Which estimator of the dispersion parameter for the Gamma family generalized linear models is to be chosen?

Transcription

1 STATISTICS Dalarna Unversty D-level Master s Thess 007 Whch estmator of the dsperson parameter for the Gamma famly generalzed lnear models s to be chosen? Submtted by: Juan Du Regstraton Number: 8096-T084 Supervsor: Md.Moudud Alam Date: June 007

2 ABSTRACT For the Gamma famly generalzed lnear models, the dsperson parameter s contaned n the varance of the model parameter estmator. So t wll affect the results of statstcal nference or any knds of tests that refer to varance. Ths paper revewed several exstng estmators of dsperson parameter va the Monte Carlo experments to see whch one s to be preferred when the sample sze s dfferent. The smulaton results show that the bas corrected maxmum lkelhood estmator performs the best n comparson wth the other methods when the sample sze s small; n large sample sze all the estmate methods perform smlar. Keywords: Generalzed lnear models; Gamma dstrbuton; Dsperson parameter; Inference on model parameters.

3 Introducton ty and constancy of varance are no longer requred n Generalzed Lnear Models (GLIM) [7]. Abandon these strct assumptons modelng become realstc and have more general felds of applcaton. Exponental famly s a class of dstrbuton famly that GLIM deals wth. Gamma dstrbuton, whch belongs to exponental famly, s one of the commonly used dstrbutons n GLIM. It s assumed to deal wth responses whch are postve and contnuous. In general, we also suppose that these varables have constant coeffcent varaton. Gamma dstrbuton s appled n many felds, for example, t s commonly used n meteorology and clmatology to represent varatons n precptaton amount [0]. It also has been found practcal applcaton as a model n studes relatng to lfe, fatgue, and relablty characterstcs of ndustral product [9].. Member of exponental famly The conventonal probablty densty functon of Gamma dstrbuton can be expressed as: y f ( y αβ) = y y Γ α β β α ;, exp( ), 0,, 0 α ( ) αβ> () Where, α and β are called the shape and the scale parameters respectvely. A dfferent parameterzaton of Gamma dstrbuton s used n generalzed lnear models for some convenence. The form of Gamma dstrbuton used n GLIM can be presented as: ν ν νy ν ( ; μν, ) exp[ ] ; 0, ν 0, μ 0 f y = y y > Γ( ν) μ μ > () Compare the two expressons above we have α = ν, β = μ/ ν. In ths form the parameter ν s called shape parameter whch determnes the shape of the

4 dstrbuton. It can be shown that the above Gamma dstrbuton belongs to the exponental famly of dstrbutons (see Appendx A for detaled dervaton). The canoncal lnk s the recprocal form of the expectaton.e. g( μ) = = E( Y ) and the μ dsperson parameter has the form a ( φ ) =. ν. Role of the dsperson parameter Accordng to the asymptotc theory, the maxmum lkelhood estmator of β follows the normal dstrbuton: ˆ β N ( β, I ), where I s the Fsher s nformaton matrx. The correspondng varance s the dagonal elements of the nverse Fsher s nformaton matrx, whch s a functon of the dsperson parameter (see Appendx C). From the calculaton (see Appendx C) we see that estmate of the model parameter does not depend on the dsperson parameter. But the effect of the dsperson parameter s obvous when we perform any knds of statstcal nference about the model parameter. Snce the expresson of standard error of the model parameter Cov ˆ β = a φ X WX, where X s the desgn T contans the dsperson parameter, ( ) ( )( ) matrx and W s the dagonal weght matrx[7]. In addton, We can see that V( y) μ =, E( y) v = μ. So V( y) = E ( y) v or V( y) = a( φ) E ( y). Thus, the varance of the random varable s also affected by the dsperson parameter. We can denote the dsperson parameter asσ =. v In applcaton, dfferent statstcal packages provde dfferent default settngs for the dsperson parameter estmator n the generalzed lnear models procedure. For example, SAS provdes MLE as the default whle R provdes method of the moment estmator as the default. Therefore, t s mportant for a data analyst to know n whch case the default settng of the respectve software package s good enough and n whch case t does not and n that case whch alternatve s to be preferred.

5 Ths paper summarzes dfferent exstng methods of estmatng the dsperson parameter and compares these estmators n terms of ther precseness and applcablty n producng vald nference about model parameters by means of Monte-Carlo experments. Ths paper s organzed n the followng way: secton lsts the dfferent estmaton methods after lterature revew; secton 3 compares the smulaton results and secton 4 draws the conclusons from the comparson of the results. Lterature Revew In order to draw a vald statstcal nference about the model parameters, frst we have to obtan a good estmate of the dsperson parameter. There are many suggestons n lteratures on Gamma dstrbuton and ts parameters estmaton. Revewed those related lteratures we found four methods to estmate the dsperson parameter.. Maxmum Lkelhood Estmator (MLE) The method of Maxmum lkelhood estmate s one of the most useful methods to estmate model parameters. McCullagh and Nelder (989) apply ths method to estmate the dsperson parameter of the gamma dstrbuton. Ths estmator s denoted as D(6+ D) σ MLE = ˆ ν 6+ D (3) Where, D= D( y; ˆ μ)/ n and D( y; ˆ μ ) s the devance of the model. Ths estmator s based on devance. It s extremely senstve to roundng errors n very small observatons and n fact devance s nfnte f any component of y s zero. If gamma assumpton s false, v / does not consstently estmate the coeffcent of Is the dfference between the calculated approxmaton of a number and ts exact mathematcal value. 3

6 varaton [7]. It s well known that maxmum lkelhood estmates maybe based when the sample sze n or the total Fsher s nformaton s small. The bas s usually gnored n practce, the justfcaton beng that t s neglgble compared wth the standard errors [].. Bas Corrected Maxmum Lkelhood Estmator (BMLE) In small or moderate sze samples, however, a bas correcton can be apprecable []. Bas corrected maxmum lkelhood estmator s obtaned by ncludng the term of order n n the expected devance. McCullagh and Nelder (989) presented the form of the bas corrected maxmum lkelhood estmator of the dsperson parameter as, 6( n k) + nd σ BMLE = D ˆ ν 6( n k ) + nd (4) Where, D = Dy ( ; ˆ μ)/ ( n k) and n s the sample sze and k s the number of parameters..3 Moment Estmator (ME) The method of Moment s another most commonly used way to estmate the unknown parameters. The moment estmator of the dsperson parameter s gven as, χ σme = = y μ μ n k ˆ ν = n k ( ˆ) / ˆ /( ) (5) Where, χ s the Pearson s Ch-square statstc, n s the sample sze and k s the number of parameters. Here, σ s consstent forσ, f β has been consstently estmated. But t s ME neffcent, partcularly for small values of the shape parameter [0], ths knd of objecton plagues moment method s applcaton. In addton, unlke the normal model, the method of moment estmator of the dsperson parameter s not unbased for the 4

7 Gamma models..4 Quas-maxmum Lkelhood Estmator (QMLE) Stacy (973) presented a set of estmators for the parameters of Gamma dstrbuton usng the method of quas-lkelhood based on the complete sample sze. The estmator of the shape parameter s appeared as the form of the nverse. We know the nverse form s equal to the dsperson parameter (α = ν, σ he gave s demoted as, = ). The estmator that ν σ n ( ) n QMLE = ln ˆ z n z ν n = (6) Where, z = y / ny, y s the sample mean. In addton, Stacy (973) has three constrans on the random sample data: ) n> ) y y j for some values of and j. 3) y > 0, for =,..., n In hs estmaton, frst he gave the log-lkelhood of the generalzed Gamma dstrbuton, and then follows the steps of maxmum lkelhood method to get the estmators. A bref note on the mathematcal dervaton of Stacy s (973) method s gven n Appendx D. For the convenence of comparson accordng to the equaton (a) n maxmum lkelhood method (see Appendx B), we make lttle change and have the equaton as, Γ '( ν ) ˆ μ logν = n log (7) Γ( ν ) y When we compare equaton (c) (see Appendx D) wth (7), we can see that they are In fact, Stacy s Quas-lkelhood s dfferent from Quas-lkelhood used n GLIM. The latter s based on varables mean and varance to buld lkelhood. But Stacy s s just a maxmum lkelhood under generalzed form of the Gamma dstrbuton. 5

8 the same. Because ˆ μ n the above equaton s the expected response of the model whle t s replaced by to the arthmetc mean y p n the quas-lkelhood method. Changng a lttle bt of the above equaton we get, y Γ '( ν ) logν = n log ( ν ) ˆ (8) Γ μ In the begnnng of the ntroducton we have shown thatα = ν. Thus, n ths progress we notce that though the estmator n Stacy s(973) paper s dfferent from the method of maxmum lkelhood, when we see the steps of the calculaton or the fnal equaton whch s used to estmate the dsperson parameter, t s clear that they are the same. The only dfference s that the fnal form Stacy gves s the unbased one, nothng new. Thus, we can see that quas maxmum lkelhood s same to the maxmum lkelhood. So, there are three methods used n ths paper to estmate the dsperson parameter v= v= x x 0.6 v= v= x x Fgure The Gamma dstrbuton wth dfferent shape parameter under μ = 6

9 3 Comparsons of dfferent estmate methods 3. Comparson of estmate results In ths secton we apply these three methods to estmate the model parameters and the dsperson parameter by Monte Carlo experments. We assgn four dfferent true values of the shape parameter v = 0.5,,,5, whch s the nverse of the dsperson parameter. We choose these values because Gamma dstrbuton wth these shape parameters has dfferent shape, whch s shown s Fgure. Four dfferent sample szes we use s: n = 0,0,50,000 functon s the nverse lnk: g ( μ ) ; form of the lnk = ; and the lnear predctor s η = α + βx, μ where α = 0.5, β =. The Monte-Carlo results are based on 0 thousand teratons. All the smulatons (mean of the estmates of dsperson parameter and the correspondng confdence ntervals) have been carred out n R.4.0 and are tabulated n Table. From Table we can see that when true value of ν decreases from to 0., bas of the smulaton results of the dsperson parameter s dfferent (see Appendx E). For example, the maxmum lkelhood estmator bas s.3%, 0.76%, 0.3%, and 0.% respectvely for the dfferent dsperson parameter v =,,0.5,0. when sample sze s 000. In other word ths means that the smulaton results are affected by the value of dsperson parameter or the shape parameter. Comparng results on each column of Table, we see the method of bas corrected maxmum lkelhood s the best among these three. Ths s very obvous when the sample sze s small. For example, when n=0 andν =, we can see that the mean of MLE s.6359, BMLE s.9055, and ME s.485. The correspondng central 95% Monte-Carlo quantle ntervals are (0.5573, 3.0), (0.669, ) and (0.500, 3.835) respectvely. Here the smulated value under the bas corrected maxmum lkelhood method (.9055) s the closest to the true dsperson parameter 7

10 value ; even though t seems that ths estmator s stll not unbased. Wth the ncreasng of the sample sze the smulaton estmator s closer to the true value. When n=0 three dfferent estmators are.809,.9435, and.688, the correspondng confdence ntervals, of course, become shorter. Table Smulaton results of dsperson parameter /v Sample sze ˆ /ν MLE BMLE ME (0.5573, 3.0).9055(0.669, ).485(0.500, 3.835) 0.809(0.958,.8444).9435(.0379, ).688(0.7564, ) (.379,.5460).959(.3675,.654).8589(.0673, 3.94) (.896,.0946).958(.80,.0974).99(.709,.330) (0.540,.679) (0.309,.894) (0.73,.93) (0.453,.476) 0.983(0.494,.5889) 0.985(0.494,.79) (0.6497,.384) (0.679,.3578) (0.5777,.5748) (0.976,.0693) (0.99,.0709) 0.998(0.8833,.308) (0.83, 0.843) 0.490(0.463, ) (0.386,.0386) (0.68, 0.764) 0.496(0.390, 0.877) (0.77, 0.907) (0.376, ) (0.395, ) (0.3034, ) (0.4589, ) (0.4597, ) (0.4489, ) (0.0448, ) 0.98(0.0558, 0.49) 0.939(0.0547, 0.438) (0.084, 0.3) 0.99(0.093, 0.340) 0.964(0.0898, ) (0.46, 0.735) 0.998(0.96, 0.839) 0.985(0.54, 0.954) (0.830, 0.68) 0.000(0.833, 0.7) 0.000(0.84, 0.98) Note:. fgures n the parenthess present 95% quantle nterval.. The estmator tabulated n ths table s the mean of the 0 thousands smulatons. Ths results are true not only n small sample szes but also n the medum sample 8

11 sze, n=50. We can get the same concluson that estmator of bas correcton maxmum lkelhood s the most approprate estmator of the true value of dsperson parameter. But n large sample sze n=000, when true value s and, the smulaton results show that ME s better than estmator of other methods. For example, whenν =, value of MLE, BMLE, and ME s.9554,.958 and.99 respectvely. And when ν = dfference between ME and BMLE s , whch s not very large. But from the length of the 95% Monte-Carlo nterval we see that BMLE s s smaller than the other two methods. For the other two values of dsperson parameter ν = 0.5, 0. the smulaton results are smlar. Except the case n large sample sze, whenν =, method of BML seems better, no matter what s the true value of the dsperson parameter. Ths can be seen from the results of Table and Appendx E. Thus, we conclude that the bas corrected maxmum lkelhood method s relatve better than the other two methods. But ME s better than BMLE when the dsperson parameter s large under the large sample sze. 3. Evaluaton of the hypothess test In the other part of the estmaton experment I have compared the estmators on the bass of ther performance n the test of sgnfcance about the model parameters α and β. We have stated n the above secton that the asymptotc dstrbuton of the model parameter s normal. The null hypothess assgns for α and β are H : α = and H ' : β = 0 respectvely. Under these two hypotheses we have, Z Z β α ˆ α α = N(0,), se.( ˆ α) ˆ β β = N(0,). se.( ˆ β ) After Smulaton of the model parameters we get the dstrbuton of test statstcs 9

12 Z (see Fgure, 3, 4). Here we just gve fgures under dsperson parameter ν = and the sample sze s n = 0,50,000 respectvely as an example. From these 3 fgures we can see that when sample sze s small densty plot of Z does not ft wth the normal curve. However, there s nearly no large dfference among those three estmaton methods. From the fgures based on Medum sample sze n=50 and large sample sze n=000, we can get the same results. And wth the ncrease of the sample sze the data fts better. Dstrbuton of and Dstrbuton of Zbeta and Method of BML /v= n=0 Method of BML /v= n=0 Dstrbuton of and Dstrbuton of Zbeta and Method of BML /v= n=50 Method of BML /v= n=50 Dstrbuton of and Dstrbuton of Zbeta and Method of BML /v= n=000 Method of BML /v= n=000 Fgure Dstrbuton of statstcs Z under method of BML Dstrbuton of and Dstrbuton of Zbeta and Method of ML /v= n=0 Method of ML /v= n=0 Dstrbuton of and Dstrbuton of Zbeta and Method of ML /v= n=50 Method of ML /v= n=50 Dstrbuton of and Dstrbuton of Zbeta and Method of ML /v= n=000 Method of ML /v= n=000 Fgure 3 Dstrbuton of statstcs Z under method of Maxmum Lkelhood 0

13 Dstrbuton of and Dstrbuton of Zbeta and Method of Moment /v= n=0 Method of Moment /v= n=0 Dstrbuton of and Dstrbuton of Zbeta and Method of Moment /v= n=50 Method of Moment /v= n=50 Dstrbuton of and Dstrbuton of Zbeta and Method of Moment /v= n=000 Method of Moment /v= n=000 Note: The broken lne s the densty of the test statstcs Z. Fgure 4 Dstrbuton of statstcs Z under method of Moment To evaluate a hypothess test the common method s to evaluate the Type-I error and the Type-II error. In ths secton we choose the Type-I error to evaluate our hypothess. Type-I error s the probablty that when the null hypothess s true but the hypothess test ncorrectly decdes to reject t. For the null hypothess n the above we calculate the probablty of the Type-I error to see the confdence about the null hypotheses. We denote probablty of the Type-I error as P (I), P (I) =P (Reject when H s true) H 0 0 = (value of test statstc s n rejecton regon when H 0 s true) For our null hypothess H : α = 0.5 and H ' β 0 =, rejecton regon s absolute 0 : value of Z (test statstcs for α or β ) statstcs lager than Z α, whch s.96 when sgnfcance level takes Thus, PI ( ) = P( Z>.96) Applyng ths defnton we get the probablty of the Type-I error under dfferent estmate methods and sample szes. The results are shown n Table.

14 Table Probablty of Type-I error P(I) MLE BMLE ME v Sample sze α β α β α β * * 0.05* 0.05* 0.05* 0.05* 0.05* * 0.05* 0.05* 0.05* 0.05* 0.05* * 0.05* 0.05* 0.05* 0.05* 0.05* Note: * means the probablty of Type-I error s non-sgnfcantly dfferent from From Table we can see that wth the ncrease of the sample sze, the probablty of commttng Type-I error both for α and β s decreasng. The largest commt probablty for α and β s 0., 0.4 respectvely; the smallest probablty for α and β s the same: 0.05, whch s equal to the sgnfcant level. Small and medum sample sze does not have the prefect probablty as large sample sze, so the

15 statstcal hypothess test does not work well under these knds of sample szes for all of the estmaton methods. But t works well under every method when sample sze s large. Thus, we conclude that at 5% sgnfcant level we can not get a reasonable hypothess test when sample sze s not large enough. Ths s consstent wth the fact that model parameters follow the asymptotc normal dstrbuton. But when we compare the error probablty for dfferent estmate methods, bas corrected maxmum lkelhood has smaller error commt probablty than the other two methods. In other word, t s also relatvely the best among these three. Ths result s concdng wth our frst one, whch obtaned from comparson of the approprateness of dfferent estmators values (see Table ). The Kolmogorov-Smrnov test s used to show whether a sample comes from a specal dstrbuton, we apply t here to check whether test statstcs Z has a standard normal dstrbuton under the null hypothess: α = 0.5, β =. The results of p-value of the KS tests for both α and β are tabulated n Appendx F. It shows that n small and medum sample sze p-values are all less than sgnfcance level 5%, so we should reject the null hypothess; when sample sze s large, p-values are larger than sgnfcance level whch means that we should accept null hypothess. The stuaton for all these three methods s the same. And ths concluson s smlar wth that we get from the probablty of the Type-I error. 4 Conclusons: Maxmum lkelhood s the most commonly used methods n parameter estmaton, but t has some dsadvantages n some specal cases especally wth small sample szes. By adjustng ts weak ponts, for example the based property, we can get better estmator. At the same tme by ntroducng other methods such as moment method, we can also avod some knd of weakness of MLE. A comparson of these rval methods for the estmaton of the dsperson parameter of Gamma dstrbuton has been studed 3

16 n ths paper va Monte Carlo experments, whch show that n small and medum sample szes, bas corrected maxmum lkelhood estmator performs better than the other two estmators. But when we do smulaton under large sample sze, those three estmators of the dsperson parameter show almost the same performance. Snce dsperson parameter affect the varance of the model parameters, we judge the approprateness of the dsperson parameter on the asymptotc standard normal dstrbuton under the null hypothess. Fgures of the densty of the standardzed form of the dsperson parameter show that they ft the normal curve well just under large sample sze. No bg advantages can be checked from ths part, results from those three methods seem very smlar. Then, comparson the probabltes of the probablty of the Type-I error and the p-values of the Kolmogorov-Smrnov test have the smlar results wth approprateness of the estmator smulaton. Thus, depend on the results of the approprateness of the smulaton, the Type-I error and p-values of Kolmogorov-Smrnov test, we conclude that n the case of small or medum sample szes bas corrected maxmum lkelhood s to be preferred. In large sample sze all those methods are smlar. So, when use statstcal software to estmate the dsperson parameter, f the sample sze s large there s nearly no bg dfference between the SAS (default, MLE) and R (default, ME). But when the sample sze s not bg enough nether of these two methods are good. Then, we should apply bas corrected maxmum lkelhood method n the statstcal software to estmate the dsperson parameter. 4

17 Reference: [] Cho, S. C. and Wette, R. (969), Maxmum lkelhood estmaton of the parameters of the Gamma dstrbuton and ther bas, Technometrcs (4). [] Cordo, G. M. and McCullagh, P. (99), Bas correcton n generalzed lnear models, J. R. Stat. Soc. B 53(3), pp [3] Davdan, M. and Carrol, R. J. (987), Varance functon estmaton, Journal of the Amercan Statstcal Assocaton 8(400), [4] Engelhardt, M. and Ban, L. J. (977), Unformly most powerful unbased test of the scale parameter of a gamma dstrbuton wth a nusance shape parameter, Technometrcs 9, pp [5] Engelhardt, M. and Ban, L. J. (978), Constructon of optmal unbased nference procedures for the parameters of the gamma dstrbuton, Technometrcs 0, pp [6] Greenwood, J. A. and Durand, D. (960), Ads for fttng the Gamma dstrbuton by maxmum lkelhood, Technometrcs, pp [7] McCullagh, P. and Nelder, J. A. (989), Generalzed Lner Models, Chapman and Hall, London. [8] Olsson, U. (00), Generalzed Lner Models: an appled approach, Studentltteratur, Lund. [9] Stacy, E. W. (973), Quas maxmum lkelhood estmator for two parameter gamma dstrbuton, IBM Res. and Devp., pp 5-4. [0] Wlks, D. L. (990), Maxmum lkelhood estmaton of the Gamma dstrbuton usng data contanng zeros, Journal of Clmate 3, pp

18 Appendx A: Detal about the gamma dstrbuton as a member of exponental famly ν ν νy ν ( ; μν, ) = exp[ ] ; 0, ν> 0, μ> 0 f y y y Γ( ν) μ μ ν y = exp + ν log( ν) ν log( μ) + ( ν )log( y) log( Γ( ν)) μ = exp{ ν y log( μ) + ν log( ν) + ( ν )log( y) log( Γ( ν))} μ Where, The canoncal parameter: θ = μ The cumulant functon: b( θ ) = log( μ) = log( θ) The dsperson parameter: a( φ) = ν The mean of the dstrbuton: ( ) '( ) Ey= b θ = μ μ V( y) = b'' a( φ) = ν The varance of the dstrbuton: ( θ) C( y; φ) = ν log( ν) + ( ν )log( y) log( Γ ( ν)) φ var( y) The coeffcent of the varaton s CV = mean( y) = ν, the parameter ν = = s the dsperson parameter. Thus the gamma dstrbutons have the CV same shape parameter then they have the same coeffcent of varaton. 6

19 Appendx B: Maxmum Lkelhood Estmate The lkelhood functon and log-lkelhood functon of gamma dstrbuton are ν y L= exp νlog μ + νlogν + ( ν ) log y log Γ( ν) μ and ν y log L= ν log μ + νlogν + ( ν ) log y log Γ( ν) μ The maxmum lkelhood estmator of ν s log L y Γ'( ν ) = log μ + logν + + log y ν μ Γ( ν) Let ths dervatve equal to zero we have Γ '( ν ) y y μ μ n logν = + log μ log y = + log Γ( ν) μ μ y We know that the rght hand sde s the devance of the gamma dstrbuton, y ˆ ˆ μ μ Dy ( ; ˆ) [ log(, ) log(, ˆ μ = yy yμ) ] = + log ˆ μ y Thus, we have the equaton, Γ '( ν ) n log ν = D( y; ˆ μ) Γ( ν ) (a) By solvng the equaton above we can get the maxmum lkelhood estmator D(6+ D) σ MLE = ˆ ν 6+ D 7

20 Appendx C: For exponental famly the densty functon for a sngle observaton s yθ b( θ) f( y; θ, φ) = exp + c y, a ( φ ) The correspondng log-lkelhood functon s ( φ ) ( φ) yθ b( θ ) l = log( L( y; θ, φ)) = + c y, a ( φ) To get the maxmum lkelhood estmate of parameter we take dervatve of respect β, accordng to the chan rule we have j l wth We know that Thus, b'( θ ) η l l dθ dμ η = β θ dμ dη β j = μ, b'' ( θ ) V( μ ) =, = V ( μ ) η = x j μ θ η = xβ, j, = xk β j βk l ( y μ) dμ = x β a φ V μ dη j ( ) ( ) In general, for n observatons we have l ( y μ) dμ = x β a φ V μ dη j ( ) ( ) j j Let the frst dervatve equal to zero we can get the maxmum lkelhood estmates of the model parameters, whch have no relaton wth the dsperson parameter. In addton, accordng to the relaton ( ) g μ models parameter, so when take second dervatve we have, = η, we know that μ s the functon of 8

21 l ( y μ) dμ = x j β j βk βk a( φ) V ( μ) dη ( y μ) dμ μ η = x j μ a( φ) V ( μ) dη η βk dμ ( y μ) = x j + x j V ' ( μ) g'' ( μ) x a( φ) V ( μ) dη a( φ) V ( μ) ( g '( μ )) g' ( μ) The nformaton matrx I s the mnus expectaton of the second dervatve, whch can be expressed as l dμ I = E = E j k β j β x x k a( φ) V ( μ) dη g' ( μ) = a( φ) V( μ) g' ( μ) Where, E( y μ) = 0. xx j k k 9

22 Appendx D (Stacy (973)) The generalzed Gamma dstrbuton s pα py p f( y, αβ,, p) = exp[ ( y β) ], y 0 pα β Compare wth the general form of ths dstrbuton we can see that t s got by settng the parameter p=. The logarthm lkelhood functon s pα py p l = log( exp[ ( y β ) ]) pα β Take the frst dervatves and let them equal to zero, n = ( β) npα + p y = 0, n = p ( β) nψ ( α) + p log y = 0, = = n n n p nαlog( β) + α y ( y β) log( y β) = 0. p The correspondng results are β = ( y ) pα ( α) α n p = ( b) Ψ log( ) = log( t ) ( c) n p Where n α = zp log( zp ) ( d ) = n y p n p = y n, = t = y y p p p, n p p p = p = = z t n y y. 0

23 Appendx E: Table of the estmators bas (%) Bas /v Sample sze MLE BMLE ME

24 Appendx F: p-value of Kolmogorov-Smrnov Tests p-value MLE BMLE ME v Sample sze Zbeta Zbeta Zbeta <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 0 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 50 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 0 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 50 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e = <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 0 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 50 <.e-6 <.e-6 <.e E-6 <.e-6 <.e <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 0 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 <.e-6 50 <.e-6 <.e-6.90e E E-.03E