆ β jj / u (V jj / wj u (V, ˆβ α wj + α (5.5 th sewness correcton, the nterval becomes

17 4 ˆ β wj (y / (u + γ (u, ˆ + / + jj j β wj (V jj u γ j (u α α 6 α 6 α (5.6 where γ j. s gven by (3.4. Kurtoss correcton could also be appled but wll not be consdered here as the correcton led to lttle mprovement n the percentle approxmatons. For one-sded (-α% confdence ntervals, the ordnary normal approxmatons to the upper confdence bounds are βˆ jj j + u α (I /, βˆ wj + u α (V jj / (5.7 for ML and LS estmaton, respectvely. th bas correcton for the ML estmators and sewness correcton for the LS estmators, the approxmate upper confdence bounds are + jj / + + j b j u α (I, βˆ wj (v jj / u α γ j (u α 6 βˆ (5.8 respectvely. hen assessng propertes of approxmatng confdence ntervals two propertes are partcularly mportant, namely ther average wdth and the devaton of the actual confdence coeffcents from ther nomnal values -α. In the present case, the wdths of the confdence ntervals are non-random and for estmaton of ß j. are j u α (I jj /, zj u (v jj, α 3j (v jj / {u + α γ j (u 6 α for ML estmaton, for LS estmaton wthout sewness correcton and for LS estmaton wth sewness correcton, respectvely. The ratos of the wdths of the confdence ntervals are therefore } j /w j (V jj /I jj /, w 3j /w j for j, respectvely. (v jj /I jj / { + γ j (u 6 α u } α (5.9 e now consder the specal case of a sngle explanatory varable g wth r r, -,.,g and x. e have

18 and w 5 w 3 {rψr (r} / w w (5. (r} / ( / ( 3/ w 3 /w {r ψ ' { + ψ (r(u u / [ g { ψ (r} ] 6 α α Table 5 shows values of { rψ '(r} / and 3/ for r (4(8(4, g 5, and α.,.5,.. The results show that for r 5, the wdths of the confdence ntervals based on LS wthout sewness correcton are consderably larger than those based on ML estmaton. Ths property s much less mared when sewness correcton s appled and when a s very small LS leads to a small reducton n the wdths of the confdence ntervals.. (5. Table 5 Rato of wdths of approxmate confdence ntervals for ß and ß based on ML and LS estmaton r 3 / : g 5 3 / { rψ ' ( r } α..5. α..5. : g Values of the actual confdence coeffcents as estmated by the smulaton nvestgaton descrbed n secton 4 are shown n tables 6 and 7 for estmaton of ß and ß, respectvely, for the two-sded case wth nomnal confdence coeffcents -α.9,.95,.99. Tables 8 and 9 show the correspondng coeffcents for the one-sded case. For ß, bas and sewness correctons are examned but for ß these correctons are zero under the gven confguraton of values for the sngle explanatory varable.

19 6 The broad conclusons reached from the results n tables 6-9 are as follows. ( For two-sded confdence nterval estmaton of ß, the ordnary normal approxmaton based on ML leads to confdence coeffcents systematcally smaller than the nomnal values, and the bas correcton leads to a worthwhle mprovement n the control of the confdence coeffcent. Usng LS estmaton sewness correcton does not lead to any mprovement, and the overall control of the confdence coeffcent s better for LS than for ML. ( For two-sded confdence nterval estmaton of ß, the procedure based on LS estmaton gves a slghtly better performance than that based on ML estmaton, for very small values of r. In general, both methods provde, very satsfactory results. ( For one-sded confdence nterval estmaton of ß, the procedure based on the ordnary ML estmator leads to confdence coeffcents whch are much larger than the nomnal values, partcularly for small values of r. th bas correcton, excellent control over the confdence coeffcents s obtaned. For the LS procedures, sewness correcton leads only to a margnal mprovement, but the control of the confdence coeffcent s very good. (v For one-sded confdence nterval estmaton of ß the LS estmaton procedure has a slghtly better performance than ML for r,. For larger values of r, both methods provde excellent control over the confdence coeffcent.

20 7 Table 6 Estmated confdence coeffcents for approxmate (-α% two-sded confdence ntervals for ß based on ( ordnary ML, ( ML wth bas correcton, ( ordnary LS, (v LS wth sewness correcton. g 5 -α.9 r ( ( ( (v α.95 ( ( ( (v α.99 ( ( , ( (v α.99 r 3 g ( ( ( (v α.95 ( ( ( (v α.99 ( ( ( (v

21 8 Table 7 Estmated confdence coeffcents for approxmate (-α% two sded confdence ntervals for ß based on ( ML, ( LS g 5 -α r ( ( ( ( ( ( g - α r ( ( ( ( ( (

22 9 Table 8 Estmated confdence coeffcents for approxmate (-α% one-sded confdence ntervals for ß based on ( ordnary ML, ( ML wth bas correcton, ( ordnary LS, (v LS wth sewness correcton g 5 -α.9 r ( ( ( (v α.95 ( ( ( (v α.99 ( ( ( (v g -α.9 r ( ( ( (v α.95 ( ( ( (v α.99 ( ( ( (v

23 Table 9 Estmated confdence coeffcents for approxmate (-α% one-sded confdence ntervals for ß based on ( ML, ( LS g 5 -α. r ( ( ( ( ( ( g -a r ( ( ( ( ( ( Tests Of Hypotheses Concernng The Regresson Coeffcents In regresson problems we are often nterested n testng the hypothess that a partcular subset of the explanatory varables have no effect. thout loss of generalty, we shall tae the subset to contan the last -l varables, so that we wsh to test the hypothess H : ß j for j l+,l+,...,. e shall wrte β ' (β', β ' where β' (β, β,...β l, β' (β l+, β l+,..., β. (6.

24 hen H : β s true,we have μ exp( β,,..., g x (6. whch, we shall refer to as the restrcted model. The model µ I. exp( β, x,..,g wll be called the full model. e frstly develop tests based on the maxmum lelhood estmators. The log-lelhood under the restrcted model s g L( β c r (log μ μ y + (6.3 where g g g c r logy log (r! logr + (r s a constant not dependng on β e let β ^ and μ ˆ exp( x ˆ β denote the ML estmates of β and μ under the full model. The estmates are gven by the soluton of the + equatons g g ' β r x, j,,...,. r x j y exp( j x ˆ (6.4 Under the restrcted model, the ML estmate βˆ s gven by the soluton of the l+ equatons g ˆ g r β r x, j,,..., ( 6. 5 x j y exp( x' j l Settng ˆ μ exp( β,,...,g, the lelhood rato statstc x' ˆ for comparng the full and restrcted models s S g r x' ˆ ˆ ˆ ˆ (β β y (μ μ. ( The statstcs S s taen to be approxmately dstrbuted as x wth -l degrees of freedom f H o s true. The statstc S taes on a partcularly smple form when the g x's are centred such that r x for j,...,. Under ths j condton use of the frst lelhood equaton n (6.4 gves

25 r y ˆ μ R, L(β c R(β ˆ ˆ + (6.7 Smlarly, use of the frst lelhood equaton n (5.5 gves r y μˆ R, L(β c R(β ˆ ˆ + (6.8 where βˆn denotes the ML estmate of ß under the restrcted model. Hence we may wrte S R(β ˆ ˆβ - Al-Abood and Young (985 show that the bas of βˆ to order R s -(+/(R when the centerng condtons for the x's hold. Ths result also holds when H o s true. Under H, the bas of βˆ s -(l+/(r to the same order of approxmaton and hence to order ( (6.9 E(S - l (6. hch agrees wth the frst moment of the approxmatng ch-square Dstrbuton e now consder the specal case when l - and we are testng H ( : ß. In ths case, S has approxmately a non-central x dstrbuton wth degree of freedom and non-centralty parameter λ β /I K (6. where I s the (+st element n the dagonal of the nverse of the nformaton matrx. The approxmate test procedure usng a double-taled test wth sgnfcance level α s reject ( H f S > X ( α (6. where x v ( α denotes the upper α% pont of the X v dstrbuton. If we let Y (S denote the power of the test based on S, then usng the results that S s aproxmately dstrbuted as U where U N(λ, and that X ( α u, we obtan the power approxmaton α γ (S φ u β (I + φ u β (I α α. (6.3

26 hen ß 3, ths power s α as requred. An alternatve test procedure can be made by tang βˆ β / I to be approxmately dstrbuted as N(,. To test H ( : ß, aganst the two-sded alternatve ß,, we use the test statstc. Z βˆ /(I and reject ( H f z > U (6.4 α The power of the test s γ (z p Z β (I > u β (I α - P Z β (I u β (I + < α - φ u β (I + φ u α β (I - α. (6.5 Asymptotcally (r -,,...,g the powers of the S and Z tests are therefore equvalent. e now develop tests whch utlse the weghted least squares estmators whch are derved from the lnear model representaton where Z Z x ' β logy - ψ(r. + logr and +,,...,g (6.6 E(ε, var(ε - ψ'(r, cov(ε,ε j. (6.7 for j,,...,. In matrx notaton we have Z X β + E, E( cov( D ε, ε (6.8 w here D dag{ Ψ' (r,..., ψ' } w (rg Under the full model, the LS estmator βˆ w s obtaned as the value of β whch mnmses R (Z β ' D ( Z β z w x and the soluton s gven; by (.5. The generalsed resdual sum of squares about the ftted full model s

27 4 rtng Z X ˆ β ( X M X' D where M I w w z here A D w D w X M R( ˆβw (Z D (Z β. 6.9 x ˆ β ' w x ( w R( βˆ X' D w z ' A. Hence Z X D w X, we obtan (6. R( ˆ β E (β' X ' A ( β w ε x ε + + E( A ε' ε tr(a D w tr( I g D w X M X'. Snce tr( g and tr (D X M X ' tr( I g w I + +, we have E R(β ˆ w g Ths result holds for all β and hence n partcular when H : β s true. Smlarly, for the restrcted model Z X β + ε the generalsed resdual sum of squares about the LS ftted model s R( ˆβ (Z x β ' D (Z X β ˆ ˆ w w w w (6. (6. here ˆβ M X ' D w ( D x' w X, X X' we may wrte D w Z. Settngs A D w D w X M R( ˆ β Z' A Z. w e Have E{R( ˆβ β' A' X β+ E( A wl ε' ε β' X' A X β+ g l A straghtforward calculaton shows that X' D w where (6.3 (6.4 β ' X' A X β β' X' A X β. (6.5

28 5 Hence we have the exact expectaton results E{R( ˆβ w H } g l. (6.6 Usng the extra sum of squares prncple, tc for testng H s a sutable test stats- S R(β ˆ w R(β ˆ w (6.7 for whch we have the exact expectaton result E(S l + β ' X' A X β. ( 6. 8 Ths gves E(S - l when H s true. The exact dstrbuton of S s unnown and an approxmaton s requred. Tang the {ε } whch are ndependently dstrbuted as loggamma random varables to be approxmately dstrbuted as N(, ψ' (r, we obtan the approxmaton S approx x ' (β' X ' A X β (6.9 where X ' v (λ represents the non-central ch-square dstrbuton wth v degrees of freedom and non-centralty parameter λ. e now consder the specal case when l. - and we are testng ( H : β. In ths case S s approxmately dstrbuted as noncentral x wth degree of freedom and non-centralty parameter λ w β /V (6.3 where V s the (+st dagonal element n the nverse of X' D X. w he test procedure s reject H ( f S > X ( α (6.3 where X V ( α s the upper α% pont of the dstrbuton of The approxmate power of the test s X V

29 6 γ (S φ u β (V + φu α β (V. (6.3 Z An alternatve test procedure s to use the test statstc ˆβ /(V w and reject ( H f Z > u. α (6.33 The exact powers of the tests based on S and Z are equal so a choce of test can be made on grounds of computatonal smplcty. In order to examne the power propertes of the tests based on the S, S, Z and Z statstcs and to assess the adequacy of the approxmatng powers gven by ((6.3,(6.5,(6.3 and (6.34, a smulaton nvestgaton has been made for the case of a sngle explanatory varable when the means {µ.} satsfy the model defned n (3.. Equal values for the shape parameters were taen wth r r (( for,...,g wth g 5,. Equally spaced values x - (g+ were used for the explanatory varable. Values ß logθ/(g- were used gvng max µ /mn µ. θ, for θ (5. The run-sze was n each case. The broad conclusons reached from the nvestgaton are ( The use of the S and S tests lead to excellent control over the sgnfcance levels for all values of r. The actual sgnfcance levels of the Z -test are much larger than the nomnal values for r,,3 but are satsfactory for larger values of r. ( For the very small values of r the power of the Z -test s greater than that of the S -test but ths seems to smply reflect the dfferences n the actual sgnfcance levels of the tests. The power dfferences between the two tests are very small for r> 3. ( The power performance of the S -test s maredly better than that of the S -test for r, but the power advantage dmnshes rapdly wth ncreasng values of r, (v The power approxmaton gven by (6.3 gves a slght overestmaton of the power, partcularly for large values of θ and small values of r. However, the results are generally very encouragng.

30 7 These fndngs are llustrated n tables,, whch show the estmated powers as obtaned by smulaton together wth approxmatng powers for the S, S and Z tests respectvely for g 5 and nomnal sgnfcance levels α.,.5,.. Table Powers of the S -test for g 5 and nomnal sgnfcance levels α as obtaned by ( smulaton, ( approxmaton (6.3 θ r ( ( ( ( ( ( ( ( ( ( α α α Table Powers of the S -test for g 5 and nomnal sgnfcance level α as obtaned by ( smulaton, ( approxmaton (6.3 θ r ( ( ( ( ( ( ( ( ( ( , α α , α

31 8 Table Powers of the Z -test for g 5 and nomnal sgnfcance level α as obtaned by ( smulaton, ( approxmaton (6.5 θ r ( ( ( ( ( ( ( ( ( ( α α , ,9 α , Goodness Of Ft Tests For The Exponental Ln Functon Fnally, we consder tests of ft for the assumed exponental ln functon for the means as gven by (. aganst general alternatves. Two tests are examned, the frst utlsng the ML estmates and provdng the lelhood rato test, the second test beng based on the LS estmates. hen no model s mposed on the means {µ.}, the ML estmates are μˆ Y,,...,g, Usng (6.3 and the frst equaton n (6.4, the lelhood rato statstc for testng the exponental ln functon s D ' βˆ log X Y. th loss of generalty we shall assume that the x s re centred such that r x j, j,...,. In ths case, D aes the smple form

32 9 D (R ˆβ g r logy. (7. Usng the well-nown result for the expectaton of a log gamma random varable, we have E(logY β logμ + r ψ(r r f the exponental ln functon s correct. logr + r 4 + Settng (r - 6 (7. E( ˆβ β ( + /R we obtan g E(D g + r 6 gnorng terms of ( 3 and smaller terms r (7.3 The usual procedure s to refer the statstc D to the ch-square Dstrbuton wth g-- degees of freedom. The form of (7.3 Suggests the use of the modfed statstc D* D/(+c, where g C r 6(g (7.4 and to tae D* as approxmately dstrbuted as ch-square wth g- degrees of freedom f the exponental ln functon s correct. To assess the effect of the modfcaton, moments and crtcal values of the null dstrbutons of the statstcs D and D * have been estmated by smulaton for the model µ. exp(ß + ß x., wth x - (g+,,...,g, for r - r (( and g 5,. The results showed that the null dstrbuton of D* approaches the X g dstrbuton much more rapdly than the dstrbuton of D. Use of D* therefore leads to much better control over the sgnfcance level of the test for small values of the shape parameter. These fndngs are llustrated n table 3 whch shows the means and varances of D and D* and n table 4 whch shows the estmated upper %, 5% and % crtcal values of the null dstrbutons of D and D* for

33 3 r. r (5,,...,g and g 5,. The estmated sgnfcance levels assocated wth the ch-square approxmatng crtcal values are shown n parentheses. Table3 Means and varances of the D and D* statstcs when r r,,...,g g 5 g mean varance mean varance r D D* D D* D D* D D* g - X Table4 Upper % ponts of the null dstrbutons of the D and D* statstcs when r r,,.,g g 5 α. α.5 α. r D D* D D* D D* 7.5(, (.8 9.3( ( (.4.6(.63 7.(.3 6.4( ( (.46.5(.7.98( (. 6.5(.96 8.(.6 7.5(.44.36(.7.3( (.8 6.7(. 8.4(.6 6.6(.6 6.6(. 8.7( (.5.(.4.3( (.49.98(..8(.88 X α ( 3 x α. g α.5 α. r D D* D D* D D* 5.89(.8 3.5( ( (.4 3.8(.3 9.9( ( (. 7.7( (.5.79( ( ( (. 6.5( (.49.43(.7.4( ( ( 6 64( ( 55 65( 8 58( (. 3.66(.4 6.4( (.54.8( (.98 ( α

34 3 If the gamma regresson model s ftted by LS usng the transformed observatons Z logy. - ψ (r + logr,,...,g, a goodness of ft statstc s provded by R (ˆ βw whch was defned n (5.8. The statstc has exact expectaton g-- and ts dstrbuton approaches the x - dstrbuton wth g-- degress of freedom as the {r } ncrease, f the assumed model s correct. Although the means of the exact dstrbuton of R (βw ˆ and the approxmaton x -dstrbuton agree, the varances are not equal. To demonstrate ths, consder the case when the shape parameters are equal, that s, r r for,...,g. In ths case the {Z } are dentcally dstrbuted as log-gamma random varables wth sewness and urtoss coeffcents gven by γ (Z 3 ( ψ (r/ g {ψ'(r, γ (Z ψ (3 (r/ {ψ '(r} g (7.5 for I,..,g (ahn (979. The LS estmator s the same as the OLS estmator of β and usng results from atqullah (96 we have g var{r(β } {ψ'(r} Var z x'β w w g (g + γ (Z (h where h s the th dagonal element n the hat matrx X(X' e therefore have X X' (7.6 Var{R( ˆβ } (g ( + c (7.7 w where ψ (3 (r ( h c (7.8 { ψ' ( r }( g The form of (7.7 leads us to consder the modfed statstc R * (β ˆ w R * (β ˆ + b, where the constants a and b are selected to gve agreement w between the exact mean and varance of R * (β ˆ and the correspondng moments w of the approxmatng X g dstrbuton. Ths gves R * (β ˆ ( + c [R(β ˆ + (g + {( + C }]. (7.9 w w

35 3 To examne the adequacy of the ch-square approxmaton to the null dstrbutons of R( βˆ and R *(βˆ, the moments and crtcal values of the dstrbutons have been estmated by smulaton for the model µ. exp(ß +ß x. wth x. (g +,,...,g and r r ( (, g 5,, The results show that the mean and varance of the dstrbuton of R * (β ˆ, as obtaned by smulaton are very close to the correspondng moments of the approxmatng x - dstrbutons. The use of the modfed statstc R * (β ˆ, leads to better control of the sgnfcance levels for small values of α and small values of r, partcularly for the larger value of g. These fndngs are llustrated n table 5 whch shows the means and varances of R(ˆ β, and R *(βˆ, and n table 6 whch shows the estmated upper %, 5 % and % crtcal values of the null dstrbutons of R(ˆ β, and R * (β ˆ, for r.r (5,,...,g and g 5,. The estmated sgnfcance levels assocated wth the ch-square approxmatng crtcal values are shown n parentheses. Table 5 Means and varances of the R( βˆ and R *(βˆ statstcs when rr,,...,g, w w g 5 g Mean Varance Mean Varance r R( βˆ R *(βˆ R( βˆ R *(βˆ R( βˆ R *(βˆ R( βˆ R *(βˆ w w w w w w w w X g

36 33 Table 6 Upper % ponts of the null dstrbuton of the R( ˆ β and R * (β ˆ w w statstc when r r,,...,g g 5 α. α.5 α -. r R(ˆ β R * (β ˆ, R(ˆ β R * (β ˆ, R(ˆ β w w R * (β ˆ, w 6.5(. 5.64( ( ( (.8.97( ( ( ( ( (.9.93( (.9 6.9( ( (.45.97(.8.93( (.6 6.4(.95 8.( (.47.7(.4.44( (.4 6.6(.95 8.( (.48.94(.3.37(. X ( α g α. α.5 α. r R(ˆ β R * (β ˆ, R(ˆ β w R * (β ˆ, R(ˆ β w R * (β ˆ, w 4.96(.3.96( (.9 5.7( (.4.76(.9 4.4(.9.96( ( ( (.7.8( (.7 3.8( ( ( (.3.93( (.3 3.8( ( ( (.3.7( (.5 3.5( ( (.53.7(.8.63(. X ( α References Al-Abood, A.M. and Young, D.H. (985. Moment propertes of maxmum lelhood and least squares estmators for an exponental regresson model wth type II censorng. Techncal Report TR/4/85, Brune Unversty. Atquallah, M. (96. The estmaton of resdual varance n quadratcally balanced least squares problems and the robustness of the F-test. Bometra, 49, Cox, D.R. and Lews, P.A. (966. The statstcal analyss of a seres of events. London, Methuen. Kahn, H.D. (979. Least squares estmaton for the nverse power law for accelerated lfe tests. Appl. Statst. 8, No., Lawless, J.F, (976. Confdence nterval estmaton n the nverse power law model. Appl. Statst., 5, No., 8-38.