Parameters Estimation of the Modified Weibull Distribution Based on Type I Censored Samples

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1 Appled Mathematcal Scences, Vol. 5, 011, no. 59, Parameters Estmaton of the Modfed Webull Dstrbuton Based on Type I Censored Samples Soufane Gasm École Supereure des Scences et Technques de Tuns BP 56, 1008 Bab Menara, Unversty of Tuns, Tunsa soufane.kasm3@gmal.com Maher Berzg École Supereure des Scences et Technques de Tuns BP 56, 1008 Bab Menara, Unversty of Tuns, Tunsa maher.berzg@gmal.com Abstract Recently, Sarhan and Zandn 009 [10] ntroduced a new three parameter dstrbuton called the Modfed Webull dstrbuton MWD whch s a general form for some well known and most commonly used dstrbutons n relablty and lfe testng such as exponental, Raylegh, lnear falure rate and classc Webull dstrbuton [7]. They nvestgated and studed some essental propertes of ths new dstrbuton. In ths paper confdence estmaton for the parameters of the MWD based on type I censored samples wth replacements and wthout replacements s developed. For llustratve purpose, the results obtaned are appled on sets of smulated data. A real data set s analyzed to see how the model works n practce. Mathematcs Subject Classfcaton: 60K10, 60K05, 6F10 Keywords: Modfed Webull dstrbuton, maxmum lkelhood estmaton, lkelhood rato, confdence regon, lnear falure rate, Webull, Raylegh, exponental dstrbuton 1 Introducton A new lfe tme dstrbuton named a modfed Webull dstrbuton MWD s recently proposed by Sarhan and Zandn [10] and Zandn and Sarhan [1]. Ths dstrbuton s a general form for some most commonly used dstrbutons

2 900 Soufane Gasm and Maher Berzg n survval analyss of techncal products, such as exponental, Raylegh, lnear falure rate and Webull dstrbuton. It s known that the exponental dstrbuton has a constant falure rate, the Raylegh dstrbuton has an ncreasng falure rate, the lnear falure rate dstrbuton has a non-ncreasng hazard falure rate and the Webull dstrbuton may have an ncreasng or decreasng falure rate. When t has an ncreasng falure rate, t starts from orgn. Ths condton s a lmtaton of the applcaton for some real relablty problems. The modfed Webull dstrbuton ntroduced by Sarhan and Zandn [10] and Zandn and Sarhan [1] s a very mportant dstrbuton; then t can be used to descrbe several relablty models. It has three parameters, one scale parameter α and two shape parameters β and γ. They used MWDα, β, γ to denote the modfed Webull dstrbuton wth three parameters α, β, γ. The probablty densty functon pdf and the cumulatve densty functon cdf of the MWDα, β, γ has respectvely the followng form: fx, θ =α + βγx γ 1 exp αx βx γ,x>0, where θ =α, β, γ,γ >0,α,β 0 such that α + β>0. F x, θ =1 exp αx βx γ,x>0. The survval functon of the MWDα, β, γ takes the followng form: Sx, θ = exp αx βx γ,x>0, and the hazard rate functon of MWDα, β, γ s hx, θ =α + βγx γ 1,x>0. We can see that the hazard functon s constant f γ = 1, ncreasng f γ>1 and decreasng f γ<1. In the case of β = 0 we obtan the exponental dstrbuton wth parameter α, by puttng α = 0 and γ = we fnd the Raylegh dstrbuton wth parameter β, f γ = a lnear falure dstrbuton wth parameter α and β s obtaned and when α = 0 the basc -parameter Webull dstrbuton wth parameters β and γ wll be derved. Fgure 1 show dfferent patterns of the hazard rate functon of MWDα, β, γ for dfferent parameters values.

3 Parameters estmaton α =0.6; β =0.4; γ =0.5 α =0.6; β =0.4; γ =1.3 α =0.6; β =0.4; γ =6 3.5 hx, α, β, γ x The hazard rate functon The man objectve of ths artcle s the development of confdence estmatons for the parameters of the MWDα, β, γ for type I censored samples wthout and wth replacements. Confdence regons for parameters of lfe tme dstrbutons are of nterest n the vew of theory and applcaton. If the parameters of the lfetme dstrbuton are estmated t s possble to calculate for nstance, the relablty of the product and the number of needed spare. Therefore t s useful to fnd smultaneous confdence regons of the parameters estmatons. In the analyzng of lfe tme data very often censored samples are observed. In these cases t s mpossble to fnd the exact dstrbuton of the pont estmator and therefore asymptotc methods are used for confdence estmaton. Ths s n compatble wth the fact that n practce mostly small samples are observed. In ths research confdence estmatons wll be found by usng the asymptotc normalty of the maxmum lkelhood estmators, ML and the asymptotc χ dstrbuton of the lkelhood rato statstc. Type I censored samples wthout replacement.1 M L estmators of the parameters Let X be the random tme to the falure wth survval functon S X x, θ and densty functon f X x, θ. In the case of type I censored wthout replacement N tems are ndependently observed and the observaton of the -th tem = 1,,N s censored at the tme T. The lkelhood functon from one observaton has the form [] L m,δ =f X x,θ δ S X T,θ 1 δ,

4 90 Soufane Gasm and Maher Berzg { 1 f a falure was observed, wth m = mnx,t and δ = 0 f the observaton was censored at tmet. If all T are equal: T = T, =1,,N the lkelhood functon has the form: { n } Lm, δ = f X x,θ S X T,θ N n, wth n=nt s a random number of observed falures. It follows then { n } Lm, θ = α + βγx γ 1 exp αx βx γ [exp αt βt γ ] N n. We obtan then the followng log-lkelhood functon: ln Lx, θ = [lnα + βγx γ 1 αx βx γ ]+N n αt βtγ. If we calculate the frst partal dervatves of the log-lkelhood functon ln Lx, θ wth respect to the parameters α, β, γ and equatng each to zero, we get then the followng system of nonlnear equatons of α, β, γ. 1 α + βγx γ 1 γx γ 1 α + βγx γ 1 x N nt =0 x γ N nt γ =0 x γ γ ln x β α + βγx γ 1 β x γ ln x β N nt γ ln T =0. To fnd out the maxmum lkelhood estmators of α, β, γ we have to solve the above system of non lnear equatons wth respect to α, β, γ. As t seems, ths system has no closed form soluton n α, β, γ. Then we have to use a numercal technque method, such as optmzaton wth constrants method, to obtan the soluton. We solve the system n three varables α, β, γ, by usng the trustregon-reflectve optmzaton method. The objectve functon to mnmze s the Eucldean norm of vector composed by the system equatons. The bound constrants are consdered allow us, one hand to select the soluton, and the second hand to control the error. In order to accelerate the convergence the gradent and the hessan of objectve functon are explctly determned.. Asymptotc Normalty of the maxmum lkelhood estmators For type I censored samples we can use asymptotc method f the number N of tems tends to nfnty. We know that f X s a random varable wth

5 Parameters estmaton 903 a three parametrc densty functon f X x, θ, twce contnuous dfferentable wth respect to the parameters α, β, γ and wth the expectaton EX > 0. Assume that the thrd absolute moment of X exsts. If the maxmum lkelhood estmators of the parameters n the case of N ndependent and dentcally dstrbuted realzatons are asymptotcally normal dstrbuted, then the parameter estmators n the case of type I censored samples are asymptotcally normal dstrbuted Kahle 1996 [6]. In Borgan 1984 [3] and Svenson 1990 [11] condtons for the asymptotc normalty are gven for type I censored samples wth and wthout replacements. It s easy to prove, that these condtons are fulflled under the above mentoned assumptons..3 Estmaton of the Fsher nformaton and asymptotc confdence bounds Because the MLE of the vector θ =α, β, γ s not obtaned n closed form, t s not possble to derve the exact dstrbuton of the MLE. In ths paragraph, we derve the approxmaton of the Fsher nformaton matrx needed to defne confdence ntervals of the parameters based on the asymptotc dstrbutons of ther MLE. Let Iθ be the Fsher nformaton matrx of the vector of unknown parameters θ =θ 1,θ,θ. Let θ 1 = α,θ = β and θ 3 = γ. The elements of the 3 3 matrx Iθ, I rs θ,r,s {1,, 3}, can be approxmated by I rs ˆθ = ln Lx,ˆθ θ r θ s. In the followng we wll fnd the second partal dervatves of the functon ln Lx, θ, for the calculus of the observed nformaton matrx of α, β and γ. α = α β α γ β = β γ γ 1 α + βγx γ 1 γx γ 1 = α + βγx γ 1 βx γ γ ln x = α + βγx γ 1 γ x γ 1 α + βγx γ 1 αx γ γ ln x = α + βγx γ 1 + = β + β x γ 1 x γ αγ ln x +α ln x βx γ 1 α + βγx γ 1 x γ ln x + β N n T γ ln T. ln x +N n T γ ln T

6 904 Soufane Gasm and Maher Berzg It follows then, the observed nformaton matrx I gven by: α α β α γ I = β α γ α β β γ γ β γ = I 11 I 1 I 13 I 1 I I 3 I 31 I 3 I 33 We can approxmate then the varance-covarance matrx, we obtan: V 11 V 1 V 13 V = V 1 V V 3 = I 1. V 31 V 3 V 33 Mller 1981 [9] proved that asymptotc dstrbuton of the MLE ˆα, ˆβ, ˆγ s gven by ˆα α V 11 V 1 V 13 ˆβ N β, V 1 V V 3. 1 ˆγ γ V 31 V 3 V 33 If we replace the parameters α, β, γ by the correspondng MLE s, we get then an estmate of V, denoted by ˆV and defned as follows: 1 Î 11 Î 1 Î 13 ˆV = Î 1 Î Î 3, where Îj = I j f we replace α, β, γ byˆα, ˆβ, ˆγ. Î 31 Î 3 Î 33 If we use the equaton 1, approxmate 1001 ν% confdence ntervals for the parameters α, β, γ are gven, respectvely as: V 11, V, V 33, ˆα ± z ν ˆβ ± z ν ˆγ ± z ν where z ν s the upper ν th percentle of the standard normal dstrbuton..4 Smultaneous confdence regon based on the lkelhood rato To mprove the confdence regons for small samples t s very useful to construct confdence regons based on the lkelhood rato. It { s known that under regularty condton [5] the log-lkelhood rato q = ln Lx, ˆθ } lnx, θ converges n dstrbuton to a central χ dstrbuton wth 3 degrees of freedom, where ˆθ =ˆα, ˆβ, ˆγ s the maxmum lkelhood estmator of the unknown.

7 Parameters estmaton 905 parameter of nterest θ =α, β, γ. The smultaneous confdence regon s defned by the nequalty q χ 1 ν,3, where χ 1 ν,3 = lnν s the 1 ν quantl of the χ dstrbuton wth 3 degrees of freedom. { The lkelhood rato takes then the form q = ln Lx, ˆθ } lnx, θ = lnν. We obtan then the followng smultaneous confdence regon: ˆα + ˆβ ˆγxˆγ 1 ln α + βγx γ 1 +α ˆα x + β x γ ˆβ xˆγ + N nt α ˆα+βT γ ˆβTˆγ = ln ν. { Remark: If we fx γ = γ the log lkelhood rato q = ln Lm, ˆθ } ln m, θ converges n dstrbuton to a central χ dstrbuton wth degrees of freedom, where ˆθ =ˆα, ˆβ s the maxmum lkelhood estmator of the unknown parameter of nterest θ =α, β. The smultaneous confdence regon s defned by the nequalty q χ 1 ν,, where χ 1 ν, = lnν s the 1 ν quantl of the χ dstrbuton wth degrees of freedom. We obtan then the followng smultaneous confdence regon: ˆα + ˆβ γx γ 1 ln +α ˆα x α + β γx γ 1 +β ˆβ x γ + N n T α ˆα+T γ β ˆβ = ln ν. By fxng β = β we obtan the followng smultaneous confdence regon: ˆα + β ˆγxˆγ 1 ln +α ˆα x γ 1 + α + βγx β x γ xˆγ + N nt α ˆα+ β T γ T ˆγ = ln ν. By fxng α = α we obtan then the followng smultaneous confdence regon: α + ˆβ ˆγxˆγ 1 ln α + βγx γ 1 + β x γ ˆβ xˆγ +N nβtγ ˆβTˆγ = ln ν. 3 Type I censored wth replacement 3.1 M L estmators of the parameters We observe now, N ndependent tems, after each falure the tem s mmedately replaced by a new one and the observaton contnued up to the tme

8 906 Soufane Gasm and Maher Berzg T,,,N. We have then N ndependent observatons of a renewal process each up to tme T,,...,N. The lkelhood functon for one realsaton of ths process, can be expressed as [8]: d L x, θ = f X x j,θ S X R,θ, where d s the number of falures of the -th realzaton of the process, x = x 1,,x d denotes the dstance between falures, S X R,θ s the survval functon of the MWDα, β, γ and d R = T s the rest-tme of the observaton. The lkelhood functon for such renewal process s gven by: x j Lx, θ = N { d α + βγx γ 1 j exp αxj βx γ } j exp αr βr γ. The log-lkelhood functon can be expressed then as: ln Lx, θ = ln α + βγx γ 1 j N d ln αx j + βx γ N j αr + βr γ. To fnd out the maxmum lkelhood estmators, we have to solve the system of non lnear equatons wth respect to α, β, γ. Ths system has no closed form soluton n α, β, γ. Then t s necessary to use a numercal technque method to solve the followng system: 1 α + βγx γ 1 j γx γ 1 j α + βγx γ 1 j β x j N R =0 N x γ j R γ =0 x γ 1 j 1 + γ ln x j α + βγx γ 1 β j x γ j ln x j β N R γ ln R =0. The problem s reduced to fndng the mnmum of constraned nonlnear multvarable norm functon. The Trust-Regon-Reflectve algorthm [4] s used to effectuate the mnmzaton. And the gradent and Hessan are calculated n order to accelerate the convergence of the algorthm.

9 Parameters estmaton Asymptotc confdence bounds In ths secton we have to derve the approxmate confdence ntervals of the parameters α, β, γ n the case of type I censored samples wth replacement, based on the asymptotc dstrbutons of the MLE of the unknown parameters α, β, γ. We now derve the observed Fsher nformaton matrx for the parameters α, β and γ. At follows we wll fnd the second partal dervatves of the functon ln Lx, θ as: α = α β α γ = = β = β γ γ I = 1 α + βγx γ 1 j γx γ 1 j α + βγx γ 1 j = βx γ 1 j 1 + γ ln x j α + βγx γ 1 j γ x γ 1 j α + βγx γ 1 j = β + β αx γ 1 j 1 + γ ln x j α + βγx γ 1 + j x γ j ln x j + x γ 1 j αγ ln x j +α ln x j βx γ 1 j α + βγx γ 1 j x γ j ln x j + β N R γ ln R. The observed nformaton matrx I for ths model s gven then by α α β α γ β α γ α β β γ γ β γ = N R γ ln R I 11 I 1 I 13 I 1 I I 3 I 31 I 3 I 33 As next we can approxmate the varance-covarance matrx V as the nverson of the observed nformaton matrx I. V 11 V 1 V 13 V = V 1 V V 3 = I 1. V 31 V 3 V 33.

10 908 Soufane Gasm and Maher Berzg It follows then that the asymptotc dstrbuton of the MLE ˆα, ˆβ, ˆγ s gven by [9]: ˆα ˆβ ˆγ N α β γ, V 11 V 1 V 13 V 1 V V 3 V 31 V 3 V If we replace the parameters α, β, γ by the correspondng MLE s, we get then an estmate of the varance-covarance matrx V, denoted by ˆV and defned as follows: ˆV = Î 11 Î 1 Î 13 Î 1 Î Î 3 Î 31 Î 3 Î 33 ˆα ± z ν 1, where Îj = I j f we replace α, β, γ byˆα, ˆβ, ˆγ. 4 By usng the equaton 3, we can approxmate 1001 ν% confdence ntervals for the parameters α, β, γ respectvely as: V 11, V, V 33, ˆβ ± z ν ˆγ ± z ν where z ν s the upper ν th percentle of the standard normal dstrbuton. 3.3 Smultaneous confdence regon based on the lkelhood rato As n paragraph {.4 the lkelhood rato takes then the form: q = ln Lx, ˆθ } lnx, θ = lnν, where ˆθ =ˆα, ˆβ, ˆγ s the maxmum lkelhood estmator of the unknown parameter of nterest θ =α, β, γ. The followng smultaneous confdence regon wll be obtaned: ˆα + ˆβ ˆγxˆγ 1 [ ] j ln α + βγx γ 1 + α ˆα x j + βx γ j ˆβxˆγ j j N [ ] + α ˆαR + βr γ ˆβRˆγ = ln ν. Remark: 1. If we fx γ = γ we obtan then the followng smultaneous confdence regon: ˆα + ˆβ γx γ 1 [ j ln + α ˆα x α + β γx γ 1 j + β ˆβ ] x γ j + j N [ α ˆαR + β ˆβ ] R γ = ln ν.

11 Parameters estmaton 909. By fxng β = β we obtan the followng smultaneous confdence regon: ˆα + β ˆγxˆγ 1 [ j ln + α ˆα x γ 1 j + α + βγx β ] x γ j xˆγ j + j N [ α ˆαR + β ] R γ Rˆγ = ln ν 3. By fxng α = α we obtan then the followng smultaneous confdence regon: ln α + ˆβ ˆγxˆγ 1 j α + βγx γ 1 + j N βr γ ˆβRˆγ βx γ j ˆβxˆγ j + = ln ν. 4 Numercal llustraton and smulaton In ths secton numercal study s consdered to apply the prevous theoretcal results to smulated lfetme data. Ths secton s devoted to ntroduce numercal results based on a large smulaton study. The smulaton has been made by wrtng some computer programs wth Matlab Type I censored samples wthout replacement To conduct a computer smulaton based on type I censored samples based on MWD we need to generate samples from the populaton MWD, so ths enhances to develop an algorthm to archve ths purpose. Thus the algorthm s a followng: Step 1. Generate a unform dstrbuted random varable U on 0,1 Step. Set U = F x, θ =1 exp αx βx γ, where θ =α, β, γ,x > 0,γ 0,α,β 0 such that α + β>0 and solve the followng equaton for x>0, βx γ + αx + ln1 U = 0, we obtan then x. Step 3. Compare x wth T,fx <T, then we have a falure and f x T then the tem s censored at tme T. Step 4. Repeat step 1 3, N tmes.

12 910 Soufane Gasm and Maher Berzg We obtan then the vector of falure tmes X =x 1,,x N and N n censored tems at tme T where N s gven and n = nt s a random number of observed falures. In the followng we present practcal applcatons of theoretcal results dscussed n the precedng sectons. Example 1 for fxed γ The followng example llustrate confdence estmaton usng a small type I censored sample wth sze N = 10 censored at tme T =0.8 and wth fxed γ = γ =. Let α =0.500, β =1.100 The observed tmes of falures are: x And the number of observed falures s n = 5. The parameter estmators are ˆα =0.470 and ˆβ = The estmatons of the Fsher nformaton are Î 11 ˆθ =0.16, Î ˆθ =0.70, Î 1 ˆθ = Substtutng the MLE of unknown parameters n the equaton 4, we get then estmaton of the varance covarance matrx as: ˆV = Therefore, the approxmaton 95% two sde confdence ntervals of the parameters α and β are [ 4.335, 5.9], [ 1.65, 3.75] respectvely. We obtan then n fgure 4.1 the confdence regon of the estmators ˆα and ˆβ based on the lkelhood rato statstc β * α Smultaneous confdence regon for N=10,50,

13 Parameters estmaton 911 Example for fxed β The followng example llustrate confdence estmaton usng a large type I censored sample wth sze N = 50 censored at tme T =0.8 and for fxed β = β =0.8. Let α =0.6,γ =3.0. The observed tmes of falures are: x x x x And the number of observed falures s n = 34. The parameter estmators are ˆα =0.6 and ˆγ =3.0. The estmatons of the Fsher nformaton are Î 11 ˆθ =30.93, Î ˆθ =1.44, Î 1 ˆθ = We obtan then n fgure 4.1 the confdence regon of the estmators ˆα and ˆγ based on the lkelhood rato statstc γ 3 * α 3 4 Smultaneous confdence regon for N=50,100. We remark that for the case N = 10 the smultaneous confdence regon s not closed. Example 3 for fxed α The followng example llustrate confdence estmaton usng a type I censored sample wth sze N = 10 censored at tme T = 1 and for fxed α = α =0.1. Let β =., γ= 1. The observed tmes of falures are: x The parameter estmators are ˆβ =.187, ˆγ = The estmatons of the Fsher nformaton are Î 11 ˆθ =1.03, Î ˆθ =0.13, Î 1 ˆθ = Substtutng the MLE of the unknown parameters n the equaton 4, we get

14 91 Soufane Gasm and Maher Berzg then estmaton of varance covarance matrx as: ˆV = We obtan then n fgure 4.1 the confdence regon of the estmators ˆβ and ˆγ based on the lkelhood rato statstc γ 1 * β Smultaneous confdence regon for N=10,50, Type I censored samples wth replacement In the followng we present practcal applcatons of theoretcal results dscussed n the precedng sectons wth three examples. Example 1 for fxed γ The followng example llustrate confdence estmaton usng a type I censored sample wth sze N = 10 censored at tme T =0.9 by gven γ = γ =1.6. Let α =0.400, β= The number of falures d and the rest tme R of the th realzaton are gven n the followng table: d R The parameter estmators are ˆα =0.450, ˆβ = The estmatons of the Fsher nformaton are: Î 11 ˆθ =0.065, Î ˆθ =0.173, Î 1 ˆθ = We obtan then n fgure 4. the confdence regon of the estmators ˆα and ˆβ based on the lkelhood rato statstc.

15 Parameters estmaton β * α Smultaneous confdence regon for N=10,50,100. Substtutng the MLE of unknown parameters n the equaton 4, we get then estmaton of varance covarance matrx as ˆV = Example for fxed β The followng example llustrate confdence estmaton usng a type I censored sample wth sze N = 10 censored at tme T =1.5 and for fxed β = β =1. Let α =.1, γ=0.9. The number of falures d and the rest tme R of the th realzaton are gven n the followng table: d R The parameter estmators are ˆα=.053, ˆγ= The estmatons of the Fsher nformaton are: Î 11 ˆθ =0.71, Î ˆθ =0.057, Î 1 ˆθ =10 4. We obtan then n fgure 4. the confdence regon of the estmators ˆα and ˆγ based on the lkelhood rato statstc. γ * α Smultaneous confdence regon for N=10,50,100.

16 914 Soufane Gasm and Maher Berzg Substtutng the MLE of unknown parameters n the equaton 4, we get then estmaton of varance covarance matrx as: ˆV = Example 3 for fxed α The followng example llustrate confdence estmaton usng a type I censored sample wth sze N = 10 censored at tme T = 1.. α = α = 0.5. Let β =1.1, γ= 1. The number of falures d and the rest tme R of the th realzaton are gven n the followng table: d R The parameter estmators are ˆβ =1.050, ˆγ = The estmatons of the Fsher nformaton are Î 11 ˆθ =0.174, Î ˆθ =0.095, Î 1 ˆθ = We obtan then n fgure 4. the confdence regon of the estmators ˆβ and ˆγ based on the lkelhood rato statstc. γ * β Smultaneous confdence regon for N=10,50,100. Substtutng the MLE of unknown parameters n the equaton 4, we get then estmaton of varance covarance matrx as: ˆV = Applcaton In ths secton we provde a data analyss to nvestgate how the MWD model works n practce and to llustrate the modelng and estmaton procedure. We present a practcal applcaton of the theoretcal results consdered n the precedng sectons. The falure tme data have been obtaned from [1] gvng the lfetme of 50 devces. Ths data are presented n table 1.

17 Parameters estmaton Falure tme data from Aarset We assume a type 1 censored sample wthout replacement. If we choose T =6.4 we obtan the followng number of observed falure n = 19. To analyze the data we have used the followng dstrbutons: Exponental dstrbuton ED, Raylegh dstrbuton RD, Webull dstrbuton WD, lnear falure rate dstrbuton LFRD and modfed Webull dstrbuton MWD. Also we compared these dstrbutons to ft the data by usng the mean square of the dfference between the emprcal cdf and the ftted cdf, denoted MSD: MSD = 1 n ˆF FE. Where ˆF and FE are the emprcal and the estmated cdf calculated at x. The estmated cdf calculated by replacng the parameters of the model adopted wth ther estmatons. The obtaned results are gven n table. The model MLE of the parameters MSD WD β,γ ˆβ =0.0078, ˆγ = LFRD α, β ˆα = 0.000, ˆβ = ED α ˆα = RD β ˆβ = NaN MWD α, β, γ ˆα =0.0133, ˆβ =0.0438, ˆγ = Results The results shown n table 5 ndcate that the model of the MWD fts the gven data better than all other models tested above. Substtutng the MLE of the unknown parameters n, we obtan then estmaton of the varance covarance matrx ˆV as ˆV = Therefore, f we use the equaton 1, the approxmaton 95% two sde confdence ntervals of the parameters α, β and γ are [ , 0.057], [ 0.03, ] and [ 0.035, ] respectvely.

18 916 Soufane Gasm and Maher Berzg 6 Concluson In ths paper we dscussed the parameter estmaton of the MWDα, β, γ based on type I censored samples wth replacements and wthout replacements. The maxmum lkelhood estmators and the confdence regons based on the lkelhood rato statstcs are obtaned. Based on the prevous analyss we can remark that the smultaneous confdence regon based on the lkelhood rato statstcs wll be smaller wth ncreasng sample sze N n both cases wth replacements and wthout replacements. Furthers, smulaton studes are developed to nvestgate the parameter estmatons of the MWD. Confdence estmatons based on the asymptotc χ dstrbuton of the lkelhood rato statstcs wll be found. Based on the MSD crtera, we found that the MWD model concord the data better than those compared dstrbutons. References [1] M. V. Aarset, How to dentfy bathtub hazard rate. IEEE transacton of relablty , [] P. K. Andersen and O. Borgan, Countng process models for lfe hstory data. Scandnavan Journal of Statstcs , [3] O. Borgan, Maxmum lkelhood estmaton n parametrc countng process models, wth applcatons to censored falure tme data. Scandnavan Journal of Statstcs , [4] T.F. Coleman and Y. L, An Interor, Trust Regon Approach for Nonlnear Mnmzaton Subject to Bounds. SIAM Journal on Optmzaton, 1996, [5] G. M. Cordero, On the correctons to the lkelhood rato statstcs. Bometrka , [6] W. Kahle, Estmaton of the parameters of the Webull dstrbuton for censored samples. Metrka , [7] J. F. Lawless, Statstcal models and methods for lfetme Data, John Wley and Sons,New York, 003. [8] R. S. Lpster and A. N. Shryayev Statstcs of random processes II, Sprnger-Verlag, New York, 000. [9] J. R. G. Mller, Survval Analyss. John Wley, New York 1981.

19 Parameters estmaton 917 [10] A. M. Sarhan and M. Zandn, Modfed Webull dstrbuton, Appled Scences , [11] A. Svenson, Asymptotc estmaton n countng processes wth parametrc ntenstes based on one realzaton, Scandnavan Journal of Statstcs , [1] M. Zandn and A. M. Sarhan, Parameters estmaton of the modfed Webull dstrbuton, Appled Mathematcal Scences , Receved: March, 011