The (P-A-L) Generalized Exponential Distribution: Properties and Estimation

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1 Iteratioal Mathematical Forum, Vol. 12, 2017, o. 1, HIKARI Ltd, The (P-A-L) Geeralized Expoetial Distributio: Properties ad Estimatio M.R. Mahmoud ad R.M. Madouh Istitute of Statistical Studies & Research, Cairo Uiversity, Egypt Copyright 2016 M.R. Mahmoud ad R.M. Madouh. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this paper, The PAL geeralized expoetial distributio is itroduced as a ew lifetime distributio. Some properties of the ew distributio are studied. The maximum likelihood estimates ad asymptotic variace-covariace matrix are obtaied. Also, approximate Bayes estimates are computed usig the Gibbs samplig procedure. Fially, a applicatio to real data set is give. Keywords: The PAL family; maximum likelihood estimates; asymptotic variace-covariace matrix; TTT plot; Kapla-Meier estimate; Bayesia estimatio 1. Itroductio Addig parameters to a existig distributio to geerate a geeralized distributio is a very commo approach for developig more flexible distributios. There are may of geeralized families that ca be obtaied by addig oe parameter such as expoetiated family ( Mudholkar ad Sirvastava, 1993), Marshall ad Olki family (Marshall ad Olki, 1997), trasmuted family ( Shaw & Buckley, 2009), The Kw-G family which ca be defied as a expoetiated family ( Cordeiro ad de Castro, 2011), The PAL family (Pappas et al., 2012) ad others. Here we will be cocered with the last oe. 2. The (P-A-L) Geeralized Expoetial Distributio Pappas et al. (2012) itroduced a ew family to geeralize a distributio by addig a shape parameter, amely PAL family. This family takes the followig form

2 28 M.R. Mahmoud ad R.M. Madouh s(x) = l {1 (1 p)s 0(x), x R, p R l (p) + {0} (1) ad whe p 1, the s s 0. The pdf ad hazard fuctio will be f(x) = (p 1)f 0 (x) {1 (1 p)s 0 (x)}l (p), (2) h(x) = (p 1)f 0 (x)h 0 (x) {1 (1 p)s 0 (x)}l [1 (1 p)s 0 (x)], (3) where f 0 ad h 0 are the pdf ad hazard fuctio of the base distributio. Pappas et al. (2012) itroduced the (P-A-L) exteded modified Weibull distributio. Al- Zahrai et al. (2015) studied the (P-A-L) exteded Weibull distributio. Now, we will take s 0 = [1 (1 e λx ) α ] which is the survival fuctio of geeralized expoetial distributio itroduced by Gupta ad Kudu (1999). So, isertig this ito (1) gives s(x) = l {1 (1 p)[1 (1 e λx ) α ]}. l (p) (4) Hece αλ(p 1)(1 e λx ) α 1 e λx f(x) = {1 (1 p)[1 (1 e λx ) α, x, λ, α > 0, p }l (p) R + {0,1} (5) The hazard fuctio is give by αλ(p 1)(1 e λx ) α 1 e λx h(x) = {1 (1 p)[1 (1 e λx ) α }l {1 (1 p)[1 (1 e λx ) α ]} (6) Figures 1-2 illustrate some of the possible shapes of the pdf ad hazard fuctio of the PAL geeralized expetial distributio for selected values of the parameters α, λ ad p.

3 (P-A-L) geeralized expoetial distributio: properties ad estimatio 29 pdf ; 3.5; ; 2.5; ;c 6; ;c 6; Figure (1): Probability desity Fuctio of the (P-A-L) Geeralized Expoetial Distributio for differet values of the parameters hf 1.1; 0.3; ; 0.8; 0.9 2; 2.8; ; 0.6; Figure (2): Hazard Fuctio of the (P-A-L) Geeralized Expoetial Distributio for differet values of the parameters. Now, to fid the raw momets, we have E(X r ) = αλ(p 1) 0 x r (1 e λx ) α 1 e λx {1 (1 p)[1 (1 e λx ) α }l (p) dx Numerical itegratio procedures ca be used to calculate the r th raw momets of the PAL geeralized expoetial distributio. Also, oe ca use Biomial expasio to obtai the followig expressio: E(X r ) = j ( 1) k+i ( j ) (α(k + 1) ) l (p)λ r j=0 k=0 i=0 (1 p) j k i α(p 1)Г(r+1) 1 (i+1) r+1.

4 30 M.R. Mahmoud ad R.M. Madouh 3. Radom Number Geeratio ad Parameter Estimatio Usig iversio method, oe ca geerates radom from the PAL geeralized expoetial with the followig formula x = 1 λ l [1 α e(u)lp p ], 1 p where u distributed as uiform distributio. Now, Parameter estimatio usig maximum likelihood ad Bayesia method will be discussed. 3.1 Maximum Likelihood Estimatio Let x 1, x 2, x 3,, x be a radom sample follow the P-A-L Geeralized Expoetial distributio. The likelihood fuctio is give by L(x; α, λ, p) = αλ(p 1)e λx i(1 e λx i) α 1 l(p) [1 (1 p)[1 (1 e λx i) α ]] (7) ad the log likelihood takes the form ll = l(α) + l(λ) + l(p 1) l(l(p)) λx i (α 1) l(1 e λx i) l ([1 (1 p)[1 (1 e λx i) α ]]) (8) Differetiatig (8) with respect to α, λ ad p, we have ll α = α + l (1 e λx i) (1 p)(1 e λx i) α l (1 e λx i) [1 (1 p)[1 (1 e λx i) α ]], ll λ = λ + (α 1) x ie λxi (1 e λx i ) α(1 p) x ie λx i(1 e λx i) α 1, [1 (1 p)[1 (1 e λx i) α ]] (9) (10)

5 (P-A-L) geeralized expoetial distributio: properties ad estimatio 31 ad ll p = p 1 pl(p) [1 (1 e λxi) α ]. [1 (1 p)[1 (1 e λx i) α ]] (11) Equatig the derivatives i (9), (10) ad (11) to zero ad solve the three oliear equatios umerically, we obtai the maximum likelihood estimators α, λ ad p. The secod derivatives of the logarithms of likelihood fuctio is give by 2 ll α 2 2 ll α λ = = p(1 p) α2 x i e λxi (1 e λx i) (1 p) α (1 e λxi) l (1 e λx i ) [1 (1 p)[1 (1 e λx i) α ]] 2, e λxi(1 e λxi) α 1 {1+αxi l(1 e λx i)} [1 (1 p)[1 (1 e λx i) α ]] +α(1 p) 2 x ie λx i(1 e λx i) 2α 1 l(1 e λx i ) [1 (1 p)[1 (1 e λx i) α ]] 2, 2 α ll p α = (1 e λxi) l (1 e λx i ) [1 (1 p)[1 (1 e λx i) α ]] 2, 2 ll p λ = αx ie λxi(1 e λxi) α 1 [1 (1 p)[1 (1 e λx i) α ]] 2, 2 ll p 2 = (p 1) 2 + ad 2 ll λ 2 = (α 1) x λ2 2 p 2 [l(p)] 2 + [1 (1 e λxi) α 2 ] [1 (1 p)[1 (1 e λx i) α ]] 2, i 2 e λx i (1 e λx i) 2 α(1 p) x i 2 e λx i(1 e λx i) α 2 (αe λx i 1) [1 (1 p)[1 (1 e λx i) α ]] + α 2 (1 p) 2 x i 2 (e λx i) 2 (1 e λx i) 2α 2 [1 (1 p)[1 (1 e λx i) α ]] 2. To obtai iterval estimatio of the parameters(α, λ, p), we first obtai the 3 3 observed iformatio matrix J(θ) which take the form:

6 32 M.R. Mahmoud ad R.M. Madouh I αα I αλ I ap J(θ) = ( I αλ I λλ I λp ) I αp I λp I pp whose elemets are the secod derivatives of the logarithms of likelihood fuctio which are give above. A 100(1-γ)% approximate cofidece iterval for each parameter θ i is give by (θ i Z γ/2 J θi θ i, θ i + Z γ/2 J θi θ i ), where J θi θ i is the (i, i) diagoal elemet of J(θ ) 1 (approximate variacecovariace matrix) for i= 1, 2, 3 ad Z γ/2 is the quatile 1 γ/2 of the stadard ormal distributio. Applicatio to Real Data Data Set: The followig data is a ucesored data set cosistig of 100 observatios o breakig stress of carbo fibers (i Gba): 0.92, 0.928, 0.997, , 1.061, 1.117, 1.162, 1.183, 1.187, 1.192, 1.196, 1.213, 1.215, , 1.22, 1.224, 1.225, 1.228, 1.237, 1.24, 1.244, 1.259, 1.261, 1.263, 1.276, 1.31, 1.321, 1.329, 1.331, 1.337, 1.351, 1.359, 1.388, 1.408, 1.449, , 1.45, 1.459, 1.471, 1.475, 1.477, 1.48, 1.489, 1.501, 1.507, 1.515, 1.53, , 1.533, 1.544, , 1.552, 1.556, 1.562, 1.566, 1.585, 1.586, 1.599, 1.602, 1.614, 1.616, 1.617, 1.628, 1.684, 1.711, 1.718, 1.733, 1.738, 1.743, 1.759, 1.777, 1.794, 1.799, 1.806, 1.814, 1.816, 1.828, 1.83, 1.884, 1.892, 1.944, 1.972, 1.984, 1.987, 2.02, , 2.029, 2.035, 2.037, 2.043, 2.046, 2.059, 2.111, 2.165, 2.686, 2.778, 2.972, 3.504, 3.863, The maximum likelihood estimates for the P A L geeralized expoetial distributio are give by α = , λ = , p = Through The popular Kolmogorov-Smirov goodess of fit test, we fit the P A L geeralized expoetial distributio to this data ad we have Kolmogorov- Smirov test statistic D= with p-value > So, we had o reaso to reject the ull hypothesis that geerated data follows the P A L geeralized expoetial distributio. I our example, the 95% cofidece itervals for the three parameters α, λ, p are listed i the followig table. Table 1: MLEs ad cofidece itervals of parameters i the case of the PAL geeralized expoetial model based o 100 breakig stress data. Parameter Estimate Stadard error 95% cofidece Iterval α (23.774, ) λ (1.5411, ) p (0.0735, )

7 (P-A-L) geeralized expoetial distributio: properties ad estimatio 33 To obtai the MLE of the survival fuctio (S(x)) of the PAL geeralized expoetial distributio, replace the parameters a, λ ad p by their MLEs a, λ, ad p i (4) as follow: l {1 (1 p ) [1 (1 e λ x ) α ]} S (x) = l (p ) Kapla ad Meier (1958) suggested the followig estimatio procedure of reliability fuctio (survival fuctio) for a oparametric model: Fix t > 0. Let t (1) < t (2) < < t () deote the recorded fuctioig times, either util failure or to cesorig, ordered accordig to size. Let J t deote the set of all idices j where t (j) t represets a failure time. Let j deote the umber of uits, fuctioig ad i observatio immediately before time t (j), j = 1,2,,. The the Kapla-Meier estimator (KME) of S(t) is defied as: S (t) = j 1 j j J t For more details about properties of Kapla-Meier estimator see Høylad ad Raussad (1994). Now, the MLE ad Kapla-Meier estimator of reliability fuctio are obtaied for our data set. The estimates are displayed graphically i the followig figures: survivalfuctio R S Figure 1: R is the MLE of reliability fuctio for complete data, S is KME of reliability fuctio for complete data, The empirical scaled TTT trasform (Aarset (1987)) ca be used to idetify the shape of the hazard fuctio. The scaled TTT trasform is covex (cocave) if the hazard rate is decreasig (icreasig), ad for bathtub (uimodal) hazard rates, the scaled TTT trasform is first covex (cocave) ad the cocave (covex).

8 34 M.R. Mahmoud ad R.M. Madouh The TTT plot for complete data is the plot of( i, G ( i )), where G ( i ) = i [ j=1 T j: +( i)t i: ] j=1 T j: for i = 1,2,,, [ i j=1 T j: + ( i)t i: ] is the total time o test at the i th failure for i = 1,2,, ad T j:, j = 1,2,,, are the order statistics of the sample. Figure 1: preset TTT of complete data. 1.0 TTT Figure1: TTT plot for the data set (complete data) As displayed i figure1: the TTT plot has icreasig shaped failure rate which agrees with the plot of the MLE of failure rate. 3.2 Bayesia Estimatio I this sectio, approximate Bayes estimates are computed usig the Gibbs samplig procedure. This procedure is used to geerate samples from the posterior distributios. The approximate Bayes estimators are obtaied uder the assumptios of o-iformative priors. We cosider the PAL geeralized expoetial model with desity fuctio (5) ad a o-iformative joit prior distributio for α, λ ad p give by: π 0 (α, λ, p) 1, (12) αλp where α, λ ad p > 0. The joit posterior distributio for these parameters ca be writte as π( α, λ, p x ) π 0 (α, λ, p)exp{l(x; α, λ, p)} (13) where l(x; α, λ, p) is the logarithm of the log likelihood fuctio give by (8), which is l(x; α, λ, p) = l(α) + l(λ) + l(p 1) l(l(p)) λx i (α 1) l(1 e λx i) l ([1 (1 p)[1 (1 e λx i) α ]]). The reparametrizatio ρ 1 = log (α), ρ 2 = log (λ), ad ρ 3 = log (p) are cosidered. Oe ca obtai from (12) a o-iformative prior for ρ 1, ρ 2, ad ρ 3 as follow π(ρ 1, ρ 2, ρ 3 ) = costat, where < ρ 1, ρ 2 ad ρ 3 <.

9 (P-A-L) geeralized expoetial distributio: properties ad estimatio 35 The covergece of the Gibbs samplig algorithm is obtaied by the choice of the values of hyper-parameters of the uiform priors. Usig the above reparamertizatio, the joit posterior distributios for ρ 1, ρ 2 ad ρ 3 is π(ρ 1, ρ 2, ρ 3 x) π(ρ 1, ρ 2, ρ 3 ). exp {ρ 1 + ρ 2 +. l[exp(ρ 3 ) 1]. lρ 3 exp(ρ 2 ). x i (exp (ρ 1 ) 1) l [1 (exp( exp(ρ 2 ) x i )] l[1 (1 exp(ρ 3 )){1 (1 e (exp(ρ 2 )x i )exp(ρ 1 ) }] } (14) Assumig the prior π(ρ 1, ρ 2, ρ 3 ) = costat, the coditioal posterior distributios used i the Gibbs samplig algorithm are give by: π( ρ 1 ρ 2, ρ 3, x ) exp {ρ 2 +. l(exp(ρ 3 ) 1). lρ 3 }, exp(ρ 2 ). x i (exp (ρ 1 ) 1) l [1 (exp( exp(ρ 2 ) x i )] l[1 (1 exp(ρ 3 )){1 (1 e (exp(ρ 2 )x i )exp(ρ 1 ) )}] π( ρ 2 ρ 1, ρ 3, x ) exp {ρ 1 +. l[exp(ρ 3 ) ]. lρ 3 (exp (ρ 1 ) 1) l (1 (exp( exp(ρ 2 ) x i ) l[1 (1 exp (ρ 3 ){1 (1 e (exp(ρ 2 )x i )exp(ρ 1 ) )}]}, ad π( ρ 3 ρ 1, ρ 2, x ) exp {ρ 1 + ρ 2 exp(ρ 2 ). x i (exp (ρ 1 ) 1) l (1 (exp( exp(ρ 2 ) x i ) l[1 (1 exp(ρ 3 )){1 (1 e (exp(ρ 2 )x i )exp(ρ 1 ) )}]}, Usig the WiBUGS software, posterior summaries of iterest ca be obtaied such as the mea; the stadard deviatio; the credible itervals ad others. A Numerical Example Cosider the data set metioed i subsectio 3.1 We cosider the PAL geeralized expoetial distributio with desity (5) uder the reparametrizatio ρ 1 = log (α),ρ 2 = log (λ), ad ρ 3 = log (p). We assume approximate oiformative prior uiform U(0,1), U(0,0.01)ad U(0,0.7) distributios for ρ 1, ρ 2 ad ρ 3 respectively. A set of Gibbs samples was geerated after a bur-i-sample of size 1000 to elimiate the iitial values cosidered for the Gibbs samplig algorithm. All the calculatios are performed usig the WiBUGS software. Oe way to assess the accuracy of the posterior estimates is by calculatig the Mote Carlo error (MC error) for each parameter. This is a estimate of the differece betwee the mea of sampled values ad the true posterior mea. The simulatio should be ru util the MC error for each parameter of iterest is less tha about

10 36 M.R. Mahmoud ad R.M. Madouh 5% of the sample stadard deviatio. Oe ca ote i our example that MC error less tha 5% of the sample stadard deviatio. Oce the covergece achieved, oe eed to ru the simulatio for a further umber of iteratios to obtai samples that ca be used for posterior iferece. The followig table lists the posterior descriptive summaries of iterest for the PAL geeralized expoetial model. Table 2: Summary results for the posterior parameters i the case of the PAL geeralized expoetial model based o 100 breakig stress data. Parameter Estimate Stadard Deviatio MC error 95% Credible Iterval α E-5 (2.699, 2.718) λ E-5 (1.000, 1.007) p E-5 (1.998, 2.014) From Tables1-2, we coclude that the usual maximum likelihood iferece usig classical asymptotic results could lead to larger cofidece itervals compared to the credible itervals come from the posterior summaries. Refereces [1] M. V. Aarset, How to Idetify a Bathtub Hazard Rate, IEEE Trasactios o Reliability, R-36 (1987), o. 1, [2] B. Al-Zahrai, P. R. D. Mariho, A. A. Fattah, A-H. N. Ahmed ad G. M. Cordeiro, The (P-A-L) Exteted Weibull Distributio: A New Geeralizatio of the Weibull Distributio, Hacettepe Joural of Mathematics ad Statistics, 45 (2015), o. 61, [3] F. M. Cordeiro ad M. de Castro, A ew family of geeralized distributios, Joural of Statistical Computatio ad Simulatio, 81 (2011), [4] R.D. Gupta, D. Kudu, Geeralized expoetial distributios, Australia ad New Zealad Joural of Statistics, 41 (1999), [5] A. Høylad ad M. Rausad, System Reliability Theory: Models ad Statistical Methods, Joh Wiley & Sos, Ic

11 (P-A-L) geeralized expoetial distributio: properties ad estimatio 37 [6] A. W. Marshall ad I. Olki, A ew Method for Addig a Parameter to a Family of Distributios with Applicatio to the Expoetial ad Weibull Families, Biometrika, 84 (1997), o. 3, [7] G. S. Mudholkar ad D. K. Srivastava, Expoetiated Weibull Family for Aalyzig Bathtub Failure-Rate Data, IEEE Trasactio o Reliability, 42 (1993), o. 2, [8] V. Pappas, K. Adamidis ad S. Loukas, A Family of Lifetime Distributios, Iteratioal Joural of Quality, Statistics, ad Reliability, 2012 (2012), Received: November 12, 2016; Published: Jauary 9, 2017

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