Bayes Estimation of the Logistic Distribution Parameters Based on Progressive Sampling

Transcription

1 Appl. Math. Inf. Sci. 0, No. 6, ) 93 Applied Matheatics & Inforation Sciences An International Journal Bayes Estiation of the Logistic Distribution Paraeters Based on Progressive Sapling A. Rashad, M. Mahoud and M. Yusuf, Departent of Matheatics, Faculty of Science, Helwan University, Ain Helwan, Cairo, Egypt. Departent of Matheatics, Faculty of Science, Ain Shas University, Cairo, Egypt. Received: 6 Mar. 06, Revised: 8 Jul. 06, Accepted: Jul. 06 Published online: Nov. 06 Abstract: In this paper we develop approxiate Bayes estiators of the two paraeters logistic distribution. Lindley s approxiation and iportance sapling techniques are applied. The Gaussian-gaa prior distribution and progressively type-ii censored saples are assued. Quadratic, linex and general entropy loss functions are used. The statistical perforances of the Bayes estiates under quadratic, linex and general entropy loss functions are copared with those of the axiu likelihood estiators based on siulation study. Keywords: Logistic distribution, progressively type-ii censoring, Gaussian-gaa prior distribution, loss functions, Lindley s approxiation, iportance sapling technique. Introduction The logistic function is one of the ost popular and widely used for growth odels in deographic studies. The logistic distribution has been applied in studies of population growth, physicocheical phenoena, bio-assay and a life test data ], and of biocheical data 6]. 8] used the logistic function as a odel for agricultural production data. ] copared between the logistic distribution and weibull distribution for odeling wind speed data. ] proposed askew logistic distribution then they derived soe properties for this distribution. Many researchers have used asyetric loss function applied to several statistical odels 4] and 3]). The noral-gaa distribution Gaussian-gaa distribution) is a bivariate four-paraeter faily of continuous probability distributions. It is the conjugate prior of a noral distribution with unknown ean and precision 5]. The Gaussian-gaa distribution has been applied in inventory control probles, the choice of a distribution to describe the deand during the lead tie tie between placeent and delivery of an order) is an iportant proble which has generated considerable research activity. This lead tie deand ay be considered a ixture of two or even three) coponents, naely, flow of orders per unit tie size of orders length of lead tie deand per unit tie 9]. The noral-gaa distribution is a generalization of noral distribution, also applied for fitting real data ], and for the easureent of efficiency in life insurance 4]. Censoring is a coon phenoenon in life-testing and reliability studies. The experienter ay be unable to obtain coplete inforation on failure ties for all experiental units. For exaple, individuals in a clinical trial ay withdraw fro the study, or the study ay have to be terinated for lack of funds. In an industrial experient, units ay break accidentally. In any situations, however, the reoval of units prior to failure is preplanned in order to provide savings in ters of tie and cost associated with testing. Progressive Type-II censoring schee can be described as follows: Suppose n units are placed on a life test and the experienter decides before hand the quantity, the nuber of failures to be observed. Now at the tie of the first failure, R of the reaining n surviving units are randoly reoved fro the experient. At the tie of the second failure, R of the reaining nr Corresponding author e-ail: ohaed.yousof@yahoo.co c 06 NSP

2 94 A. Rashad et al.: Bayes estiation of the logistic... units are randoly reoved fro the experient. Finally, at the tie of the th failure, all the reaining surviving units R = nr...r are reoved fro the experient. Progressive Type-II censoring schee consists of, and R,...,R, such that R...R = n. The failure ties obtained fro a progressive Type-II censoring schee will be denoted by x,...,x. In this paper, we propose different ethods to estiate the paraeters of logistic distribution with Gaussian-gaa prior distribution based on progressive type-ii censoring schee. The paper consists of five sections: In section, we present soe basic concepts which will be used through out this paper. Also it shows the historical survey on soe studies in theoretical and application which have been ade on progressive censoring. Finally, it contains a description of under-study proble. In section, we use the Maxiu Likelihood Estiators MLEs) of the unknown paraeters based on progressively type-ii censoring saples. In section 3, we provide a Bayesian ethod to estiate these paraeters. Also the reliability function and hazard rate function, using progressive type-ii censoring saples is discussed. Based on the square error loss function, linear-exponential loss function, and general entropy loss function. In the Bayesian ethod we propose two approaches to approxiate the posterior: Lindley s approxiation and iportance sapling technique. In section 4, to deonstrate the iportance of the results obtained in the preceding sections, siulation studies are conducted. Using Monte Carlo ethod, with fixed saple size n the total ites put in a life test), with constant censoring schee. In section 5, concluding rearks on siulation study. Maxiu Likelihood Estiators MLEs) In this section, we derive the MLEs of the unknown paraeters based on progressively type-ii censoring saples. Assue the failure tie distribution to be the logistic distribution with probability density function pdf) f x; µ,)= e xµ) e xµ) < µ <, > 0, ) ; <x<,.) and the corresponding cuulative distribution function cdf) is given by Fx; µ,)= e xµ)..) Based on the observed saple x <... < x fro a progressive type-ii censoring schee, R,...,R ), the likelihood function can be written as Lx; µ,)=c fx i )Fx i )] R i,.3) where c=nnr )...nr...r ), f.) and F.) are given by.) and.) respectively. Then Lx; µ,)= c e x i µ)r i ) The log-likelihood function can be written as logl]=l=logc] log] log e x i µ) ) ] Ri ) e x i µ) ) Ri ). R i )x i µ) l=logc] log] R i )x i µ) R i )log e x i µ) ]..4) The MLEs of the unknown paraeters can be obtained by differentiating the log-likelihood function.4) with respect to the unknown paraeters and equating to zero, we get R i ) ˆ ˆ e x i ˆµ) ˆ R i ) e x i ˆµ) ˆ ˆ R i )x i ˆµ) ˆ = 0, e x i ˆµ) ˆ e x i ˆµ) ˆ R i )x i ˆµ) ˆ = 0..5) The solution of the non-linear equations.5) is ˆµ, ˆ. The MLEs of the reliability function, and the hazard rate function are given as ˆRt)= t ˆµ e ˆ, Ĥt)= ˆ e 3 Bayes Estiates for the Unknown Paraeters µ and t ˆµ) ˆ ). In this section Bayesian estiation of the paraeters of the logistic distribution is obtained. Also the reliability function and hazard rate function, using progressive type-ii censoring saples is discussed. Quadratic, linex, and general entropy loss functions are used. c 06 NSP

3 Appl. Math. Inf. Sci. 0, No. 6, ) / 95 Assuing that the joint inforative prior distribution for µ and is a Gaussian-gaa distribution, given by ϕµ,)= γα λ Γα) α e γ e λµδ) ; µ, ), π 0, ), 3.) <δ <,λ > 0,α > 0,γ > 0. By using equations.3) and 3.) we get the joint posterior distribution for µ and as follows ϕ µ, x)= 0 ϕµ,)lx µ,) ϕµ,)lx µ,)dµd = α eγ e λµδ) e e x i µ) ) Ri ) α eγ e λµδ) e 0 e x i µ) ) Ri ) d dµ x i µ)r i ) x i µ)r i ) 3.) Integration in equation 3.) cannot be obtained in a closed for, so we solve it nuerically. In the following subsections we derive Bayesian estiators for location and scale paraeters, the reliability function, and the hazard rate function under soe loss functions. 3. Bayesian Estiators Under Square Error Loss Function. Bayesian estiator for location paraeter µ ˆµ sq = Eµ)= 0 µ α eγ e λµδ) e x i µ) ) Ri ) α e γ e λµδ) e x i µ)r i ) 0 e x i µ) dµd dµd. e x i µ)r i ) ) Ri ). 3.3) Provided that Eµ) exists and is finite. This integration cannot be solved analytically, so we use Lindley s Bayes approxiation 7]. Let uµ, ) be a function of µ and, and we want to find Bayes estiator for it, based on ϕµ, ) as a prior distribution. The log-likelihood function for the logistic distribution based on progressive type II censored saples is given by.4), Bayes estiate of uµ, ) using Lindley approxiation is obtained as follows: uµ,)ϕµ,)l x µ,)dµd 0 E uµ,) x)=. 0 ϕµ,)l x µ,)dµd Let Qµ,) = logϕµ,)] E uµ,) ) uµ,) i j u i j u i Q j )τ i j i i, j,k,w =,,Q = Qµ,) µ u = uµ,),u = uµ,) µ j ]) L i jk u w τ i j τ kw, 3.4) k w µ,) ML,Q = Qµ,),u = uµ,),u = uµ,), µ,u = uµ,), µ L = l µ,l = l µ,l = l,l = 3 l µ 3,L = 3 l µ, L = 3 l µ,l = 3 l 3. Calculate the eleents of atrix{l ij } = l µ l µ l µ l ] = ] τ τ, by using τ τ Matheatica progra we can calculate the inverse atrix, and find the values of τ i j. Substitution in equation α γλ µδ), 3.4), Q =λ µ δ),q = u = µ, the Bayesian estiator for location paraeter µ is given as ˆµ sq µq τ Q τ L τ 3L τ τ L τ τ τ ) L τ τ. Bayesian estiator for scale paraeter Substitution in equation 3.4), α γλ µδ) Q = λ µ δ),q =,u =, the Bayesian estiator for scale paraeter is given as ˆ sq Q τ Q τ L τ τ L τ τ τ ) 3L τ τ L τ 3. Bayesian estiator for reliability function Rt) c 06 NSP

4 96 A. Rashad et al.: Bayes estiation of the logistic... Substitution in equation 3.4), Q = α γλ µδ) λ µ δ),q =,u = Rt),the Bayesian estiator for reliability function Rt) is given by ˆR sq Rt) Q u τ] u τ )Q u τ u τ ) u τ u τ u τ L u τ ) u τ τ L u τ τ τ ) ) 3u τ τ L u τ τ τ ) ) 3u τ τ L u τ τ u τ ) 4. Bayesian estiator for hazard rate function Ht) Substitution in equation 3.4), α γλ µδ) Q = λ µ δ),q =,u = Ht), the Bayesian estiator for hazard rate function Ht) is given by Ĥ sq Ht) Q u ] τ u τ )Q u τ u τ ) u τ u τ u τ L u τ u τ τ )L u τ τ τ ) 3u τ τ )L u τ τ τ ) 3u τ τ )L u τ τ u τ ) 3. Bayesian Estiators Under Linear-Exponential Loss Function LINEX). Bayesian estiator for location paraeter µ ˆµ LINEX = c logeecµ )] Provided that Ee cµ ) exists and is finite. Substitution in equation 3.4), α γλ µδ) Q = λ µ δ),q =,u = e cµ, the Bayesian estiator for location paraeter µ is given as e cµ cq e cµ τ cq e cµ τ c ˆµ LINEX e cµ τ c log ce cµ L τ 3ce cµ L τ τ ce cµ L τ τ τ ) ce cµ L τ τ. Bayesian estiator for scale paraeter Substitution in equation 3.4), α γλ µδ) Q = λ µ δ),q =,u = e c, the Bayesian estiator for scale paraeter is given as e c cq e c τ cq e c τ ce c L τ τ ˆ LINEX c log ce c L ) c e c τ τ τ τ 3ce c L τ τ ce c L τ 3. Bayesian estiator for reliability function Rt) Substitution in equation 3.4), Q = α γλ µδ) λ µ δ),q =,u = e crt), the Bayesian estiator for reliability function Rt) is given by e crt) Q u ) τ u τ ) u τ Q u τ ] u τ u τ ˆR LINEX u c log τ L u τ u τ τ ) L u τ τ τ ) 3u τ τ ) L u τ τ τ ) 3u τ τ )L u τ τ u τ ) 4. Bayesian estiator for hazard rate function Ht) substitution in equation 3.4), Q = α γλ µδ) λ µ δ),q =,u = e cht), the Bayesian estiator for hazard rate function Ht) is given by e cht) Q u ) τ u τ ) u τ Q u τ ] u τ u τ Ĥ LINEX u c log τ L u τ u τ τ ) L u τ τ τ ) 3u τ τ )L u τ τ τ ) 3u τ τ )L u τ τ u τ ) 3.3 Bayesian Estiators Under General Entropy Loss Function. Bayesian estiator for location paraeter µ ˆµ Gentropy =Eµ q )] q Provided that Eµ q ) exists and is finite. Substitution in equation 3.4), Q =λ µ δ), c 06 NSP

5 Appl. Math. Inf. Sci. 0, No. 6, ) / 97 α γλ µδ) Q =,u = µ q, the Bayesian estiator for location paraeter µ is given as µ q qq µ q τ qq µ q τ qq)µ q τ ˆµ Gentropy qµ q L τ 3qµ q L τ τ L qµ q τ τ τ )) qµ q L τ τ. Bayesian estiator for scale paraeter substitution in equation 3.4), Q =λ µδ),q = α γλ µδ),u = q, the Bayesian estiator for scale paraeter is given as q qq q τ qq q τ qq) q τ ˆ Gentropy q q L τ τ q q L τ τ τ ) 3q q L τ τ q q L τ 3. Bayesian estiator for reliability function Rt) Substitution in equation 3.4), Q = α γλ µδ) λ µ δ),q =,u = Rt)) q, the Bayesian estiator for reliability function Rt) is given by Rt)) q Q) u τ u τ ) u τ Q u τ ] u τ u τ u τ ˆR Gentropy L u τ u τ τ ) L u τ τ τ ) 3u τ τ )L u τ τ τ ) 3u τ τ )L u τ τ u τ ) 4. Bayesian estiator for hazard rate function Ht) Substitution in equation 3.4), Q = α γλ µδ) λ µ δ),q =,u = Ht)) q, the Bayesian estiator for hazard rate function Ht) is q q q given by Ht)) q Q) u τ u τ ) u τ Q u τ ] u τ u τ u τ Ĥ Gentropy L u τ u τ τ ) L u τ τ τ ) 3u τ τ )L u τ τ τ ) 3u τ τ )L u τ τ u τ ) It is worth noting that when the value q=, the general entropy loss function is the sae as the squared error loss function. 3.4 Iportance Sapling Technique Iportance sapling is the general technique of sapling fro one distribution to estiate an expectation under a different distribution. In Bayesian analyses, given a likelihood Lθ) for a paraeter vector θ, based on data X and a prior ϕθ), the posterior is given by ϕ θ) = C Lθ)ϕθ), where the noralizing constant C = Lθ)ϕθ)dθ is deterined by the constraint that the density integrate to. This noralizing constant often does not have an analytic expression. General probles of interest in Bayesian analyses are coputing eans and variances of the posterior distribution, and also finding quantities of arginal posterior distributions. In general let gθ) be a paraetric function for which gθ)= gθ)ϕ θ X)dθ, 3.5) needs to be evaluated. In any applications,3.5) cannot be evaluated explicitly, and it is difficult to saple directly fro the posterior distribution, so iportance sapling can be applied. Saples can be drawn fro a distribution with density qθ). In this case, if θ,θ,...,θ N is a rando saple fro qθ) then 3.5) can be estiated with gθ)= N gθ i )w i q, 3.6) N w i where w i = Lθ i)ϕθ i ) qθ i ) and the sapling density qθ) need not be noralized. This technique is described in detail 0].We generate a saples fro noral-gaa distribution with paraeters δ = 0., λ =, α =, and γ = ). We use the following procedure:. Generate Gaaα,γ) and. c 06 NSP

6 98 A. Rashad et al.: Bayes estiation of the logistic... µ Noral δ, λ ).. Repeat this procedure to obtain, µ ),..., N, µ N ). 3. The approxiate value of 3.5) can be obtained by 3.6). 4 Siulation studies To deonstrate the iportance of the results obtained in the preceding sections, siulation studies are conducted. For this purpose, by using Monte Carlo ethod, with fixed saple size n the total ites put in a life test), with constant censoring schee, where R = R = R 3 =... = R, where is the saple size of progressively censored fro the saple of size n. For exaple if the R s are ones, n ust be even and is half the value of n. The following algorith is used to generate saple based on progressive type-ii censoring schee, based on any continuous df F, see 3]..Generate independent Unifor 0,) observations W,...,W..Set V i = W /γ i i, γ i = i R j )for j=i,,...,. 3.U i = V V...V i,,,...,. 4.Set X i = F U i ), then X i, for i =,,...,, is the progressive type-ii censoring schee based on the df F. 5.We repated steps,,3 and 4 000) ties, for different values of n and. Estiation average = 000 θ i 000, ean square error 000 ˆθ i θ i) 000, where, θ is the paraeter and θ is the = estiator. All the coputations are prepared by Matheatica 9. Since the non-linear equations.5) are not solvable analytically, nuerical ethods can be used, as Newton Raphson ethod with initial values closed to real values of the paraeters. Throughout this section we will use the following abbreviations:.ml : eans that the estiate by using the MLE),.B Sq : eans that the estiate under squared error loss function, 3.B Lx,c=5 : eans that the estiate under linex loss function at c=5, 4.B Lx,c=8 : eans that the estiate under linex loss function at c=8, 5.B Lx,c=0 : eans that the estiate under linex loss function at c=0, 6.B Ge,q=0 : eans that the estiate under general entropy loss function at q = 0, 7.B Ge,q=8 : eans that the estiate under general entropy loss function at q = 8, 8.B Ge,q=0 : eans that the estiate under general entropy loss function at q = 0. Fro the siulation studies we noted that:.in general, the Bayesian estiators have ean square error less than that of the MlE..Increasing the saple size leads to decrease ean square error and increase the accuracy of estiators. 3.The estiate of µ under general entropy loss function is the best, and the iportance sapling technique is ore accurate than Lindley s Bayes approxiation. Also by decreasing the value of the paraeter, the accuracy of estiates increases and ean square error decreases. 4.For the paraeter, the estiate under squared error loss function is the best, and it followed by the general entropy loss function. 5.The estiate of the reliability function Rt) under linex loss function is the best at = 0.8, and the general entropy loss function is the best at = 0.7,0.9, especially when the value q=8, and it followed by the MLE. 6.For the hazard rate function Ht), the estiate under linex loss function is the best at = 0.8, and the general entropy loss function is the best at = 0.7, 0.9. Also the iportance sapling technique is ore accurate than Lindley s Bayes approxiation especially in case of the estiate under squared error loss function. 5 Concluding rearks In this paper, MLE and Bayesian estiation of the two paraeters, reliability, and hazard rate functions for the logistic distribution using Lindley s approxiation and iportance sapling technique, based on progressively type-ii censoring saples are obtained. We assued Gaussian-gaa prior distributions for the paraeters. Coputer siulation study is perfored, and show that increasing the saple size leads to decrease ean square error and increase the accuracy of estiators. The estiate of µ under general entropy loss function is the best, and the iportance sapling technique is ore accurate than Lindley s Bayes approxiation. For the paraeter, the estiate under squared error loss function is the best. The estiate of the reliability function Rt) under linex loss function is the best at = 0.8, and the general entropy loss function is the best at = 0.7,0.9, especially when the value q = 8, and it followed by the MLE. For the hazard rate function Ht), the estiate under linex loss function is the best at = 0.8, and the general entropy loss function is the best at = 0.7, 0.9. Also the iportance sapling technique is ore accurate than Lindley s Bayes approxiation especially in case of the estiate under squared error loss c 06 NSP

7 Appl. Math. Inf. Sci. 0, No. 6, ) / 99 Table : The average, ean square error, when n=00, =00, schee00*) and µ = 0. c 06 NSP

8 300 A. Rashad et al.: Bayes estiation of the logistic... Table : The average, ean square error, when n=00, =50, schee50*) and µ = 0. c 06 NSP

9 Appl. Math. Inf. Sci. 0, No. 6, ) / 30 function. The siulation also stresses the iportance of linex and general entropy loss functions as shown in the case studied. Acknowledgeent The authors are grateful to the anonyous referee for a careful checking of the details and for helpful coents that iproved this paper. References ] Alzaatreh, A., Faoye, F. and Lee, C. 04). The gaa-noral distribution:properties and applications. Coputational Statistics & Data Analysis, 69, ] Balakrishnan N. 99). Handbook of the Logistic Distribution, Marcel Dekker, Inc., New York, Basel. Hong. Kong. 3] Balakrishnan, N. and Aggarwala, R. 000). Progressive Censoring: Theory, Methods and Applications, Birkhauser, Boston. 4] Bekker, A., Roux, J. J. J. and Mostert, P. J. 000). A generalization of the copound Rayleigh distribution: using bayesian ethods on cancer survival ties, Counications of Statistics-Theory and Methods, 9: 7, ] Bernardo, J.M. and Sith, A.F.M. 993). Bayesian Theory, Wiley. 6] Gupta, S. S., Qureishi, A. S., and Shah, B. K. 967). Best Linear Unbiased Estiators of the Paraeters of the Logistic Distribution Using Order Statistics, Mieograph Series Departent of Statistics, Purdue University, 9:, ] Lindley, D. V. 980). Approxiate Bayesian Methods, Trabajos de Estadistica Y de Investigacion Operativa, 3:, ] Oliver, F. R. 964). Methods of Estiating the Logistic Growth Function, Applied Statistics, 3:, ] Ord, J.K. and Bagchi, U. 983). The truncated noralgaa ixture as a distribution for lead tie deand. Naval Research Logistics Quarterly, 30, ] Rubinstein, R. Y. 98). Siulation and the Monte Carlo Method, Wiley, New York. ] Scerri, E. and Farrugia, R. 996). Wind data evaluation in the Maltese Islands, Renewable Energy, 7:, ] Subrata, C. Partha J. H. and M. Masoo Ali. 0). Anew skew logistic distribution and its properties, Pak. J. Statist, 8:4, ] Wen, D. and Levy, M. S. 00). Adissibility of bayes estiates under blinex loss for the noral ean proble, Counications in Statistics-Theory and Methods, 30:, ] Yuengert, A. M. 993). The easureent of efficiency in life insurance:estiates of a ixed noral-gaa error odel. Journal of Banking & Finance, 7, A. Rashad is Professor of Matheatical Statistics at Helwan University, Egypt. He is referee of several international journals in the frae of atheatical statistics. He has published research articles in reputed international journals of atheatical and engineering sciences. His ain research interests are: Bayesian analysis, inforation theory, reliability theory and statistical inference. M. Mahoud is Professor of Matheatical Statistics at Ain Shas University, Egypt. He is referee of several international journals in the frae of atheatical statistics. He has published research articles in reputed international journals of atheatical and engineering sciences. His ain research interests are: Bayesian analysis, statistical inference and distribution theory. M. Yusuf is PhD student of Matheatical Statistics at Helwan University, Egypt. He is referee of several international journals in the frae of atheatical statistics. His ain research interests are: Bayesian analysis, order statistics and statistical inference. c 06 NSP