Statistical Inferences for Lomax Distribution Based on Record Values (Bayesian and Classical)

Transcription

1 Joural of Moder Applied Statistical Methods Volume Issue Article --0 Statistical Ifereces for Lomax Distriutio Based o Record Values (Bayesia ad Classical Parviz Nasiri Uiversity of Payame Noor, Tehra, Ira Sama Hosseii Uiversity of Payame Noor, Tehra, Ira Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Nasiri, Parviz ad Hosseii, Sama (0 "Statistical Ifereces for Lomax Distriutio Based o Record Values (Bayesia ad Classical," Joural of Moder Applied Statistical Methods: Vol. : Iss., Article. DOI: 0.3/jmasm/330 Availale at: This Regular Article is rought to you for free ad ope access y the Ope Access Jourals at DigitalCommos@WayeState. It has ee accepted for iclusio i Joural of Moder Applied Statistical Methods y a authorized editor of DigitalCommos@WayeState.

2 Statistical Ifereces for Lomax Distriutio Based o Record Values (Bayesia ad Classical Cover Page Footote The authors are thakful to the referees ad editor for their valuale commets. This work was partially supported y the Tehra Payame Noor Uiversity (PNU through a grat. This regular article is availale i Joural of Moder Applied Statistical Methods: iss/

3 Joural of Moder Applied Statistical Methods Copyright 0 JMASM, Ic. May 0, Vol., No., //$9.00 Statistical Ifereces for Lomax Distriutio Based o Record Values (Bayesia ad Classical Parviz Nasiri Sama Hosseii Uiversity of Payame Noor, Tehra, Ira A maximum likelihood estimatio (MLE ased o records is otaied ad a proper prior distriutio to attai a Bayes estimatio (oth iformative ad o-iformative ased o records for quadratic loss ad squared error loss fuctios is also calculated. The study cosiders the shortest cofidece iterval ad Highest Posterior Distriutio cofidece iterval ased o records, ad usig Mea Square Error MSE criteria for poit estimatio ad legth criteria for iterval estimatio, their appropriateess to each other is examied. Key words: Lomax distriutio; record values, maximum likelihood estimatio, method of momet, Bayesia estimatio, shortest iterval, highest posterior desity (HPD iterval, quadratic loss fuctio, squared error loss fuctio, prior desity, posterior desity, simulatio, MSE. Itroductio Let X, X, X3, e a sequece of idepedet ad idetically (iid radom variale with cumulative distriutio (cdf fuctio F(x ad proaility desity (pdf f(x For defie T (, T( + { j X j XT ( } mi :. The sequece { X T ( } is kow as a upper record value statistic ad the sequece { T ( } is kow as a record time sequece (Arold, Balakrisha & Nagaraja, 99. Chadler (9 was oe of the first to study record theory ad he defied a mathematical model for record values. Record values arise aturally i may applicatios ivolvig data Parviz Nasiri is a Associate Professor i the Departmet of Statistics Uiversity of Payame Noor, Tehra, Ira. him at: Pasiri@yahoo.com. Sama Hosseii is a Lecturer i Departmet of Statistics. him at: S.hosseii.stat@gmail.com. relatig to weather, sports, ecoomics ad life testig studies. May authors have studied records ad their associated statistics as well as iferece-ased testig o records. Some of the est examples may e foud i the works of Balakrisha, Arold, Nagaraja (99, Ahsaullah (99 ad Nevzoroz (9. Sevgi, et al. (00 examied the relatioship etwee order statistics ad records. Mohammad (00 ad Balakrisha (99 examied the recurret relatios etwee the momets for the geeralized expoetial ad Lomax distriutios. Ahsaullah (9 studied record values received from Lomax distriutio, ad Ahsaullah ad Hollad (99 discussed oth scale ad locatio estimatio of the distriutio of geeralized extreme values ased o records. Asgharzadeh (009 discussed oth MLE ad Bayesia estimatio ased o record values ad Cha (99 presets iterval estimatio accordig to records for groups of scales ad locatios. Solima ad Ad Ellah (00 compared Bayesia ad No-Bayesia estimatio ased o records. The Lomax distriutio plays a importat role i reliaility. Cosider the oeparameter Lomax distriutio with pdf 9

4 LOMAX DISTRIBUTION INFERENCES FROM BAYESIAN AND CLASSICAL VALUES + f ( x; ( + x x 0, > 0, ad cdf ( ( F x; + x x 0, > 0. ( ( T( T( T( ( T( ( T( i f x, x,, x f x ; h x ;, where f( x. F( x T( i ( T( i; h x T( i i ( A applicatio of the Lomax distriutio i receiver operatig characteristic (ROC was preseted y Campell ad Rataparkhi (993. Distriutioal properties ad recurrece relatio momets of record values was studied y Balakrisha (99 ad Ahsaullah (99. Much work has ee doe with respect to estimatig the parameters usig oth classical ad Bayesia techiques, ad parametric ad oparametric iferece ased o record values have also ee studied extesively (for example, see Ahmadia, et al., 009; Solima & Al- Aoud, 00; Baklizi, 00. This study has several compoets: It cosiders Lomax parameter estimatio ased o record values. It estimates the parameter usig maximum likelihood ad method of momet (MME ased o record values. It uses a appropriate selectio of desity fuctio for a prior distriutio to derive a Bayesia estimatio ased o record values. For the latter, y applyig a appropriate selectio for the prior desity, the society parameter is cotrolled; this meas that the Mea Square Error MSE of the Bayesia estimatio is cotrolled y cotrollig the parameters of this distriutio. Fially, it derives the shortest iterval estimatio ad Highest Posterior Desity (HPD iterval estimatio ased o record values. Examples are used to illustrate the various compoets. Poit Estimatio of Parameter: The Method of Maximum Likelihood Estimatio If XT(, XT(,, XT( represets the first upper record values from the Lomax distriutio i (, the the joit distriutio of X, X,, X is T( T( T( Thus, for the Lomax distriutio, T( ( + x f ( xt(, xt(,, xt(, ( + x T( i i (3 ad the log likelihood fuctio is ( T( T( i ( L l l + x l + x. i ( The maximum likelihood estimatio (MLE ased o records ca e otaied from ( as ad ( + x T ( l + 0 l( + xt(. ˆ ( Usig (, the margial pdf of X T( ca e derived as f ( xt( therefore, ( ( x T! + ( xt( + (l( + (, ( ˆ MLE Var mle ˆ., ( coversely, if (3 is rewritte as 0

5 NASIRI & HOSSEINI ( T(, T(,, T( f x x x exp( l l ( + xt( l( xt( i, + i the l( + x is a complete sufficiet statistic for parameter. Therefore ˆMLE ased o record is the equal to the Uiformly Miimum- Variace Uiased Estimator (UMVUE for parameter. Poit Estimatio of Parameter: The Method of Momet Estimatio The MME, first itroduced y Pearso (9, was oe of the first methods used to estimate the society parameter (for additioal details ad a example see Pearso, 9. The Lomax parameter is estimated y the MME ased o record values y usig the desity fuctio (, which results i ( ( E X T. Next, solvig the equatio ( T( E X ( X yields a MME ased o record values, where is average of the first records ( X (, X (,, X (. Thus, T T T ˆMME. ( + X Bayesia Estimatio of Parameter The Bayesia estimator of is otaied ased o record values uder the two followig loss fuctios: ad L ˆ, ˆ, ( ( ( ˆ ( ˆ L,, ( where ˆ is a estimator of. Assumig a iverse Weiull distriutio IWD ( γ, β, c, the prior for is cojugated as such that β γ β π( ( + exp, Γ( γ β E ( Var β, γ ( ( γ β ( γ (9 where γ > 0, β > 0. Note that ~ gamma( γ, β. This prior desity has a advatage over other priors ecause it is easy to use ad the parameter ( γ, β ca e chose such that prior precisio for the true value of is fulfilled ecause Bayesia estimatios are fuctios of ( γ, β, therefore, the precisio of the Bayesia estimatios caot e cotrolled y alterig the prior distriutio parameters. Comiig likelihood fuctio (3 with prior desity (9, the posterior desity of is otaied as ( x π l ( x ( β + + T Γ( γ l ( xt( β l ( + xt( + β exp where x x (,, x (, > 0. Note that T T + γ

6 LOMAX DISTRIBUTION INFERENCES FROM BAYESIAN AND CLASSICAL VALUES ( γ β ( x,, ~, L. T( xt( gamma xt( Bayesia Estimator of Uder Quadratic Loss Fuctio The posterior distriutio of is IWD( + γ, β + l ( + x, c, where IWD is Iverse Weiull Distriutio (i other words where W is a Weiull variale ad the Bayes estimator of is ased o record values uder a quadratic loss fuctio (, for example ˆ,, as give y Berger (9 is ˆ, ( ω ( μ ( (, (,, T T T( E ω ( X (, X (,, X T T T( E X X X ( E X, X,, X E X, X,, X β + L +. + γ + ( ( ( T T T T( T( T( ( xt( (0 Bayesia Estimator of Uder Squared Error Loss Fuctio Cosiderig the posterior distriutio of ad loss fuctio (, the Bayes estimator ased o record values, for example, ˆ,, is give as (Berger, 9: ˆ ( μ(,,, ( T(, T(,, T( ( + xt( + β E X X X, T( T( T( E X X X L. + γ As a result, the Bayesia estimatio is formed as a differetia comiatio of oth prior distriutio ad sample distriutio as: ˆ, ( xt( l + γ β +. + γ + γ γ ( Iterval Estimatio of Based o Record Values: The Shortest Iterval Estimatio To otai the shortest ( α % cofidece iterval estimatio ased o record values, a pivot quatity is chose as a fuctio of a miimal sufficiet statistic for parameter ( ˆMLE such that l Q ( + X T ( From ( it is clear that the distriutio of Q is χ for ay costats a ad, hece, ( < < ( P a Q f t dt α.. Q a ( Algeraic maipulatio results i the cofidece iterval ( + xt( ( + xt( l l < < a thus, the legth of iterval is otaied as L l( + x T (. a, (3 To miimize (3 ad satisfy (, a ad are selected usig the Lagrage multipliers method ψ ( a,, λ l( + xt( + λ fq( t dt ( α. a a After derivatio y λ, a, ad, the followig results:

7 NASIRI & HOSSEINI π fq a l + a l ( ( xt( ( + xt( t dt α ( ( f t dt α Q λ f a 0 a. a f ( a f ( Q Q + λ f 0 ( Accordigly, a ad must satisfy ( to yield the shortest iterval estimatio for : ( < < ( P a Q f t dt a Q α, fq a fq ( ( a ( Iterval Estimatio of Based o Record Values: Highest Posterior Desity (HPD Estimatio After otaiig the posterior distriutio π ( XT(, XT(,, XT(, the prolem of the likelihood that the parameter lies withi the iterval [ cl, c U ] arises. Bayesias call the iterval ased o the posterior distriutio a credile iterval; the iterval [ cl, c U ] is said to e a ( α % credile iterval for if cu cl ( T( T( T( π X, X,, X d α. ( The Highest Posterior Desity (HPD regio is give y { A : π ( X,,, T( XT( XT( c} where c is chose so that cu cl ( ( ( T T T( π X, X,, X d α ( c X, X,, X ( ( ( π( c X, X,, X L T T T U T( T( T( ( The HPD iterval estimatio is optimal i the sese that it results i the shortest iterval. Let λ, y this assumptio the posterior distriutio of λ is Gamma ( + γ,l( + x + β. After algeraic maipulatio, a HPD estimatio ( α % for parameter ased o records is give y * Γ ( γ, AcL, AcU cl + γ + α, Γ( + γ cu exp, where A β l ( xt( + + ad ( ca ca L U ( * Γ is the geeralized icomplete Gamma fuctio. Therefore HPD iterval estimatio ased o record values ca e otaied as:,. cu cl ( Simulatio ad Examples MSE ad Bias To illustrate the estimatio techiques developed, cosider the followig simulated data from the Lomax distriutio:

8 LOMAX DISTRIBUTION INFERENCES FROM BAYESIAN AND CLASSICAL VALUES This data was otaied y usig the trasformatio xi, where u ( ui i is a uiformly distriuted radom variale. If oly the upper record values have ee oserved, these are: for a o-iformative prior distriutio with γ 0, β, ad γ, β. for a iformative prior distriutio. Results from equatios (, (0 ad ( for the parameter computed for,,,, are preseted i Tale. Iterval Estimatio Results from usig equatios ( ad ( for the parameter, λ computed for,,,, are preseted i Tales ad 3. Tale : Estimatio, Bias ad MSE Numer of Records ( Estimate Bias MSE ˆMLE ˆ, No-Iformative ˆ, No-Iformative ˆ, Iformative ˆ, Iformative

9 NASIRI & HOSSEINI Tale: Shortest ( α % Cofidece Iterval Estimatio Based O Record Values Numer of Records ( Iterval Legth ( α % Cofidece % % % Tale3: Highest Posterior Distriutio (HPD ( α % Iterval Estimatio Based O Record Statistics Numer of Records ( λ ( λ, λ L U [, ] λ λ Legth U L ( α % Cofidece % % %

10 LOMAX DISTRIBUTION INFERENCES FROM BAYESIAN AND CLASSICAL VALUES ˆ MLE Figure : MSE s of the Estimators, ˆ,, ad ˆ, Iformative ad No-Iformative Figure : Legths of the Shortest Iterval ad Highest Posterior Distriutio (HPD Estimatios Based o Record Statistics for 90% Cofidece

11 NASIRI & HOSSEINI Figure 3: Legths of the Shortest Iterval ad Highest Posterior Distriutio (HPD Estimatios Based o Record Statistics for 9% Cofidece Figure : Legths of the Shortest Iterval ad Highest Posterior Distriutio (HPD Estimatios Based o Record Statistics for 99% Cofidece

12 LOMAX DISTRIBUTION INFERENCES FROM BAYESIAN AND CLASSICAL VALUES Figure : Legths of the Shortest Iterval ad Highest Posterior Distriutio (HPD Estimatios Based o Record Statistics for 90%, 9% ad 99% Cofidece Coclusio MLE ad Bayesia estimatios ased o record values were otaied. For the Bayes estimatios, i order to cotrol the passive parameter of society, the prior distriutio was assumed to e Gamma. I additio, Bayes estimatios were otaied for two types of loss fuctios ad, with a view of prior estimatio, usig a iformative posterior desity fuctio, HPD estimatios were otaied i a theoretic way (see Tale 3. Coversely, the shortest cofidece iterval was otaied usig a MLE ased o records ad equatio ( (Tate & Klett, 99; see Tale for results. Theoretical results of the study are explaied umerically y simulatio i the followig ways: Tale shows that a iformative Bayesia estimatio ased o records uder squared error loss fuctio has the lowest MSE compared to the iformative Bayesia estimatio, which is ased o records uder a quadratic loss fuctio with a oiformative Bayesia estimatio uder a squared error loss fuctio.. This is also compared to a MLE ased o records; comparisos are show i Figure. Cofidece itervals ad their legths for record umers,,,, ad cofidece levels 90%, 9% ad 99% were otaied. The loger the, the shorter the iterval distace (see Tale 3. Comparig Tales ad 3, it the poit at which HPD estimatios have a shorter legth tha the cofidece iterval with optimal legth is oserved. This compariso is illustrated i Figures, 3 ad for various cofidece levels; Figure shows the compariso for all levels.

13 NASIRI & HOSSEINI Ackowledgmets The authors are thakful to the referees ad editor for their valuale commets. This work was partially supported y the Tehra Payame Noor Uiversity (PNU through a grat. Refereces Ahmadia, J., Jozai, M. J., Marchadc, E., & Parsia, A. (009. Bayes estimatio ased o k-record data from a geeral class of distriutios uder alaced type loss fuctios. Joural of Statistical Plaig ad Iferece, 39, 0-9. Ahmed A., Solima, A. H., Ad Ellah, K. S. Sulta. (00. Compariso of estimates usig record statistics from Weiull model: Bayesia ad o-bayesia approaches. Computatioal Statistics & Data Aalysis,, 0-0. Ahsaullah, M. (99. Record values of the Lomax distriutio. Statistica Neerladica,, -9. Ahsaullah, M. (99. Itroductio to record values. Needham Heights, MA: Gi Press. Ahsaullah, M., & Hollad, B. (99. O the use of record values to estimate the locatio ad scale parameters of the geeralized extreme value distriutio. The Idia Joural of Statistics,, Arold, B. C., Balakrisha, N., & Nagaraja, H. N. (99. Records. Caada: Wiley. Asgharzadeh, A. (009. O Bayesia estimatio from expoetial distriutio ased o records. Joural of the Korea Statistical Society, 3, -30. Baklizi, A. (00. Likelihood ad Bayesia estimatio of Pr(X < Y usig lower record values from the geeralized expoetial distriutio. Computatioal Statistics ad Data Aalysis,, Balakrisha, M. (99. Relatios for sigle ad product momets of record values from Lomax distriutio. The Idia Joural of Statistics,, 0-. Berger, J. O. (9. Statistical decisio theory ad Bayesia aalysis. New York, Spriger. Campell, G., & Rataparkhi, M. V. (993. A applicatio of Lomax distriutio i receiver operatig characteristic (ROC curve aalysis. Commuicatios i Statistics,, -. Cha, P. S. (99. Iterval estimatio of locatio ad scale parameters ased o record values, Statistics & Proaility Letters, 3, 9-. Mohammad, Z. R. (00. Ifereces for geeralized expoetial distriutio Based o record statistics. Joural of Statistical Plaig ad Iferece, 0, Nevzorov, V. B. (00. Records: Mathematical theory. USA: America Mathematical Society. Pearso, K. (9. Cotriutios to the mathematical theory of evolutio. Phil. Tras. Royal Soc. Sevgi, Y. O., Mohammad, A., Fazil A. A., & Fuda, A. (00.Switchig record ad order statistics via radom cotractios. Statistics & Proaility Letters, 3, 0-. Solima, A. A., & Al-Aoud, F. M. (00. Bayesia iferece usig record values from Rayleigh model with applicatio. Europea Joural of Operatioal Research,, 9-. Tate, & Klett. (99. Optimal cofidece itervals for the variace of a ormal distriutio. Joural of the America Statistical Associatio, (, -. 9