VOL. 3, NO. 11, November 2013 ISSN ARPN Journal of Science and Technology All rights reserved.

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1 VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. Usg Square-Root Iverted Gamma Dstrbuto as Pror to Draw Iferece o the Raylegh Dstrbuto Chrs Bambey Guure Departmet of Bostatstcs, School of Publc Health Uversty of Ghaa, Lego-Accra Ghaa ABSTRACT Based o a complete falure tme data, the maxmum lkelhood ad Bayesa estmator uder squared error ad geeral etropy loss fuctos for the scale parameter ad the relablty fucto of the Raylegh dstrbuto are derved. Assessmets betwee the estmators are vestgated through a smulato study. The results dcate that, Bayes estmator uder the squared error loss fucto performs better tha the others havg obtaed ad compared the estmators usg mea squared errors ad absolute bases of the estmated values. Keywords: Iverted Gamma, Squared Error ad Geeral Etropy Loss fuctos, Bayesa Iferece, Smulato Study. INTRODUCTION The Raylegh dstrbuto s a specal case of the two parameter Webull dstrbuto ad a sutable model for lfe testg studes. Polovko (968) ad Dyer ad Whsead (97) demostrated the mportace of ths dstrbuto electro vacuum devces ad commucato egeerg. The cumulatve dstrbuto fucto (c.d.f.), the relablty fucto ad the desty fucto of the Raylegh dstrbuto are defed as x F( x) exp, x [0, ), () Rx ( ) exp () ad x x f( x; ) exp, x 0, 0 > () Its hazard rate s a learly creasg fucto of tme. Hece, whe the falure tmes are dstrbuted accordg to the Raylegh dstrbuto, the relablty fucto decreases at a much hgher rate tha that of the expoetal relablty fucto does, Km ad Ha (009). I may lfe testg studes, t s commo that the lfetmes of some test uts may be recorded exactly, dcatg that all the uts have faled. Ifereces for the Raylegh dstrbuto have bee dscussed by several authors. Harter ad Moore (965) derved a explct form for the MLE of based o type II cesored data. Dyer ad Whsead (97) cosdered the best lear ubased estmator of based o type II cesored sample. Balakrsha (989) showed that a approxmate MLE s as effcet as the best lear ubased estmator. Bayesa estmato ad predcto problems for the Raylegh dstrbuto based o doublycesored sample have bee cosdered by Feradez (000) ad Raqab ad Mad (00). Wu et al (006) have also cosdered the Bayesa estmator ad predcto tervals for future observatos based o progressvely type II cesored samples. Km ad Ha (009) cosdered MLE, approxmate MLE ad Bayes estmato procedures for the scale parameter based o a multply type II cesored sample. Guure et al (0) cosdered Bayesa ferece based o Webull model for terval-cesored survval data. I ths paper, our ma object s to study the maxmum lkelhood estmato ad Bayes estmato procedures for the scale parameter ad the relablty Rx ( ) of the Raylegh dstrbuto based o a complete sample Accordg to complete samples, Surles ad Padgett (00) showed that the two-parameter geeralzed Raylegh dstrbuto s qute effectve modelg stregth of data ad geeral lfetme data. The rest of ths paper s orgazed as follows. I Secto, the MLE of the parameter ( ) ad the relablty Rx ( ) based o a complete falure data sample are preseted. I Secto, the Bayes estmator uder squared error loss ad geeral etropy loss fuctos are troduced. Comparsos amog the estmators are coducted through smulatos secto 4, results are secto 5 ad secto 6 cocluso.. MAXIMUM LIKELIHOOD ESTIMATOR I ths secto we cosder the maxmum lkelhood estmator (MLE) of the Raylegh dstrbuto. Let x,..., x be a radom sample of sze from a Raylegh dstrbuto, the the lkelhood fucto ca be wrtte as; 0

2 VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. f( x ; ) exp ad the log-lkelhood wrtte as x log( ) x (4) (5) The ormal equato becomes; x d 0 d mplyg that ˆ MLE x, (6) Therefore maxmum lkelhood estmate of the relablty fucto from [6] s ˆ( Rx, ) exp ˆ x MLE. BAYESIAN ESTIMATION Sce, s the parameter beg sort for, accordg to Bayesa t s a radom varable, we therefore, cosder the atural cojugate famly of pror dstrbutos for used Feradez (000), as p θ ( ) exp, > 0, (8) where the shape parameter p > 0 ad scale parameter θ > 0. Ths desty s kow as the square-root verted gamma dstrbuto. For θ 0, ( ) reduces to a geeral class of mproper prors ad f p 0 ad θ 0, the a mproper pror for s the Jeffreys (96) pror. Note that fα, the the desty fucto of α has a gamma dstrbuto wth parameters p ad θ. Combg equatos (4) ad (8), the posteror desty fucto of ca be obtaed as (7) ( x) ( ) exp ( ) exp (9) d We ca therefore obta the posteror desty fucto uder the squared error loss fucto from above wth respect to the parameter. The squared error loss fucto s smply the posteror mea. Hece, we have ( x) u( ) ( ) ex p d ( ) exp d where u( ) represets the loss fucto. (0) Whe we cosder the Bayes estmate of the relablty fucto uder ths loss fucto the posteror desty wll be x x u exp ( ) exp d Rx ( ) ( ) exp d () The geeral etropy loss fucto s asymmetrc ature, that, t s used to determe the degree of overestmato ad uderestmato of a fucto of terest. It s a geeralzato of the etropy loss fucto. The Bayes estmator of, deoted by ˆ BG s gve as ˆ ( ) k k, provded (.) BG fte. E E exst ad s Therefore, the posteror desty fuctos of the parameter ad the relablty fucto are gve respectvely as ( x) k u[ ] ( ) exp d d ( ) exp 04

3 VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. ad Rx ( ) BG k x x x () u exp ( ) exp d ( ) exp d For the relablty fucto x exp x u u exp,, x 4 x x exp x exp u 4 6 Cosderg the geeral etropy loss fucto, we have for the parameter ( ), ( ), ( ) k k k u u k u k x () The above equatos caot be obtaed explctly hece we apply a umercal approach proposed by Ldley (980) to approxmate the rato of two tegrals such as (6). Ths has bee used by several authors to obta the approxmate Bayes estmators. For detals see Ldley (980) ad Press (00). Based o Ldley s approxmato, the approxmate Bayes estmators of ad x exp uder the squared errors loss fucto s gve accordg to Guure et al (0a) Hece ad ad the pror s [ ] ˆ u uδ uρδ 0uδ x 0 4, u, u, u 0 ( ) δ 0 4 x 0 5 p θ ( p ) exp ρ p θ exp p θ θ exp p θ exp ad for the relablty fucto k kx exp u exp, u ad k k 4 kx exp k x exp u SIMULATION STUDY A smulato study was carred out to determe the best estmator for the scale parameter ad the relablty fucto of the Raylegh dstrbuto wth complete falure data. We report the results for 0.5,.0 ad.5 ad that of 5, 50 ad 00. I ths secto our ma am s to compare the Bayes estmator wth the classcal maxmum lkelhood estmator. To make the comparso more meagful, we assume the oformatve pror o the Raylegh parameter by takg θ p 0 of the square-root verted gamma dstrbuto. Ths makes t a mproper pror but the posteror dstrbuto s proper. We compare the MLEs wth the Bayes estmates terms of bases ad mea squared errors (MSE) for dfferet sample szes ad for dfferet parameter values. All the computatos are performed usg the R programmg laguage whch s freely avalable ole. We have cosdered for geeralty the geeral etropy loss fucto parameter to be k ± 0.8. Note that the geerato of R( ) s very smple. If U follows a uform dstrbuto [0, ], the x log( U) follows R( ). Therefore, f oe has a good uform radom umber geerator, the the geerato of Raylegh radom devate s mmedate. For each sample sze we compute the MLEs of the scale parameter ad the relablty fucto ad also k 05

4 VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. the Bayes estmates usg Ldley s approxmato. We replcate the process 000 tmes ad obta the mea squared error ad the absolute bas of the estmates. The results are reported Tables ad. 5. RESULTS Table : Mea Squared Error for ˆ ad Rx ˆ( ) of the Estmators ˆmle ˆbs ˆ ge Rx ˆ( )mle Rx ˆ( )bs Rx ˆ( )ge k0.8 k-0.8 k0.8 k Table : Absolute Bas for ˆ ad Rx ˆ( ) of the Estmators ˆmle ˆbs ˆ ge Rx ˆ( )mle Rx ˆ( )bs Rx ˆ( )ge k0.8 k-0.8 k0.8 k mle maxmum lkelhood,bs Bayes squared error ad ge Bayes geeral etropy loss 6. CONCLUSION We obtaed the Bayesa estmato approach usg square-root verted gamma pror from whch we had a o-formatve pror for the scale parameter of the Raylegh dstrbuto whch was employed usg squared error ad geeral etropy loss fuctos va Ldley approxmato. Comparsos are made betwee the estmators based o smulato study wth mea squared errors ad absolute bases. Table, shows the mea squared error values of the scale parameter ad the relablty fucto. It s bee observed that, the Bayes o-formatve pror estmator has the smallest mea squared error values uder the squared error loss fucto tha the others to a very large exted. As the sample sze creases Bayes estmator uder the geeral loss fuctos performs better tha MLE but they all have ther MSE values decreasg correspodgly. The Absolute Bas of the estmated values are preseted Table. We observe that all the estmators also have ther absolute bas values decreasg as the sample sze creases but aga Bayes estmator wth respect to squared error loss gves very mmal bas tha the others. REFERENCES [] Balakrsha, N. (989). Approxmate MLE of the scale parameter of the Raylegh dstrbuto wth cesorg. IEEE Trasactos o Relablty, 8, [] Guure, C. B.,Ibrahm, N. A., Adam M. B., Bosomprah S. ad Al Omar, A. M. (0a). Bayesa Parameter ad Relablty Estmate of Webull Falure Tme Dstrbuto. Bullet of the Malaysa Mathematcal Sceces Socety. I Press. [] Guure, C. B., Ibrahm, N. A. ad Adam, M. B. (0). Bayesa Iferece Based o Webull Model for Iterval-Cesored Survval Data. Computatoal ad Mathematcal Methods Medce, Vol. 0, Artcle ID

5 VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. [4] Dyer, D. D., & Whsead, C. W. (97). Best lear estmator of the parameter of the Raylegh dstrbuto-part I: Small sample theory for cesored order statstcs. IEEE Trasactos o Relablty,, [0] Km, C., & Ha, K Estmato of the scale parameter of the Raylegh dstrbuto wth multply type II cesored sample, Joural of Statstcal Computato ad Smulato, 79, [5] Ferádez, A. J. (000). Bayesa ferece from type II cesored Raylegh data. Statstcs ad Probablty Letters, 48, [6] Ferádez, A. J. (004). O estmatg expoetal parameters wth geeral type II progressve cesorg. Joural of Statstcal Plag ad Iferece,, [7] Harter, H. L., & Moore, A. H. (965). Pot ad terval estmators, based o m order statstcs, for the scale parameter of a Webull populato wth kow shape parameter. Techometrcs, 7, [8] Jeffreys, H. (96). Theory of Probablty. Oxford: Claredo Press. [9] Km, C., & Ha, K Estmato of the scale parameter of the Raylegh dstrbuto uder geeral progressve cesorg, Joural of the Korea Statstcal Socety. 8, [] Ldley, D.V., 980. Approxmate Bayesa method. Trabajos Estadst., - 7. [] Press, S.J., 00. The Subjectvty of Scetsts ad the Bayesa Approach.Wley, NewYork. [] Polovko, A. M. (968). Fudametals of Relablty Theory. New York: Academc Press. [4] Raqab, M. Z., & Mad, M. T. (00). Bayesa predcto of the total tme o test usg doubly cesored Raylegh data. Joural of Statstcal Computato ad Smulato, 7, [5] Surles, J.G. ad Padgett, W.J. (00), \Iferece for relablty ad stress-stregth for a scaled Burr Type X dstrbuto", Lfetme Data Aalyss, vol. 7, [6] Wu, S.-J., Che, D.-H., & Che, S.-T. (006). Bayesa ferece for Raylegh dstrbuto uder progressve cesored sample. Appled Stochastc Models Busess ad Idustry,,