The Truncated Lindley Distribution: Inference and Application

Transcription

1 J. Stat. Appl. Pro. 3, No. 2, (2014) 219 Joural of Statistics Applicatios & Probability A Iteratioal Joural The Trucated Lidley Distributio: Iferece ad Applicatio Sajay Kumar Sigh, Umesh Sigh ad Vikas Kumar Sharma Departmet of Statistics ad DST-CIMS, Baaras Hidu Uiversity, Varaasi, Pi , Idia Received: 22 Apr. 2014, Revised: 1 Ju. 2014, Accepted: 2 Ju Published olie: 1 Jul Abstract: This paper itroduced the trucated versios of the Lidley distributio ad studied the characteristics of the proposed distributios with showig the mootoicity of the desity ad hazard fuctios. The statistical proprieties such as momets, quatile fuctio ad order statistics are also discussed. The maximum likelihood estimators are costructed for estimatig the ukow parameters of the upper, lower ad double trucated Lidley distributios. A set of real data cotaiig the stregths of the glass of aircraft widow, is cosidered to show the applicability of the trucated Lidley distributios. Keywords: Trucated Lidley distributio, Momets, Quatile fuctio, Order statistics, Maximum likelihood estimator, 62F10 1 Itroductio The trucated distributios are quite effectively used where a radom variable is restricted to be observed o some rage ad these situatios are commo i various fields. For istace, i survival aalysis, failures durig the warraty period may ot be couted. Items may also be replaced after certai time followig the replacemet policy, so that failures of the item are igored. May researchers, therefore beig attracted to the problem of aalysig such trucated data ecoutered i various disciplies, proposed the trucated versios of the usual statistical distributios. [2] discussed the applicatio of the trucated versio of the Birbaum-Sauders (BS) distributio to improve a forecastig actuarial model ad particularly for modellig data from isurace paymets that establish a deductible. [1, 17] discussed the applicatio of the trucated Pareto distributio to the statistical aalysis of masses of stars ad of diameters of asteroids. The trucated Weibull distributio has bee foud beig applied i the various fields such as to aalyse the diameter data of trees trucate data specific threshold level, to predict the height distributio of small trees based o icomplete laser scaig data, to modellig the diameter distributio of forest, to characterize the observed Portuguese fire size distributio, to seismological data, o the developmet of the pit depths o a water pipe etc. For more detail o the trucated Weibull distributio ad related refereces readers may refer to book [13] covered the subject of Weibull distributio ad recetly published article [18] based o the trucated Weibull distributio. From the above commetary ad moitorig the wide applicability of the trucated distributios, we proposed the trucatio i the Lidley distributio. The Lidley distributio is mixture of expoetial(θ) ad gamma(2, θ) distributios with their mixig proportios are (1/(1 + θ)) ad (θ/(1 + θ)), respectively ad was first proposed by [11] as couter example of the fiducial statistics. [8] have give the extesive mathematical treatmets to study the various properties of the Lidley distributio ad advocated the use of Lidley distributio over the expoetial distributio cosiderig the waitig times before service of the baks customers. Oe of the mai reasos to prefer the Lidley distributio over the expoetial distributio is its time depedet icreasig failure rate which is commo practice i the survival aalysis. Sice last decade, Lidley distributio has bee attractig the attetio of the researchers, scietists ad the reliability probatioers, ad may author exteded it to the various parsimoious distributios. To ame a few extesios, three parameter geeralized Lidley [16], the geeralized Lidley [14, 15], exteded Lidley [3], weighted Lidley [7], power Lidley [6], expoetial Poisso-Lidley [4] ad the trasmuted Lidley [12] distributios. Correspodig author vikasstats@rediffmail.com

2 220 S. K. Sigh et. al. : The Trucated Lidley Distributio: Iferece... Some extesios of the Lidley distributios e.g. power Lidley ad geeralized Lidley distributios etc. are the good competitors of the Weibull distributio ad ca be quite effectively used to model the real pheomeo where the Weibull distributio seems to be icompatible to the real data. I this directios, oe ca also study the properties of trucated versios of these Lidley s geeralizatios as the alterative models to the trucated Weibull distributio i the literature. Therefore, this article aims to start the discussios with itroducig the cocept of the trucatio i oe parameter Lidley distributio. The rest of the paper is arraged i the followig sectios. I sectio 2, the trucated versios of the Lidley distributio, amed as the upper trucated Lidley (UTL), lower trucated Lidley (LTL), double trucated Lidley (DTL) distributios are itroduced. Particularly, the flexibility of the UTL distributio has bee show demostratig the characteristics of the probability desity (pdf) ad hazard fuctios with differet combiatio of the values of its parameters. The momets, quatile fuctio ad order statistics of the UTL distributio are derived i sectio 3. I sectio 4, the method of the maximum likelihood is applied to obtai the estimates of the parameters of the UTL, LTL ad DTL distributios. I sectio 5, a set of real data is modelled through the differet distributios ad their applicability are compared. Fially, the paper is cocluded i sectio 6. 2 The trucated Lidley distributios A distributio G(x;Θ) is said to be a double trucated distributio over the iterval [ν,ζ] if it has the cumulative distributio fuctio (cdf) defied as ad probability desity fuctio (pdf) is G(x;Θ)= F(x;Θ) F(ν;Θ), ν x ζ, <ν < ζ < (1) F(ζ;Θ) F(ν;Θ) g(x;θ)= f (x;θ), ν x ζ, < ν < ζ < (2) F(ζ;Θ) F(ν;Θ) where, f (x;θ) ad F(x;Θ) are the pdf ad cdf of the baselie model ad Θ R deotes the vector parameters of base lie model. Here, three cases ca be recogized as (i)whe ν = 0 ad ζ, it reduces to baselie model. (ii)whe ν = 0, it is called the upper trucated distributio of the baselie model. (iii)whe ζ, it is called the lower trucated distributio of the baselie model. I this article, we cosider the Lidley distributio as a baselie model with the followig distributio fuctio ( F(x;θ)=1 1+ θx ) e θx,x>0,θ > 0 (3) 1+θ Usig (1) ad (3), the double trucated Lidley distributio is defied as g D (x;θ)= θ 2 (1+x)exp( θx) ; 0 ν x ζ < (4) (1+θ) F(ζ;θ) F(ν;θ) I the followig sectios, we will oly discuss the properties of the upper trucated Lidley distributio ad the same procedure ca be applied to study the properties of the lower trucated Lidley distributio as well as double trucated Lidley distributio. The upper trucated Lidley distributio has the followig pdf is give by g U (x;θ)= θ 2 (1+x)exp( θ(x ζ)) ; 0 x ζ (5) (1+θ)(exp(θζ) 1) θζ It is deoted by UTL(θ,ζ). Note that the above pdf will behave like as (i) d dx g(x;θ)=g (x)= θ 2 (1 θ θx)exp( θ(x ζ)) (1+θ)(exp(θζ) 1) θζ (ii)whe θ 1, g (x)<0, it idicates that g(x) is decreasig i x. (iii)whe θ < 1, g(x) is ui-modal ad mode values is x Mo =(1 θ)/θ, see Figure (1).

3 J. Stat. Appl. Pro. 3, No. 2, (2014) / The correspodig hazard fuctio at epoch t is give by [9] used the term η(x)= f (x) f(x) It followed that H(t;θ)= θ 2 (1+ t)exp( θt) ; 0 t ζ (6) (1+θ) F(ζ;θ) F(t;θ) to determie the mootoicity of the hazard fuctio. For UTL distributio, we get η UT L (x)= g (x) g(x) = f (x)/f(ζ) f(x)/f(ζ) = f (x) f(x) = η L(x) (i)h(0)=θ 2 /[(1+θ)(1 exp( θζ)) θζ exp( θζ)] (ii)h(ζ)=, i.e. as t ζ, H(t) (iii)η L (x)= 1 > 0 x, it implies that the hazard rate fuctio of UTL distributio is icreasig i x ad θ, see Figure 1+x 2 (2). 3 Statistical properties 3.1 Momets ad related measures The rth momet uder the upper trucated Lidley distributio is defied as The rth momet ca also be writte as Particularly, if r = r + 1, we have E[X r ]= θ 2 1 (1+θ) F(ζ;θ) zeta 0 x r (1+x)e θx dx (7) µ r = θ 2 ϕ r (θ,ζ)+ϕ r+1 (θ,ζ),r= 1,2,... (8) 1+θ F(ζ,θ) µ r+1 = θ 2 ϕ r+1 (θ,ζ)+ϕ r+2 (θ,ζ) 1+θ F(ζ,θ) (9) From (8) ad (9), we get µ r+1 = µ r 1+k r+1 (θ,ζ) 1+kr 1,r= 1,2,... (10) (θ,ζ) where, k i (θ,ζ)= ϕ i+1(θ,ζ) ϕ i (θ,ζ),,2,... ( ) ad ϕ 1 (θ,ζ)= 1 e θζ (1+θζ) /θ 2, ( ϕ j (θ,ζ)= jϕ j 1 (θ,ζ) ζ j e θζ) /θ, j = 2,3,... The mea ad variace of UTL distributio ca be easily calculated by θ [(θ+ 2)ϕ 1 (θ,ζ) ζ 2 e θζ] µ = [ ( ) (θ+ 1) 1 e θζ θζe θζ]

4 222 S. K. Sigh et. al. : The Trucated Lidley Distributio: Iferece... Fig. 1: The desity fuctio of UTL distributio for give θ = 0.5, 1 & 1.5 ad ζ = 10 Fig. 2: The hazard fuctio of UTL distributio for give θ = 0.5, 1 & 1.5 ad ζ = 10 ad [ ] σ 2 = V(X)=E[X E(X)] 2 = µ 2 µ 2 1 respectively. We calculated the mea ad variace of ULT distributio for give values of θ ad ζ ad preseted i Table 1. It is observed from Table 1 that mea ad variace decrease as θ icreases while ζ is kept fix. For fixed values of θ, mea ad variace icrease iitially as ζ icreases ad stabilise at a poit. It is due fact that there is o mass to be trucated from the data after a certai poit for a give value of θ. The skewess ad kurtosis of the distributio ca be simply verified by usig the followig relatioship (µ 3 3µµ 2 + 2µ3 ) 2 S k = ( µ 2 µ2) 3 (µ 4 4µµ 3 + 6µ2 µ 2 3µ4 ) K t = ( µ 2 µ2) 2 The skewess ad kurtosis of the UTL distributio are sketched i Figure 3 with respect to its parameters θ ad ζ.

5 J. Stat. Appl. Pro. 3, No. 2, (2014) / Fig. 3: The skewess (lower) ad kurtosis (upper bold) fuctio of UTL distributio with varyig θ ad T Table 1: Mea ad variace for various choices of the values of θ ad ζ. θ ζ=5 ζ=10 ζ=15 ζ=20 ζ=25 µ σ 2 µ σ 2 µ σ 2 µ σ 2 µ σ Quatile fuctio The quatile fuctio is used to describe the percetiles of the distributio ad obtaied as the solutio of the followig equatio G(ξ τ,θ)=τ,τ (0, 1) (11) From (3) ad (11), we have (1+θ+ θξ τ )e (1+θ+θξ τ) = (1+θ)(τF(ζ;θ) 1) exp(1+θ) To solve the above equatio for ξ τ, [10] itroduced the use of Lambert W fuctio for the geeratio of radom variables with Lidley or Poisso-Lidley distributio. The Lambert W fuctio is a multivalued complex fuctio defied as the solutio of the equatio: W(z)exp(W(z))=z, (13) where, z is a complex umber. Now, form (12) ad (13), we obtaied (1+θ+ θξ τ )= W 1 ( (1+θ)(τF(ζ;θ) 1) exp(1+θ) where, W 1 is egative brach of the Lambert W fuctio. Thus, ξ τ = 1 1 θ 1 ( ) (1+θ)(τF(ζ;θ) 1) θ W 1 exp(1+θ) As ζ, from the above equatio (15), we get the quatile fuctio of Lidley distributio derived by [10] as ξ τ = 1 1 θ 1 ( ) (1+θ)(τ 1) θ W 1 exp(1+θ) ) (12) (14) (15) (16)

6 224 S. K. Sigh et. al. : The Trucated Lidley Distributio: Iferece... The media of the UTL distributio ca obtaied as Md x = 1 1 θ 1 ( ) (1+θ)(F(ζ;θ) 2) θ W 1 2exp(1+θ) (17) 3.3 Order statistics I this subsectio, we derive the pdf of the sth(1 s ) order statistics X s:, g s: say, is defied as g s: (t)= 1 B(s, s+1) g(t)gs 1 (t){1 G(t)} s, (18) where, B(s, s + 1) is the beta fuctio. Expadig the biomial expasio, we get s ( )( ) 1 g s: (t)= B(s, s+1) ( 1) i s F(t,θ) s+i f (t,θ) i F(ζ,θ) F(t,θ) where, f (t,θ) ad F(t,θ) are the pdf ad cdfs of the Lidley distributio. For s=1, particularly the pdf of the first order statistics is give by 1 ( ) g 1: (t)= ( 1) i 1 f (t,θ)f i (t,θ) i F i+1 (20) (ζ,θ) Substitutig, the pdf ad cdfs of the Lidley distributio, we obtaied g 1: (t)= θ 2 1 ( 1) i (1+ t)e 1+θ θt( 1 ( 1+ θt ) 1+θ e θt ) i ( ( ) i+1 (21) B( i,i+1) 1 1+ θζ 1+θ )e θζ Similarly, the pdf of X : is give by g : (t)= θ 2 (1+ t)e θt( 1 ( 1+ 1+θ θt ) e θt ) 1 ( ( ) (22) (1+θ) 1 )e θζ 1+ θζ 1+θ The mea ad variace of the sth order statistics ca be obtaied by usig the formulae used i sectio 3.1 for UTL distributio. (19) 4 Maximum likelihood estimatio I this sectio, we describe the procedure to obtai the maximum likelihood estimates (MLE) of the parameters of UTL as well as lower trucated Lidley (LTLD) ad double trucated Lidley (DTLD) distributios based o the radom sample x = {x 1,x 2,,x } of size, so that these distributios ca be effectively used to model the real problems depedig upo the ature of the data. We fitted these distributios to a set of real data i ext sectio. 4.1 MLEs for UTLD Let x be a iid (idepedet ad idetically distributed) sample of size from UTL distributio. The likelihood fuctio based o the observed sample x is give by [ θ 2 L(θ,ζ x)= (1+θ)(exp(θζ) 1) θζ ] (1+x i )e θ (x i ζ) It is to be oted here that S= x i is the joit sufficiet statistics for θ ad ζ. The correspodig log-likelihood equatio is give by ll=2l(θ) l[(1+θ)(exp(θζ) 1) θζ]+ l(1+x i ) θ (23) x i + θζ (24)

7 J. Stat. Appl. Pro. 3, No. 2, (2014) / Note that i the above log-likelihood equatio (27), it is ot possible to get a estimate of ζ i terms of observed sample sice ζ is free from x. Now, from the order statistics, let x (1) < x (2) < < x () be the order sample correspodig to x 1,x 2,,x. The, the MLE ˆζ of ζ ca be take as ˆζ = max(x 1,x 2,,x ) i.e. ˆζ = x () largest observatio. Oce, we get the MLE of ζ, the MLE ˆθ of θ ca be obtaied as the solutio of the followig o-liear equatio: ( ) 2 (1+ζ) e θζ 1 + θζe θζ θ (1+θ) ( e θζ 1 ) 1 θζ x i + ζ = 0 (25) I order to solve the above equatio, we eed to use the iterative procedure like Newto s method. 4.2 MLEs for LTLD The likelihood fuctio based o x from LTL distributio is give by The log-likelihood equatio is give by L(θ,ν x)= θ 2 (1+θ+ θν) (1+x i )e θ (x i ν) (26) ll=2l(θ) l(1+θ+ θν)+ l(1+x i ) θ x i + θν (27) Similarly from the above subsectio, the maximum likelihood estimate of ν will be ˆν = mi(x i );, 2,..., smallest observatio. The maximum likelihood estimate ˆθ of θ ca be uiquely determied by solvig the followig log-likelihood equatio 2 θ (1+ν) (1+θ+ θν) 1 x i + ν = 0 (28) Applyig some mathematical treatmets o equatio (28) yields, ˆθ = ( x 2ν 1)+ x x(2ν+ 3) 4ν(ν+ 1)+1 2(1+ν)( x ν) (29) where, x is the mea of the observed sample. 4.3 MLEs for DTLD The likelihood fuctio uder the assumptio of the double trucated Lidley distributio for the radom variable X, is give by L(θ,ν,ζ x)= θ 2 φ(θ) (1+x i )e θ x i (30) ( where, φ(θ)=(1+θ) e θν e θζ) ( + θ νe θν ζe θζ). The correspodig log-likelihood fuctio is give by ll=2l(θ) l(φ(θ))+ l(1+x i ) θ x i (31) For give MLEs of ν ad ζ as ˆν = x (1) ad ˆζ = x (), respectively, the MLE of θ ca be obtaied by solvig the followig log-likelihood equatio 2 θ φ (θ) x=0 (32) φ(θ) where, φ (θ)= dφ(θ) dθ.

8 226 S. K. Sigh et. al. : The Trucated Lidley Distributio: Iferece... 5 Real data modellig I this sectio, we verified that the trucatio of the Lidley distributio improves its applicability takig the stregth data of glass of the aircraft widow which is reported by [5]. The data are give as: The summary of the above data is give by 18.83, 20.80, , 23.03, 23.23, 24.05, , 25.5, 25.52, 25.80, 26.69, , 26.78, 27.05, 27.67, 29.90, 31.11, 33.20, 33.73, 33.76, , 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, , 45.29, Uits Miimum 1st Qu. Media Mea 3rd Qu. Maximum We fitted the data by expoetial, Weibull, Lidley, ad lower, upper ad double trucated Lidley distributios. The distributio fuctio of the Weibull model is defied as F(x)=1 exp( θx p );θ, p>0 To compare the goodess-of-fit of above models, we used the Akaikes iformatio criterio (AIC), Corrected Akaikes iformatio criterio (AICC), Bayesia iformatio criterio (BIC) ad Kolmogorov-Smirov (K-S) statistic, which are calculated form the followig formulae AIC= 2log(L)+2k, AICC=AIC+ 2k(k+1) ( k 1), BIC= 2log(L)+k log() ad D=sup F (x) F 0 (x). x where, k is the umber of parameters, is the sample size ad F (x) is the empirical distributio fuctio. Based o the data, the fittig summary icludig the estimates of the parameters, log-likelihood, AIC, AICC, BIC ad KS statistics values have bee summarised i Tables 2. The probability-probability (P-P) plots for various distributios based o real data are plotted i Figure 4. Figure 5 shows the log-log plot of the survival fuctio of the cosidered models based o the real data. The above study clearly idicate that the double trucated Lidley distributio gives reasoable fit to the data. From Figure 5, we observed that the usual distributios such as Exp, Lidley ad Weibull are tryig to capture the data from 0 (zero) as they support the whole positive real lie. Whereas, the upper trucated ad the lower trucated Lidley distributios capture oly right ad left tails of the data respectively. The performaces based o used criterio Table 2: Maximum likelihood estimates, AIC, AICC, BIC ad KS statistics values uder cosidered models based o real data Distributio Estimates LogL AIC AICC BIC KS Exp(α) ( ˆα)= W(α,p) ( ˆα, ˆp)=( , ) LD(θ) ( ˆθ)= UTLD(θ,ζ) ( ˆθ, ˆζ)=( , ) LTLD(θ,ν) ( ˆθ, ˆν)=( ,18.83) DTLD(θ,ν,ζ) ( ˆθ, ˆν, ˆζ) =(0.0539,18.83,45.381) (AIC & BIC etc.) of the differet trucated forms of the Lidley distributio ca be diagrammatically show as Worst Lidley UT Lidley LT Lidley DT Lidley Best

9 J. Stat. Appl. Pro. 3, No. 2, (2014) / Fig. 4: The probability-probability (P-P) plots of various distributios based o real data Fig. 5: Log-log plot of the survival fuctio of various models based o real data 6 Coclusios I this article, we itroduced the trucated Lidley distributios called upper trucated, lower trucated ad double trucated Lidley distributio. Particularly, the properties of the upper trucated Lidley distributio such as momets, quatile fuctio ad order statistics are discussed. The maximum likelihood estimators are costructed for estimatig the ukow parameters of the upper trucated Lidley as well as lower trucated ad double trucated Lidley distributios. The goodess-of-fits of the expoetial, Weibull, Lidley ad trucated (lower, upper, double) Lidley distributios have bee compared through the AIC, AICC, BIC ad KS statistics ad foud that the double trucated Lidley distributio fits well the data of the widow stregths. Fially, it is cocluded that the trucated distributios ca be quit effectively used to model the real problems ad so we ca recommed the use of the trucated Lidley distributios i various fields icludig egieerig, medical, fiace ad demography where such type of trucated data are commoly ecoutered. Refereces [1] I. B. Aba, M. M. Meerschaert, A. K. Paorska, Parameter estimatio for the trucated Pareto distributio, Joural of the America Statistical Associatio 101 (2006) [2] S. E. Ahmed, C. Castro-Kuriss, E. Flores, V. Leiva, A. Sahueza, A trucated versio of the birbaum-sauders distributio with a applicatio i fiacial risk, Pak. J. Statist. 26 (2010) [3] H. S. Bakoucha, B. M. Al-Zahrai, A. A. Al-Shomrai, V. A. Marchi, F. Louzada, A exteded Lidley distributio, Joural of the Korea Statistical Society Article i press, doi: /j.jkss

10 228 S. K. Sigh et. al. : The Trucated Lidley Distributio: Iferece... [4] W. Barreto-Souzaa, H. S. Bakouchb, A ew lifetime model with decreasig failure rate, Statistics 47 (2013) [5] E. Fuller-Jr, S. Friema, J. Qui, G. Qui, W. Carter, Fracture mechaics approach to the desig of glass aircraft widows: A case study, SPIE Proc 2286 (1994) [6] M. Ghitay, D. Al-Mutairi, N. Balakrisha, L. Al-Eezi, Power lidley distributio ad associated iferece, Computatioal Statistics ad Data Aalysis 64 (2013) [7] M. Ghitay, F. Alqallaf, D. Al-Mutairi, H. Husai, A two-parameter weighted Lidley distributio ad its applicatios to survival data, Mathematics ad Computers i Simulatio 2011 (2011) [8] M. Ghitay, B. Atieh, S. Nadarajah, Lidley distributio ad its applicatio, Mathematics ad Computers i Simulatio 78 (2008) [9] R. Gupta, System ad Bayesia Reliability: Essays i Hoour of Professor R.E Barlow. Series o Quality, Reliability ad Egieerig Statistics, chap. Nomootoic failure rates ad mea residual life fuctios, World Scietific Press, Sigapore, 2001, pp [10] P. Jorda, Computer geeratio of radom variables with Lidley or Poisso Lidley distributio via the Lambert W fuctio, Mathematics ad Computers i Simulatio 81 (2010) [11] D. Lidley, Fiducial distributios ad Bayes theorem, Joural of the Royal Statistical Society: Series B 20 (1958) [12] F. Merovci, Trasmuted Lidley distributio, Iteratioal Joural of Ope Problems i Computer Sciece ad Mathematics 6 (2013) [13] D. P. Murthy, M. Xie, R. Jiag, Weibull Models, Joh Wiley & Sos, [14] S. Nadarajah, H. Bakouch, R. Tahmasbi, A geeralized Lidley distributio, Sakhya B - Applied ad Iterdiscipliary Statistics 73 (2011) [15] S. K. Sigh, U. Sigh, V. K. Sharma, Expected total test time ad Bayesia estimatio for geeralized Lidley distributio uder progressively Type-II cesored sample where removals follow the beta-biomial probability law, Applied Mathematics ad Computatio 222 (2013) [16] Y. Zakerzadeh, A. Dolati, Geeralized Lidley distributio, Joural of Mathematical Extesio 3 (2009) [17] L. Zaietti, M. Ferraro, O the trucated Pareto distributio with applicatios, Cetral Europea Joural of Physics 6 (2008) 1 6. [18] T. Zhag, M. Xie, O the upper trucated Weibull distributio ad its reliability implicatios, Reliability Egieerig ad System Safety 96 (2011)