Central Coverage Bayes Prediction Intervals for the Generalized Pareto Distribution

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1 Statistics Reseach Lettes Vol. Iss., Novembe Cental Coveage Bayes Pediction Intevals fo the Genealized Paeto Distibution Gyan Pakash Depatment of Community Medicine S. N. Medical College, Aga, U. P., India Abstact The Genealized Paeto model is consideed undelying model fom which obsevables ae to be pedicted by Bayesian appoach. The Type II censoed data fom the model is consideed fo the Cental Coveage Bayes pediction technique. Both the known and unknown cases of the paametes have been consideed hee. A simulation study also has been caied out fo illustating the pefomances of the pocedues. Keywods Genealized Paeto Model; Cental Coveage Bayes Pediction; Type II Censoing Intoduction The Paeto model plays an impotant ole in socio economic studies. It is often used as a model fo analysing aeas including city population distibution, stock pice fluctuation, oil field locations and militay aeas. It has been found to be suitable fo appoimating the ight tails of distibution with positive skewness. The Paeto distibution and thei close elatives povide a vey fleible family of fat tailed distibutions, which may be used as a model fo the income distibution of highe income goup. The Paeto distibution has a deceasing failue ate, so it has often been used fo model suvival afte some medical pocedues (the ability to suvive fo a longe time appeas to incease, the longe one suvives afte cetain medical pocedues). An impotant objective of a life-testing epeiment is to pedict the natue of futue sample based on a cuent sample. Pediction of mean, the smallest o the lagest obsevation in a futue sample, has been a topic of inteest and impotance in the contet of quality and eliability analysis. The objective of the pesent pape is to pedict the natue of the futue behaviou of the obsevation when sufficient infomation of the past and the pesent behaviou of an event o an obsevation is known o given fo the genealized Paeto model. Unde Type II censoing scheme we obtain the Cental Coveage Bayes pediction length of intevals. Both known and unknown case fo the paamete has been consideed. A good deal of liteatue is available on pedictive infeence fo futue failue distibution unde diffeent citeion. Few of those who have been etensively studied pedictive infeence fo the futue obsevations fo the Paeto model ae Anold & Pess (989), Ouyang & Wu (994), Mousa (), Soliman (), Nigm et al. (3), Wu et al. (4), Raqab et al. (7). Model and Pio Distibutions The pobability density function and cumulative density function of the consideed genealized Paeto model ae given as f ( ;, ) F ( ;, ) ; < <, >, (.). (.) Hee, is known as the shape paamete and as the scale paamete. Suppose n items ae put to test unde model (.) without eplacement and test teminates as soon as th the,,..., be the item fails ( n). If obseved failue items fo the fist components, then the likelihood function fo fist failue items (,,..., ) is T L ( ) ep T, (.3)

2 Statistics Reseach Lettes Vol. Iss., Novembe whee T log i (n ) log (i) () (i) and T log. i When the shape paamete is known, the scale paamete has a conjugate pio density and in pesent case it is taken as inveted Gamma distibution with the pobability density function g ( ) / e Γ ( ) ; >, >, >. (.4) We believe, as stated in Anold and Pess (983) that fom a Bayesian view point, thee is clealy no way in which one can say that one pio is bette than othe. It is moe fequently the case that, we select to estict attention to a given fleible family of pios, and we choose one fom that family, which seems to match best with ou pesonal beliefs. Also the main eason fo its geneal acceptability is the mathematical tactability esulting fom the fact that the inveted gamma distibution is the conjugate pio fo the scale paamete. Now, the coesponding posteio density fo the paamete is obtained as Γ ( ) ( ). π ep ( ) (.5) In most pactical applications with pope infomative infomation on the Paeto scale paamete, the pio vaiance is usually finite. Now we povide the pio infomation in the case when the shape paamete is unknown, which is moe common in pactice. It is known that in this case the Paeto distibution does not have a continuous conjugate joint pio distibution, although thee eist a continuous-discete joint pio distibution. The continuous component of this distibution is elated to the scale paamete, and the discete one is elated to the shape paamete. In pesent case when the paamete is unknown, the joint pio density fo both the paametes and is taken as / e (, ) ; Γ ( ) ϕ g >, ϕ,(,, ϕ) >. (.6) The joint posteio density fo the consideed paametes and is obtained as π (, ) ( ϕ e ) e e Γ ( ) T T ; d. (.7) Cental Coveage Bayes Pediction Inteval (Paamete is Known) Let,,..., be the fist obseved failue items fom a sample of size n unde the Type II censoing scheme fom the model (.). If Y ( y, y,..., y m) be the second independent andom sample of futue obsevations fom the same model. Then the Bayes pedicative density of the futue obsevation Y is h y and obtained by simplifying denoted by ( ) h ( y ) f ( y ;, ) h π ( ) y ( y ). log ( y) log ) (3.) In the contet of Bayes pediction, we say that ( l ),l is a ( ε ) % pediction limits fo the futue andom vaiable Y, if ( < Y < l ) ε. P l (3.) Hee l and l ae said to be lowe and uppe Bayes pediction limits fo the andom vaiable Y and ε is called the confidence pediction coefficient. The Cental coveage Bayes pediction lowe and uppe limits ae obtain by solving following equality P d ε C C (3.3) ( Y l ) P ( Y l ). Hee, l C and l C ae said to be lowe and uppe Cental coveage Bayes pediction limits fo the andom vaiable Y. Using (3.) and (3.3), the lowe and uppe Bayes pediction limits of Y ae obtained as

3 Statistics Reseach Lettes Vol. Iss., Novembe / ( ) )( ε ) ε l { e } ; ε (3.4) C and / ( ) )( ) ε ε l { e } ; ε.(3.5) C Theefoe, the Cental coveage Bayes pediction length of inteval fo the given futue obsevation Y is I l l. (3.6) C C C Numeical Analysis To assess and study the popeties of the Bayes pediction length of inteval unde the Cental coveage the andom samples ae geneated as follows:. Fo the puposefully given values of the pio paametes and, geneate by using the pio density g ( ). The consideed values of pio paametes ae (, ) (,.5), (4,.5) and (8,.5). Hee smalle values wee conside fo the pio paametes to eflect little pio infomation wheeas lage eflect the highe pio infomation. Decisively, we kept the pio means the same as the oiginal means, although we have seen (not epoted hee) that the oveall behaviou emains the same.. Using obtained in step (), and the consideed values of.5,.,.5 geneates the, andom samples of size n( ) fom the consideed model (.). 3. The selected set of censoed sample size 4, 6, 8, with the level of significance ε 99 %, 95 %, 9 % ; the Cental coveage Bayes pediction length of intevals has been obtained and pesented in Table. 4. We obseve fom the table that the length of the inteval tend to be close as inceases o decease when othe paametic values ae fied. Simila tend has also been seen when combination of the pio paamete inceases. It is also noted that when confidence level deceases the length of intevals also deceases. Remak: In the case when the censoed sample size ( ), the censoing citeion is educes to the complete sample size citeion and hence the esult ae valid fo complete sample case. Cental Coveage Bayes Pediction Inteval (Paamete is Unknown) When both paametes ae consideed to be unknown, the Bayes pedicative density fo the futue obsevation Y is denoted by h ( y ) and obtained by simplifying h ( y ) f ( y ;, ) π (, ) d d h ( y ) log ( y ) log ) ( ) d. T e ( y ) (5.) Using (5.) and (3.3), the lowe and uppe Bayes pediction limits fo Y ae does obtained by solving following equations ε e and T ( ) T ε e l C log d T ( ) T l Fo the futue obsevation Y, the Cental coveage C log d. Bayes pediction length of the inteval is I C l l. (5.) C C Numeical Analysis When both paametes consideed as the andom vaiable, a simulation study also has been caied out to study the popeties of the Bayes pediction inteval as follows: 3

4 Statistics Reseach Lettes Vol. Iss., Novembe. Fo the simila set of the consideed values of the pio paametes and, as taken in Section 4, the values of has been geneated.. Fo the given values of pio paamete ϕ (.5,., 5.), of fom the unifom U (,/ ϕ ). geneate a andom value 3. Using the above geneated values of and obtained in steps () & () we geneates the, andom samples of size n( ) fom the consideed model (.). 4. Fo the simila set of selected values fo and ε (Section 4), the Bayes pediction lengths of intevals have been obtain and pesented them in the Table. 5. It has been obseved fom the Table that, all the popeties of the Bayes pediction length of intevals ae simila to the case when one paamete is known. Conclusions The Genealized Paeto model is consideed hee fom the obsevables is to be pedicted by Bayesian appoach. The Type II censoed data is consideed fo the Cental Coveage Bayes pediction technique unde both known and unknown cases of the paametes. Fo illustating the pefomances of the pocedues a simulation study has been caied out unde the puposefully choosing the values of the hype paametes. The conjugate pio fo the scale paamete is consideed hee on the basis of the geneal acceptability. In case when the shape paamete is unknown (a moe common in pactice), a continuousdiscete joint pio distibution is consideed. Fo all the consideed values it is obseved that the cental coveage Bayes pediction lengths of intevals ae shote. Also, the length tends to be shotest with deceasing in the sample size o the confidence level. Acknowledgement The authos would like to thank the leaned efeees fo thei suggestions which had impoved significantly the ealie daft of the manuscipt. REFERENCES [] Ali-Mousa, M. A. M. (). Infeence and pediction fo Paeto pogessively censoed data. Jounal of Statistics, computation and Simulation 7, [] Anold B. C. and Pess S. J. (983). Bayesian infeence fo Paeto populations. Jounal of Econometics,, [3] Anold, B. C. and Pess, S. J. (989). Bayesian estimation and pediction fo Paeto data. Jounal of Ameican Statistical Association, 84, [4] Nigm, A. M., Al-Hussaini, E. K. and Jaheen, Z. F. (3). Bayesian one sample pediction of futue obsevations unde Paeto distibution. Statistics, 37 (6), [5] Ouyang, L. Y. and Wu, S. J. (994). Pediction intevals fo an odeed obsevation fom a Paeto distibution. IEEE Tansactions on Reliability, 43 (), [6] Raqab, M. Z., Ahmadi, J. and Doostpaast, M. (7). Statistical infeence based on ecod data fom Paeto model. Statistics, 4 (), 5-8. [7] Soliman, A. A. (). Bayes pediction in a Paeto Lifetime Model with andom sample size. Jounal of the Royal Statistical Society, Seies D, 49 (), 5-6. [8] Wu, J. W., Lu, H. L., Chen, C. H. and Yang, C. H. (4). A note on the pediction intevals fo a futue odeed obsevation fom a Paeto distibution. Quality & Quantity, 38, Gyan Pakash Pesently woking as an Assistant Pofesso in Statistics, in the depatment of Community Medicine, S. N. Govenment Medical College, Aga, U. P., India 4

5 Statistics Reseach Lettes Vol. Iss., Novembe TABLE BAYES PREDICTION LENGTH OF THE INTERVAL WHEN IS KNOWN.5.5.5, 99% 95% 9% 99% 95% 9% 99% 95% 9%, , , , , , , , , , , , TABLE BAYES PREDICTION LENGTH OF THE INTERVAL WHEN IS UNKNOWN φ.5 φ. φ 5., 99% 95% 9% 99% 95% 9% 99% 95% 9%, , , , , , , , , , , ,