Modified Moment Estimation for a Two Parameter Gamma Distribution

Transcription

1 IOSR Joural of athematcs (IOSR-J) e-issn: , p-issn: X. Volume 0, Issue 6 Ver. V (Nov - Dec. 04), PP odfed omet Estmato for a Two Parameter Gamma Dstrbuto Emly rm, Abel Ouko, Cheruyot W. pkoech 3 Techcal Uversty of eya Departmet of athematcs ad Statstcs The East Afrca Uversty, eya Departmet of athematcs 3 aasa ara Uversty, eya Departmet of athematcs ad Physcal Sceces Abstract: A study of a modfcato of momet estmato of a two parameter gamma dstrbuto has bee carred out. The scale parameter has bee estmated usg the odfed omet Estmates (Es) ad ther performace s compared to the axmum Lkelhood Estmates (LEs). odfed omet Estmates are easy to compute relatve to axmum Lkelhood Estmates. To adjust ther bas, the Jackkfe techque has bee used. The effcecy of these estmators s compared by calculatg ther respectve varaces. The R-has bee used to carry out the ote Carlo smulato of data. The results obtaed show that the odfed omet Estmates perform as well as the axmum Lkelhood Estmates whe bas as well as varato of data s take to accout. eywords: Gamma dstrbuto, odfed omet Estmates, axmum Lkelhood Estmates I. Itroducto Several authors have cosdered the problem of estmatg the parameter of the gamma dstrbuto. Fsher (9) showed that the ethod of omets may be effcet for estmatg a two-parameter gamma dstrbuto ad suggested the use of axmum Lkelhood (L) method. edall ad Stuart (977) showed that effcecy of the estmated shape parameter ( ) of a gamma dstrbuto by the method of momets may be as low as percet. The ma task s to fd the modfed momet estmators whch are close to effcecy ad easy to calculate ad vestgate ther performace by comparg them wth LEs. We shall adjust the bas of these estmators to get LEs ad Es by use of Jackkfe techque. The gamma dstrbuto s a two parameter famly of cotuous probablty dstrbutos. It has a scale parameter α ad a shape parameter λ. The gamma dstrbuto s frequetly used as a probablty model for watg tmes. For stace, lfe testg, the watg tme utl falure s a radom varable that s frequetly modeled usg the gamma dstrbuto. The gamma dstrbuto ca also be used to model compoets that have two causes of falure such as sudde catastrophc falures ad wear out falures. A radom varable X s sad to have a gamma dstrbuto wth parameters α ad λ, f ts probablty desty fucto s of the form x x/ f ( x,, ) e x>0 ( ) Where α>o s the scale parameter ad λ>0 s the shape parameter. The gamma fucto x 0 x e dx s equal to the factoral fucto recurrece relato (Hogg ad Crag, 978) ( )!!, where a postve teger. ca be obtaed from the Two-parameter gamma dstrbuto s a popular dstrbuto for aalyzg lfetme data. It has lots of applcatos dfferet felds other tha lfetme dstrbutos. The two parameters of a gamma dstrbuto (scale ad shape) ad because of these parameters, the dstrbuto has a bt of flexblty to aalyze ay postve real data. It has creasg as well as decreasg falure rate depedg o the shape parameter. There are DOI: / Page

2 odfed omet Estmato for a Two Parameter Gamma Dstrbuto specfcally three basc shapes of the dstrbuto depedg o whether λ<, λ= ad λ>. Gamma hazard fucto s capable of represetg these three shapes. The sum of depedet ad detcally dstrbuted gamma radom varables has a gamma dstrbuto. If a system has oe compoet ad -spare parts ad f the compoet ad each spare part have depedet ad detcally dstrbuted gamma lfetme dstrbutos, the the lfetme dstrbuto of the system also follows a gamma dstrbuto. Probablty Desty Fucto (pdf) Table. Propertes of gamma dstrbuto x x/ (,, ), 0 f x e x ( ) Cumulatve Desty Fucto (cdf) Parameter Restrcto ea x( ) Fx ( ) ( ) >0 >0 Varace where x t x( ) t e dt 0 II. Jackkfe Techque The basc dea behd Jackkfe techque s to systematcally re-compute the statstc estmate leavg out oe observato at a tme. A sample of sze s selected ad the the Es ad LEs of - observatos are computed. Let J be the estmate obtaed by leavg out the th observato ad the usg the - observatos, the Jackkfe estmate s gve by where ( ) j... X III. Bas of a estmator Bas of a estmator s the dfferece betwee the expected value of the estmator ad the true value of the parameter beg estmated. A estmator wth zero bas s called ubased. Otherwse the estmator s sad to be based. If s a estmator of the parameter, the the bas of a estmator s defed as Bas E E( ) Where E( ) s the expected value ad θ s the true value IV. Effcet estmator Let T ad T be ay two ubased estmators of a parameter. We defe effcecy of T relatve to T var( T ) by eff (T/T) = ad say that T s more effcet tha T f eff (T/T)>. var( T ) If T s a most effcet estmator of θ the the effcecy of ay other ubased estmator T of θ relatve to T s var( T ) gve by eff(t/t) = var( T ) j j 4. Asymptotc effcecy DOI: / Page

3 odfed omet Estmato for a Two Parameter Gamma Dstrbuto A ubased estmator T s asymptotcally most effcet f lm eff (T/T) = where T s the most effcet estmator of θ. V. Cosstecy A cosstet sequece of estmators s a sequece of estmators that coverge probablty to the quatty beg estmated as the dex (usually the sze of the sample) grows wthout boud. I other words, creasg the sample sze creases the probablty of the estmator beg close to the populato parameter θ. A sequece of estmators {T ; 0} s a cosstet estmator for θ f ad oly f, for every ϵ > 0, o matter how small, we have lm Pr T y VI. ote Carlo Smulato ote Carlo Smulato s a scheme employg umbers for solvg problems where passage of tme plays o substatve role. Here parameter estmates wll be calculated, aalyzed ad compared wth the true parameters of the two parameter gamma dstrbuto. VII. axmum lkelhood estmato method. The two parameter gamma dstrbuto has a desty fucto x x/ (,, ) e x 0, 0.. f x Let x,x x (>) represet a radom sample of values of X The log lkelhood fucto s gve by log log ( ) log... l x x Takg partal dervatves wth respect to α ad λ l (, ) log ( ) log x DOI: / Page l g(, ) x..... v Where d ( ) log d s the dgamma fucto The maxmum lkelhood estmates (LEs) of the parameters of the gamma dstrbuto are the values of the parameter that produce the maxmum jot probablty desty for the observed data X. The gamma dstrbuto s a regular umodal dstrbuto ad ts LEs are derved by fdg the locatos of the fucto peaks wth the estmated parameters (, ). x Equatos () ad (v) gves the LEs as ad Y l ( ) where Y (l x l x) Aalytcally the explct expresso for the LEs caot be obtaed ad ther solutos requre umercal techques. Two methods of umercal solutos have bee developed. 7. Newto-Raphso ethod Solvg equato (v) for α we get x v

4 odfed omet Estmato for a Two Parameter Gamma Dstrbuto x deotg the sample mea. Substtutg ths to the secod equato to get a expresso terms of λ oly we get l x ( ) l( x ) 0 Ths ca be rearraged to gve x l( ) ( ) l( ) 0...( v) x x( x) deotg the geometrc mea Equato (v) ca be solved umercally usg Newto Raphso teratos. The LEs are therefore gve by equato (v) ad the soluto of equato (v). Ths method has bee explored by Cho ad Wette (969) 7. LE Scorg ethod. Cho ad Wette (969) cosdered a maxmum lkelhood scorg method ad oted that although ths proposed techque volves more computato tha the Newto Raphso method, t s appealg because t provdes a statstcal crtero for stoppg the teratos. VIII. ethod of omet estmators. The method of momets cossts of equatg sample momets wth uobservable populato quattes for the quattes to be estmated. Suppose X, X,, X are depedet detcally dstrbuted radom varables wth a gamma pdf x x f ( x,, ) e x 0 ( ) Cosderg the momet geeratg fucto for the gamma dstrbuto we get the followg. X(t) =E (etx) tx x e x exp( ) dx ( t) ( ) Computg the populato momets for the gamma we have U ` (, ) E[ X ] Ad the secod populato momet s U ` (, ) E[ X ] ( ) 0 These are the frst two populato momets. Let the frst ad secod sample momets are ad respectvely. Equatg the populato momets wth the sample momets we get where = ( ) x = x Solvg for ad we get the followg as the method of momet estmators of ad DOI: / Page

5 odfed omet Estmato for a Two Parameter Gamma Dstrbuto for IX. odfed momet estmato We ca ow preset a class of method of momets estmators of (, ), a lmtg member of whch are easly computed ad hghly effcet. Let X, X, X be a sample from equato (). From equato (v), has the soluto X L v L From () ad terms of Y l X l X (l X s the average of the logarthms of the data values), L s obtaed from the followg equato. Y l ( ) v L Ths s obtaed as follows From equato () ad solvg λ we get, log ( ) log X 0 log ( ) log X log log X ( ) But log x logx ad X log log x ( ) log X log X log ( ). Ths reduces to Y log ( ). L L The resultg estmates are asymptotcally ormally dstrbuted ad asymptotcally effcet. X L L.e. N(0, I ( ))... x where ( ) ad I( ) s the Fsher formato oe observato. A computatoal dffculty assocated wth dgamma fucto poses a challege ad a search for ts approxmato s evtable. Thor (958) proposed a approxmato obtaed by replacg ψ(α) by ts secod order asymptotc expaso. A more popular product of the search seems to be the estmator of Greewood ad Durad (960), who approxmated Y by f GD y f( y), y f( y),0.577 y 7 DOI: / Page

6 odfed omet Estmato for a Two Parameter Gamma Dstrbuto where f(y) = ( y y)/y f(y) = ( y y)/( y + y) y To obta L ofte starts wth the stadard method of momet estmates /...( ) L X s x s / X x where s s the sample varace These estmates arse from the followg observato. From the ethod of omets t follows that From LEs we got the followg X E( X ) E( X ) E( X ) E( X ) E( X ) VarX If X G(, ) the E X E X ad.. x Var X k E X E X Geerally for >0 we get x X Cor, X If = the we get equato (x) whch employs two momets, otherwse equato (x) employs three momets as oted by Wes et al (003) who also dd a passage to the lmt as k ted to zero ad obtaed EX.... (xv) Cov l X, X Geerally for > we employ hgher momets ad get the estmates as follows k E X E X / k X Cor For = we propose the modfed momet estmators as follows EX [ ] EX ( ) var X E X, X for >0 DOI: / Page

7 odfed omet Estmato for a Two Parameter Gamma Dstrbuto X. Smulato Smulato expermet was carred out to compare the performace of the odfed omet Estmators ad axmum Lkelhood Estmators for the two-parameter gamma dstrbuto. The ote Carlo smulato was performed for the samples of szes =0, 30, 50,00 ad geerated values of α ad λ. To reduce the bas the Jackkfe techque was appled to the smulated values. λ 3 6 XI. Results Table : Costraed estmates of α=.0 whe λ=, 3 ad 6 E LE JE JLE Var-E Var-LE E E E E E E E E - odfed omet Estmates LE - axmum Lkelhood Estmates JE - Jackkfe odfed omet Estmates JLE - Jackkfe axmum Lkelhood Estmates Var-E - Varace of odfed omet Estmates Var-LE - Varace of axmum Lkelhood Estmates Fgure. Estmates of α= for λ= DOI: / Page

8 varace odfed omet Estmato for a Two Parameter Gamma Dstrbuto Fgure. Estmates of α= for λ=3 Fgure 3. Estmates of α= for λ=6.00e-0.80e-0.60e-0.40e-0.0e-0.00e E E E-0.00E E sample sze varmme varmle Fgure 4. Varace of estmates of α= for λ= DOI: / Page

9 varace odfed omet Estmato for a Two Parameter Gamma Dstrbuto.80E-0.60E-0.40E-0.0E-0.00E E E E-03.00E E sample sze varmme varmle Fgure 5. Varace for estmates of α= for λ=3 Table shows the estmates of α based o dfferet sample szes. It s evdet that whe the sample sze s small (say 0) smulated at, the estmates of α= gves values for both Es ad LEs. The estmates of Es ad LEs are so close ad cocde wth small eglgble dffereces as the sample sze creases. It also shows how the estmates of α perform as λ=3 for varous sample szes. The Es asymptotcally coverge to the true value. It s evdet that the estmates of α based o LEs are more based as compared to the Es. The Jackkfe bas reducto method works very well ths case.from the smulated results, t s clear that the performace of Es are almost for dfferet szes s desrable. The varaces of Es are so small as compared to the LEs. It s evdet that the Es are hghly favorable as estmates of a two parameter gamma dstrbuto. The table also shows the estmates of α perform as λ=6 for varous sample szes. For large sample szes the Es almost coverge to the true value. It also shows the varace of the estmates of α for λ= ad λ=3. It s clear that the varace of the Es s small as compared to the LE couterpart. XII. Cocluso The classcal approach to estmatg the parameters of the two parameter gamma dstrbuto has largely cetered o approxmatg the soluto the lkelhood equato. Here the estmates of the two parameter gamma dstrbuto have bee obtaed usg modfed momet estmato, whch requres o umercal teratos ad are easy to mplemet. The bases of Es ad LEs were adjusted usg Jackkfe techque. The estmates of Es work well as the LEs especally whe bas s take to accout for dfferet sample szes. Refereces []. Cho, S.C ad Wette R., 969. axmum lkelhood estmato of parameters of the Gamma Dstrbuto ad ther bas. Techometrcs, []. Fsher, R. A., 9. O the mathematcal foudatos of theoretcal statstcs, Phlosophcal trasactos of the Royal Socety of Lodo A, [3]. Greewood, J.A ad Durad, D., 960. Ads for fghtg the gamma dstrbuto by maxmum lkelhood. Techometrcs,, [4]. Hogg, R.V ad Crag, A.T., 978. Itroducto to mathematcs statstcs.4th edto, NewYork, acmlla. [5]. edall, ad Stuart, A., 977. The advaced theory of statstcs. Charles Grff ad compay lmted. [6]. Thor, H.C., 958. A ote o the gamma dstrbuto, othly Weather Revew, 86, 7- [7]. Wes, D.P. Cheg ad Beauleu N.C (003), A class of methods of momets estmators for the two-parameter gamma famly. Pak.J.Statstcs 9, 9-4. DOI: / Page