A continuous probability model allowing occurrence of zero values: Polynomial-exponential

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1 A cotiuous probability model allowig occurrece of zero values: Polyomial-epoetial Christophe Cheseau, Hassa Bakouch, Pedro Ramos To cite this versio: Christophe Cheseau, Hassa Bakouch, Pedro Ramos. A cotiuous probability model allowig occurrece of zero values: Polyomial-epoetial. 7. <hal-57656> HAL Id: hal Submitted o 3 Aug 7 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 A cotiuous probability model allowig occurrece of zero values: Polyomial-epoetial Christophe Cheseau, Hassa S. Bakouch, Pedro L. Ramos 3 Uiversité de Cae Basse-Normadie, LMNO, Départemet de Mathématiques, UFR de Scieces, F-43, Cae. Departmet of Mathematics, Faculty of Sciece, Tata Uiversity, Tata, Egypt 3 Istitute of Mathematical Sciece ad Computig, Uiversity of São Paulo, São Carlos, Brazil ARTICLE HISTORY Compiled August, 7 ABSTRACT This paper deals with a ew two-parameter lifetime distributio with icreasig, decreasig ad costat hazard rate. This distributio allows the occurrece of zero values ad ivolves the epoetial, liear epoetial ad other combiatios of Weibull distributios as submodels. May statistical properties of the distributio are derived. Maimum likelihood estimatio of the parameters is ivestigated with a simulatio study for performace of the estimators. Two real data sets are aalyzed for illustrative purposes ad it is oted that the distributio is a highly alterative to the gamma, Weibull, Logormal ad epoetiated epoetial distributios. KEYWORDS Hazard rate; Two-parameter distributios; Reliability ad statistical measures; Maimum Likelihood Estimatio; Data applicatios.. Itroductio with motivatios I aalysis of the lifetime data, mootoe hazard rates are commo. Such data ca be modelled usig the log-ormal, Weibull ad gamma distributios. The Weibull distributio is more popular tha log-ormal ad gamma because the survival ad hazard rate fuctios of the last two distributios have ot a closed form ad hece umerical itegratios are required. Gupta ad Kudu [6] itroduced the epoetiated epoetial EE) distributio as a etesio to the epoetial distributio ad also as a alterative to the gamma distributio. Further developmets o the epoetiated epoetial distributio ca be see i [7]. I may practical applicatios, cotiuous probability models that allow occurrece of zero values have vast importace, for eample i forecast models whe we observe the mothly raifall precipitatio, it is commo i dry periods the o occurrece of precipitatio, therefore the occurrece of zero values ca be observed i differet measures, such as the average, maimum ad miimum. I survival aalysis, we may Correspodig author: Pedro Luiz Ramos, pedrolramos@usp.br

3 observe data with istataeous failure due costructio problem. Aother eample of zero occurrece is the hydrologic data i arid ad semiarid regios, like aual peak flow discharges. Moreover, zero occurrece ca be met i may areas, e.g. maufacturig defects, medical cosultatios, hydrology, ecology ad ecoometrics. The above four distributios log-ormal, Weibull, gamma ad epoetiated epoetial) do ot provide the characteristic of zero occurrece. Therefore, we itroduce a ew cotiuous probability model that allows occurrece of zero values ad it shall be called Polyomial-epoetial PE) distributio. This model represets a powerful alterative to the metioed distributios above ad also a etesio to epoetial, liear epoetial ad other combiatios of Weibull distributios as it shall be see later. The model is specified i terms of the cumulative distributio fuctio cdf) as: F ) e λ α,, + )\{}, where α >, λ > ad F ) if < ) ad Figure presets plots of the cdf for differet values of α ad λ. Note that, there is a discotiuity whe, to avoid that we ca cosidered the cotiuous etesio for give by F ) { e λ α,, + )\{}, e λα,. ) Moreover, a sigificat accout of mathematical properties for the ew distributio are provided ad the hazard rate fuctio has costat, icreasig or decreasig shape, which makig the PE distributio a alterative to the metioed distributios above. Aother attractive feature of the PE distributio is that it has closed form epressios for its cdf ad hazard rate fuctio, which is ot the case, for the log-ormal ad gamma distributios. Further, the distributio has several particular sub-models. For α, the PE distributio gives the epoetial ad whe α, it reduces to the oe parameter liear epoetial distributio. Also, whe α is a iteger, we ca epress F ) as F ) ep λ α ), k >, ad hece the survival fuctio k S) F ) of the PE distributio is the product of the survival fuctios of Weibull distributios with parameters λ, ), λ, ),..., ad λ, α) with respect to value of k, which meas the distributio has a epoetial geeral polyomial. Also, we ca ote F ) give by ) has the form: F ) GH)), where G) is the cdf associated to the epoetial distributio with parameter λ > ad H) α with H) α) is a positive icreasig fuctio with H) ad lim + H) +. The iferetial procedure for the parameters of PE distributio is preseted usig the maimum likelihood estimatio. It is show that oe of the estimators ca be obtaied i closed-form ad this allows us to obtai the estimates solvig a simple oe o-liear equatio. The performace of the MLEs are compared usig etesive umerical simulatios. The paper is orgaized as follows. I Sectio we itroduce the PE distributio. Sectio 3 is devoted to some of its mathematical properties. Estimatios of the parameters via the maimum likelihood method are ivestigated i Sectio 4. A simulatio

4 aalysis is give i Sectio 5. I Sectio 6 we apply our proposed model i two real data sets. Fially i Sectio 7 we coclude the paper. F) α.5 α. α.5 α. α.8 α.6 F) α. α.5 α. α.99 α.95 α Figure. λ. Cumulative desity fuctio shapes for PE distributio cosiderig differet values of α ad fied. Polyomial-epoetial distributio The associated probability desity fuctio pdf) of the cdf give by the equatio ) is f) λ αα+ α + ) α + ) e λ α,, + )\{}, ) with the cotiuous etesio for : f) λαα+) e λα. Figure presets some plot of the PE distributio for differet values of α ad λ, ad showig various shapes of the desity fuctio with left skewess. f) α3., λ.5 α., λ.5 α., λ.5 α.8, λ.5 α.4, λ.5 α., λ.5 f) α5, λ. α5, λ.5 α5, λ. α5, λ.5 α5, λ. α5, λ Figure. Desity fuctio shapes for PE distributio cosiderig differet values of α ad λ. As we kow, may distributios such as log-ormal, Weibull, gamma ad epoetiated epoetial, to list a few, do ot allow occurrece of zero values. I this regard, 3

5 the followig remark shows that the PE distributio ca be used as a model with occurrece of zero values. Let us observe that f) λ > for all α > ad λ >. Further, it follows from equatio 3) that f) λα α e λα as. Therefore, the upper tail behavior of the pdf is a product of a polyomial power ad a epoetial polyomial power decay, both of them deped oly o α. Obviously, larger values of α lead to faster decay of the upper tail, which iterprets α as a shape parameter. The hazard rate fuctio hrf) is give by h) λ αα+ α + ) α + ),, + )\{}, with the cotiuous etesio for give by h) λαα+). The study of the behavior of the hazard fuctio is ot a easy task. Glaser s [5] lemma is difficult to be implemeted sice ηt) d dt logft)) does ot has a simple form. However, from graphical aalysis we observed that the hazard fuctio presets a decreasig hazard rate for α > ad λ >, icreasig hazard rate for α < ad λ > ad costat rate for α, ad for this purpose some plots of the hazard fuctio with various values for the parameters α ad λ are preseted i Figure 3. h) 3 4 α3. α. α. α.8 α.4 α. h) α. α.5 α. α.99 α.95 α Figure 3. Hazard fuctio shapes for PF distributio for λ ad cosiderig differet values of α. Moreover, ote that, h) λ as, ad h) λα α as. Hece, we coclude that, the lower tail of the hazard rate fuctio is a costat, while its upper tail is a polyomial which allows for icreasig, decreasig ad costat hazard rate shapes. Remark. Usig the idicator fuctio: A ) if A ad elsewhere, we have followig aalytic epressios for F ), f) ad h): F ) e λ α ),+ )\{}) e λα) {}),+ ) ), 4

6 f) λ αα+ α + ) α + ) e λ ) λαα + ) {}) e λα,+ ) ), ),+ )\{}) α 3) ad h) λ αα+ α + ) α + ) ),+ )\{}) λαα + ) These aalytic epressios will be useful i the et. ) {}),+ ) ). 3. Mathematical properties 3.. Some useful epasios The result below presets a polyomial epasio of the cdf F ) give by ). Propositio 3.. We have the followig epasio, for, + )\{}, where F ) k l j ] A k,l,j [ ) k k+αl+j,) ) + αl j,+ ) ), A k,l,j k l ) k j ) k! )l+j λ k. Proof of Propositio 3.. First of all, let us ow ivestigate a epasio for e λ α by distiguishig the case [, ) ad the case >. If [, ), ote that α α ) give e λ α k ) λ α k k! k l j k l ) k j. The epoetial ad biomial series k k! λ)k k α ) k ) k ) k! )l+j λ) k k+αl+j. If >, ote that α α ). It follows from the epoetial ad biomial series that e λ α k ) λ α k k! k l j k l ) k j 5 k k! λk α ) k ) k! )l+j λ k αl j. ) k

7 Hece e λ α Therefore F ) e λ α k l j ) A k,l,j ) k k+αl+j,) ) + αl j,+ ) ). k l j This complete the proof of Propositio 3.. ) A k,l,j ) k k+αl+j,) ) + αl j,+ ) ). The result below presets a epasio of the pdf f) give by ) via polyomial ad the epoetial fuctio e λ, which will be importat to esure the permutatio of sum ad itegral i several probabilistic quatities. Propositio 3.. We have the followig epasio, for, + )\{}, f) where k l j ] B k,l,j [ ) k R k,l,j ),) ) + S k,l,j ),+ ) ) e λ, B k,l,j k l ) k j ) k! )l+j λ k+, 4) ad R k,l,j ) α k+α )l+j+α+ α + ) k+α )l+j+α + k+α )l+j S k,l,j ) α k+α )l j+α α + ) k+α )l j +α + k+α )l j. Proof of Propositio 3.. Let us observe that λ α α ) λ + λ λ α + λ. Therefore, we ca epress the pdf f) as f) λα α+ α + ) α + )e λ Let us ow ivestigate a epasio for ad >. ) e λ α α ) e λ. by distiguishig [, ) 6

8 If [, ), it follows from the epoetial ad biomial series that α ) e λ k! ) k k ) k λ α k! λ)k k α ) k ) k+ k l j k l ) k j ) k! )l+j λ) k k+α )l+j. Hece If >, epoetial ad biomial series give α ) e λ ) k k! ) λ α k! λk k α ) k k k k l j k l ) k j ) k+ ) k! )l+j λ k k+α )l j. α ) e λ ) ) k k l j k! )l+j λ k k l j [ ] ) k k+α )l+j,) ) + k+α )l j,+ ) ). Owig to this equality, we obtai the desired epasio: f) k l j ] B k,l,j [ ) k k+α )l+j,) ) + k+α )l j,+ ) ) α α+ α + ) α + ) e λ ) B k,l,j ) k R k,l,j ),) ) + S k,l,j ),+ ) ) e λ. k l j This eds the proof of Propositio Momets ad momet geeratig fuctio Here ad after, we cosider a radom variable X followig the PEα, λ) distributio with α > ad λ >. We defie the upper icomplete gamma fuctio as Γs, ) + t s e t dt ad the lower icomplete gamma fuctio as γs, ) ts e t dt, s >,. 7

9 Let r. Usig the otatios ad the result of Propositio 3., the r-momets of X is give by EX r ) r f)d k l j B k,l,j ) k U k,l,j,r + V k,l,j,r ), where U k,l,j,r r R k,l,j )e λ d ad V k,l,j,r + r S k,l,j )e λ d. We have U k,l,j,r α + r+k+α )l+j+α+ e λ d α + ) r+k+α )l+j e λ d γr + k + α )l + j + α +, λ) α λ r+k+α )l+j+α+ α + ) γr + k + α )l + j +, λ) + λ r+k+α )l+j+. O the other had, we have V k,l,j,r α r+k+α )l j+α e λ d α + ) r+k+α )l j e λ d Γr + k + α )l j + α, λ) α λ r+k+α )l j+α α + ) Γr + k + α )l j, λ) + λ r+k+α )l j. r+k+α )l+j+α e λ d + The momet geeratig fuctio of X is give by, for t < λ, Mt) Ee tx ) e t f)d k l j γr + k + α )l + j + α +, λ) λ r+k+α )l+j+α+ r+k+α )l j +α e λ d Γr + k + α )l j + α, λ) λ r+k+α )l j +α B k,l,j ) k U k,l,j t) + V k,l,j t) ), where Uk,l,j t) et R k,l,j )e λ d ad Vk,l,j t) + e t S k,l,j )e λ d. We have Uk,l,j t) α k+α )l+j+α+ e λ t) d α + ) + k+α )l+j e λ t) d γk + α )l + j + α +, λ t) α λ t) k+α )l+j+α+ α + ) γk + α )l + j +, λ t) + λ t) k+α )l+j+. k+α )l+j+α e λ t) d γk + α )l + j + α +, λ t) λ t) k+α )l+j+α+ 8

10 O the other had, we have V + k,l,j t) α k+α )l j+α e λ t) d α + ) + + k+α )l j e λ t) d Γk + α )l j + α, λ t) α λ t) k+α )l j+α α + ) Γk + α )l j, λ t) + λ t) k+α )l j. + k+α )l j +α e t λ) d Γk + α )l j + α, λ t) λ t) k+α )l j +α 3.3. O other meas ad momets The followig result proposes a epasio of the primitive t r f)d, with t >. It will be useful i the et. Propositio 3.3. For ay r ad t >, we have t r f)d k l j where B k,l,j is defied by 4), ad ) B k,l,j ) k Uk,l,j,r t) + V k,l,j,r t) [,+ )t), 5) Uk,l,j,r + k + α )l + j + α +, λ mit, )) t) αγr λ r+k+α )l+j+α+ γr + k + α )l + j + α +, λ mit, )) α + ) + λ r+k+α )l+j+α+ γr + k + α )l + j +, λ mit, )) λ r+k+α )l+j+ Vk,l,j,r + k + α )l j + α, λt) t) αγr + λ r+k+α )l j+α Γr + k + α )l j, λt) λ r+k+α )l j. α + ) Γr + k + α )l j + α, λt) λ r+k+α )l j +α The proof of Propositio 3.3 follows from Propositio 3. with Uk,l,j,r t) mit,) r R k,l,j )e λ d ad Vk,l,j,r t) t r S k,l,j )e λ d. The epressios of these itegrals i terms of upper icomplete gamma fuctio ad the lower icomplete gamma fuctio is obtaied proceedig as Subsectio 3.. Several crucial coditioal momets use the itegral t r f)d for various values of r. The most useful of them are preseted below. For ay t >, The r-th coditioal momets of X is give by, EX r X > t) F t) + t r f)d EX r ) F t) t ) r f)d. 9

11 The r-th reversed momets of X is give by Let µ EX). EX r X t) F t) The mea deviatios of X about µ is give by t δ E X µ ) µf µ) r f)d. µ f)d The mea deviatios of X about the media M is give by M η E X M ) µ f)d. Residual life parameters ca be also determied usig EX r ) ad t r f)d for several values of r. I particular, The mea residual life is defied as Kt) EX t X > t) St) ad the variace residual life is give by V t) V arx t X > t) EX ) St) The mea reversed residual life is defied as EX) t Lt) Et X X t) t F t) ad the variace reversed residual life is give by t ) f)d t ) f)d t tkt) [Kt)]. t f)d W t) V art X X t) tlt) [Lt)] t + F t) t f)d Stress-stregth reliability Let X be a radom variable followig the PEα, λ ) distributio with pdf deoted by f X ) ad Y be a radom variable followig the PEα, λ ) distributio with cdf deoted by F Y ), with α >, λ > ad λ >. The the stress-stregth reliability is defied by R P X > Y ). Sice the itegral o, + ) of the pdf ) with

12 parameters α, λ + λ ) deoted by f ) is equal to oe, we have R P X > Y ) + λ λ + λ + f X )F Y )d α α+ α + ) α + λ ) e λ+λ) α d + f )d λ λ + λ. This result is of iterest i a parametric estimatio cotet; oly λ ad λ eed to be estimated to have a estimatio of R the maimum likelihood estimators for λ ad λ yield the maimum likelihood estimator for R by the plug-i method) Order statistics distributios We ow itroduce order statistics ad preset some of their properties i our mathematical framework geeral results ca be foud, for istace, i [4]). Let X, X,..., X be i.i.d. radom variables followig the PEα, λ) distributio with α > ad λ >. Let us cosider its order statistics as X :, X :,..., X :, i.e., for ay i {,..., }, X i: {X,..., X } with X :... X : so X : X ) ifx,..., X ) ad X : X ) supx,..., X )). Let us ow preset some importat distributios related to X :, X :,..., X :. Some importat of them ivolvig our distributio are preseted below. The geeral epressio of the cdf of X i: is give by F Xi: )! i )! i)! i ) i ) k k i + k [F )]i+k, R. k Hece, for ay, + )/{}, we have F Xi: ) For the case, we have F Xi: )! i )! i)! i ) i ) k [ e λ k i + k k! i )! i)! i ) i ) k k i + k k The geeral epressio of the pdf of X i: is give by f Xi: ) Thus, for ay, + )/{}, we have f Xi: ) α ] i+k. [ e λα] i+k.! i )! i)! [F )]i [ F )] i f), R.! [ i )! i)! λ e λ α ] i α α+ α + ) α + ) e λ i+) α.

13 For the case, we have! [ f Xi: ) e λα] i λαα + ) e λα i+). i )! i)! As i Remark, oe ca epress f Xi: ) i a oe form usig,+ )\{} ) ad {} ). For i < j ad i < j, the geeral epressio of the joit pdf of X i:, X j: ) is give by f Xi:,X j:) i, j )! i )! j)! j i ) [F i)] i [F j ) F i )] j i [ F j )] j f i )f j ). For the case i, j ), + ) /{, )}, we have f Xi:,X j:) i, j )! i )! j)! j i ) λ αα+ i [ e λi α i i ] i [ e λi α i i α + ) α i + )αα+ j α + ) α j + ) i ) j ) e λi α i α e λj j j i λ j+)j ] j i α j j. The epressio of f Xi:,X j:) i, j ) for i, j ), + ) /{, )} ca be set i a similar maer, usig the values of F ) ad f). I the followig propositio, we provide the asymptotic distributios of the etreme values X : ad X :, ad show that they are epoetial ad Gumbel distributios, respectively, which adapt the stadards of the asymptotic distributio of etremes. Propositio 3.4. Let X ) be a sequece of i.i.d. radom variables followig the P Eα, λ) distributio, the X ) coverges i distributio to a radom variable X havig the epoetial distributio ) of parameter λ. ) X) α log) λ coverges i distributio to a radom variable X havig the Gumbel distributio of parameters ad λ. Proof of Propositio 3.4. Let us prove the two poits i tur. Sice X,..., X are i.i.d., usig stadard mathematical argumets, for, + )/{}, the cdf of X ) is give by F X) ) F )) e λ ) α. So lim + F X) ) e λ F X ). This eds the proof of the first poit. Agai, sice X,..., X are i.i.d., usig stadard mathematical argumets, for

14 ) { log) λ, + / F X α log) ) λ ) F log) λ + log) }, the cdf of X α ) log) λ λ ) )) α e is give by λ+ log) λ ) α + log) λ log) + α λ ). Therefore, whe +, several equivaleces give F X α log) ) λ ) e log e λ ) e e λ. Hece lim + F X α log) ) λ ) e e λ F X ). The secod poit is proved Stochastic orderig The orderig mechaism i life time distributios ca be illustrate by the cocept of stochastic orderig. See, for istace, []. This subsectio presets the basic of this cocept, with a result usig the proposed distributio. A radom variable X is said to be stochastically smaller tha a radom variable Y i the stochastic order X st Y ) if the associated cdfs satisfy: F X ) F Y ) for all. hazard rate order X hr Y ) if the associated hrfs satisfy: h X ) h Y ) for all. likelihood ratio order X lr Y ) if the ratio of the associated pdfs give by fx) f Y ) decreases i. Importat equivaleces eist; whe the supports of X ad Y have a commo fiite left ed-poit, the we have: X lr Y X hr Y X st Y. Propositio 3.5. Let X be a radom variable followig the PEα, λ ) distributio with pdf deoted by f X ) ad Y be a radom variable followig the PEα, λ ) distributio with pdf deoted by f Y ), with α >, λ > ad λ >. If λ > λ, the we have X lr Y. Proof of Propositio 3.5. For ay, + )/{}, we have f X ) f Y ) λ e λ λ) α λ ad fx) f Y ) λ λ e λ λ)α. As metioed i Itroductio, the fuctio H) α is icreasig fuctio of. Ideed, we have H ) +αα+ +α) α ) ad a study of fuctio shows that + α α+ + α) α for ay α >. Hece, if λ > λ, fx) f Y ) decreases i ad X lr Y. Propositio 3.5 is proved. Remark. Oe ca prove that λ > λ implies that X hr Y,which follows immediately from the defiitio ): for ay, + )/{}, we have h X ) 3

15 λ α α+ α+) α + α λ α+ α+) α + ) h ) Y ), ad for, we have h X ) λ αα+) λαα+) h Y ) Record values distributios Let us ow focus our attetio o record values ad preset some of their properties i our mathematical cotet geeral results ca be foud, for istace, i []). Let X, X,..., X be i.i.d. radom variables followig the PEα, λ) distributio with α > ad λ >. We defie a sequece of record times U) as follows: U), U) mi{j; j > U ), X j > X U ) } for. We defie the i-th upper record value by R i X Ui), with R X. The geeral epressio of the cdf of R i is give by i [ log F ))] k F Ri ) F )), R. k! k Hece, for ay, + )/{}, we have For the case, we have F Ri ) e λ α i k i F Ri ) e λα [ ] λ α k. k! k The geeral epressio of the pdf of R i is give by λα) k. k! f Ri ) [ log F ))]i f), R. i )! Hece, for ay, + )/{}, we have f Ri ) Note that, for, we have i )! λi i α ) i α α+ α + ) α + ) ) i+ e λ α. f Ri ) α + i )! λα)i e λα. The geeral epressio of the joit pdf of R,..., R ) is give by, for,..., ) R with <... <, f R,...,R ),..., ) f ) h k ). k 4

16 For the case,..., ), + ) /{,..., )}, we have + αα+ α + ) α + f R,...,R ),..., ) λ ) e λ α k α α+ k α + ) α k + k ). The epressio of f R,...,R ),..., ) for,..., ), + ) /{,..., )} ca be set i a similar maer, usig the values of f) ad h). For i < j ad i < j, the geeral epressio of the joit pdf of R i, R j ) is give by f Ri,R j) i, j ) [ log F i))] i i )! For the case i, j ), + ) /{, )}, we have f Ri,R j) i, j ) [ log F i))] i i )! [ α i )! λj i i i αα+ i [ )] j i log F i) F j) h i )f j ). j i )! [ log )] j i F i) F j) j i )! ] i j i )! h i )f j ) [ j α j j i α i ] j i i α + ) α i + )αα+ j α + ) α j + ) α i ) j ) e λj j j. The epressio of f Ri,R j) i, j ) for i, j ), + ) /{, )} ca be set i a similar maer, usig the values of F ), f) ad h). 4. Maimum likelihood estimatio Let X, X,..., X be a radom sample with commo distributio the PEα, λ) distributio with α > ad λ >. Let θ α, λ) be the parameter vector ad,,..., be the observed values. The the likelihood fuctio associated to,..., is give by Lθ) i λ αα+ i α + ) α i + i ) e λi α i i ),+ )\{} i) λαα + ) For the set of simplicity, let us set u i,+ )\{} i ), v i {} i ), ad v i. The log-likelihood fuctio ca be epressed as i lθ) logλ) λ u i + i α α+ i α + ) α i u i log + ) i ) i α i u i i i + v i logλ) + logα) + logα + ) log)). i i 5 e λα ) {} i). i u i +

17 The oliear log-likelihood equatios lθ) θ are give by lθ) α u i α i )[ + α log i )] log i ) i α α+ i i α + ) α i + λ + α + ) v i, α + i α i u i i i log i) i 6) ad lθ) λ λ i u i i α i i. 7) Note that after some algebraic maipulatios i 7) we have that λ i u i α i. 8) i i Replacig 8) i 6) the maimum likelihood estimates of α ad λ are determied by solvig the oe liear equatio 6). Sice it does ot admit ay eplicit solutio, umerical procedures ca be used. Uder mild coditios the maimum likelihood estimators are asymptotically ormal, with a asymptotic variace-covariace matri depedig o the Fisher iformatio matri. Crucial quatities to determie the etries of this matri are the secod partial derivatives of the log-likelihood fuctio give by l θ) α u i α log i ) i )[ + α log i )] log i ))α α+ i α + ) α i + ) i α α+ i i α + ) α i + ) u i α α i [ i )[ + α log i )] log i )] i α α+ i i α + ) α i + ) α ) i λ u i i i log i)) α + α + ) v i, i i ad l θ) λ λ, lθ) λ α α i u i i i log i). i 5. Simulatio aalysis I this sectio a simulatio study is preseted to compare the efficiecy of the maimum likelihood method. The compariso is performed by computig the Bias ad the 6

18 mea square errors MSE) give by Biasα i ) N Biasλ i ) N N ˆα i α, i N ˆλ i λ, i MSEα i ) N MSEλ i ) N N ˆα i α), j N ˆλ i λ), j where N is the umber of estimates obtaied through the MLE. The 95% coverage probability of the asymptotic cofidece itervals are also evaluated. Here we epect that the most efficiet estimatio method returs both Bias ad MSE closer to zero. Additioally, for a large umber of eperimets, usig a 95% cofidece level, the frequecies of itervals that covered the true values of α ad λ should be closer to 95%. The programs ca be obtaied, upo request. The values of the PE were geerated cosiderig the followig algorithm: ) Geerate U i Uiform, ), i,..., ; ) Fid i from the solutio of F i ) u i, i,..., ; The simulatio study is performed uder the assumptio.5, ) ad 4, ), N, ad, 35,..., 46). The chose values allow us to obtai data with both icreasig α < ) ad decreasig α > ) hazard rate. It is importat to poit out that, the results of this simulatio study were similar for differet choices of α ad λ. Figures 4 ad 5 preset the Bias, the MSE ad the coverage probability with a 95% cofidece level of the estimates obtaied through the MLE for differet samples of size. Bias α) MSE α) CP α) Bias λ) MSE λ)....3 CP λ) MLE Figure 4. MLE. Bias, MSEs related from the estimates of α 3 ad λ for N simulated samples uder the From the obtaied results, we ca coclude that as there is a icrease of both Bias ad MSE ted to zero, i.e., the estimator are asymptotic efficiecy. Moreover, the coverage probability of the cofidece levels ted to the omial value assumed.95. Therefore, the MLE showed to be a good estimator for the parameters of the PE distributio. 7

19 Bias α) MSE α) CP α) Bias λ)..4.8 MSE λ)..4.8 CP λ) MLE Figure 5. MLE. Bias, MSEs related from the estimates of α.9 ad λ. for N simulated samples uder the 6. Applicatio to real data I this sectio, we illustrate the fleibility of our proposed distributio by cosiderig two real data sets. The results obtaied from the PE distributio are compared with oes of the Weibull, Gamma, Logormal ad the EE distributios, ad oparametric survival fuctio Here, differet discrimiatio criterio are cosidered based o log likelihood fuctio. Let k be the umber of parameters to be fitted ad ˆθ the MLEs of θ, the discrimiatio criterio methods are respectively: Akaike iformatio criterio AIC lˆθ; ) + k; Corrected Akaike iformatio criterio AICC AIC + k k + ))/ k ); Haa-Qui iformatio criterio HQIC lˆθ; ) + k log log)); Cosistet Akaike iformatio criterio CAIC lˆθ; ) + k log) + ). The best model is the oe which provides the miimum values of these criteria. The Kolmogorov-Smirov KS) test is also cosidered aimig to check the goodess of the fit for the models. This procedure is widely kow ad based o the KS statistic D sup F ) F ; θ), where sup is the supremum of the set of distaces, F ) is the empirical distributio fuctio ad F ; θ) is the cdf of the fitted distributio. Uder a sigificace level of 5% if the data comes from F ; θ) ull hypothesis), the hypothesis is rejected if the P-value is smaller tha.5. The et subsectios give a descriptio of the used data ad their aalysis uder the metioed distributios above. 6.. Air coditioig system data The data have bee preseted by Proscha [9] ad further aalyzed by Adamidis ad Loukas []. Table cosists of the umber of successive failures of the air coditioig system of each member of a fleet of 3 Boeig 7 jet airplaes. Table displays the MLEs, stadard-error ad 95% cofidece itervals for α ad λ. Table 3 presets the results of AIC, AICc, HQIC, CAIC criteria, for the compared 8

20 Table. Data set related to the umber of successive failures of the air coditioig system of each member of a fleet of 3 Boeig 7 jet airplaes distributios. Table. MLE, Stadard-error ad 95% cofidece itervals for α ad λ. θ MLE S. error CI 95% θ) α ;.3) λ ;.8) Table 3. Results of AIC, AICc, HQIC, CAIC criteria ad the p-value for the KS test for the compared distributios cosiderig the umber of successive failures of the air coditioig system of each member of a fleet of 3 Boeig 7 jet airplaes. Test PE Weibull Gamma Logormal EE AIC AICc CAIC HQIC P-value I Figure 6, we have the TTT-plot, the survival fuctio adjusted by the compared distributios ad the o-parametric survival fuctio. Comparig the empirical survival fuctio with the adjusted models we observe a goodess of the fit for the PE distributio, which is cofirmed from differet discrimiatio criterio methods as the PE distributio has the miimum value for all statistics ad the largest for the P-value. Cosequetly, we coclude that the data related to the umber of successive failures of the air coditioig system of each member of a fleet of 3 Boeig 7 jet airplaes ca be described by the PE distributio. 9

21 St) Empirical PE Weibull Gamma Logormal EE ht) Failure Failure Figure 6. Survival fuctio adjusted by the compared distributios ad a o-parametric method cosiderig the data sets related to the umber of successive failures of the air coditioig system of each member of a fleet of 3 Boeig 7 jet airplaes. 6.. Mothly raifall data I this subsectio, we cosidered the data set firstly preseted i Bakouch et al. [3]. The data set is related to the total mothly raifall durig September at São Carlos located i southeaster Brazil. Such city has a active idustrial profile ad high agricultural importace where the study of the behavior of dry ad wet periods has proved to be strategic ad ecoomically sigificat its developmet. Table 4 presets the data related to the total mothly raifall mm) durig September at São Carlos. Table 4. The data set related to the total mothly mm) raifall durig September at São Carlos Nadarajah ad Haghighi [8] observed that maimum likelihood estimate of the shape parameter is o-uique for the Gamma, Weibull ad Geeralized epoetial distributios if data set cosists of zeros ad therefore oe of these three distributios ca fit this kid of data set. O the other had the PE distributio is defied as, which allow us to use the origial values i the presece of zero. Table 5 displays the MLE, stadard-error ad 95% cofidece itervals for α ad λ. Table 6 presets the results of the P-value for the KS test for the compared distributios. Table 5. MLE, Stadard-error ad 95% cofidece itervals for α ad λ. θ MLE S. error CI 95% θ) α ;.69) λ.44..;.9)

22 Table 6. Results of KS test for the compared distributios cosiderig the data set related to the total mothly raifall durig September at São Carlos. Test PE Weibull Gamma Logormal EE P-value I the Figure 7, the survival fuctio adjusted by the compared distributios ad the Kapla-Meier estimator. St) Empirical PE Weibull Gamma Logormal EE ht) Failure Failure Figure 7. Survival fuctio adjusted by the compared distributios ad a o-parametric method cosiderig the data set related to the total mothly raifall durig September at São Carlos. The adjusted models whe compared to the empirical survival show a goodess of the fit for the PE distributio. Additioally, this result is corroborated by the P-value of the KS test. Therefore, our proposed distributio ca be used to describe the data related to the total mothly raifall durig September at São Carlos.

23 7. Cocludig remarks I this paper, we itroduced a ew two-parameter distributio called polyomial epoetial distributio, which geeralizes the ordiary epoetial, liear epoetial ad other combiatios of Weibull distributios, because survival fuctio of the PE distributio represets the product of the survival fuctios of Weibull distributios with parameters λ, ), λ, ),..., ad λ, α). The ew distributio could be a alterative model for lifetime data, specially for the presece of istataeous failures iliers), sice stadard distributios such as Gamma, Weibull, Logormal ad epoetiated epoetial may ot be suitable. We provided a mathematical treatmet of the ew distributio. The estimatio of parameters was discussed by the maimum likelihood approach. Simulatio studies were performed to assess the performace of the maimum likelihood estimators. We fitted the proposed distributio to two real data sets ad compared its fit to those of commoly kow lifetime distributios, establishig that the ew model ca be a good competitor for the latter. We hope that the proposed distributio may be used i wide applicatios as well as lifetime modelig. Future studies ca be ivestigated by usig other baselie fuctios G), see Itroductio sectio. Refereces [] K. Adamidis ad S. Loukas, A lifetime distributio with decreasig failure rate, Statistics & Probability Letters ), pp [] M. Ahsaullah, Record statistics, Nova Sciece Publishers, 995. [3] H. Bakouch, S. Dey, P. Ramos, ad F. Louzada, Biomial-epoetial distributio: Differet estimatio methods with weather applicatios, TEMA to appear) 7). [4] H.A. David ad H. Nagaraja, Wiley series i probability ad statistics, Order Statistics, Third Editio 3), pp [5] R.E. Glaser, Bathtub ad related failure rate characterizatios, Joural of the America Statistical Associatio 75 98), pp [6] R. Gupta ad D. Kudu, Epoetiated epoetial family: A alterative to gamma ad weibull distributios, Biometrical Joural 43 ), pp [7] R. Gupta ad D. Kudu, Geeralized epoetial distributio: Eistig results ad some recet developmet, J. Stat. Pla. If. 37 7), pp [8] S. Nadarajah ad F. Haghighi, A etesio of the epoetial distributio, Statistics 45 ), pp [9] F. Proscha, Theoretical eplaatio of observed decreasig failure rate, Techometrics 5 963), pp [] M. Shaked, J.G. Shathikumar, ad Y. Tog, Stochastic orders ad their applicatios, SIAM Review ), pp