Estimation for Unknown Parameters of the Extended Burr Type-XII Distribution Based on Type-I Hybrid Progressive Censoring Sheme M.M. Amein Department of Mathematis Statistis, Faulty of Siene, Taif University, Hawia (888), Taif, Kingdom of Saudi Arabia. Department of Mathematis, Faulty of Siene, Al-Azhar University, Nasr City (11884), Cairo, Egypt. Neveen Sayed-Ahmed Department of Mathematis Statistis, Faulty of Siene, Taif University, Hawia (888), Taif, Kingdom of Saudi Arabia. Department of Statistis, Faulty of Commere, Al-Azhar University (Girls Branh), Nasr City, Cairo, Egypt. Abstrat: In this paper, Bayesian non-bayesian estimations are used to make point interval estimations for extended Burr Type-XII distribution in ase of Type- I hybrid progressive ensoring sheme. Markov Chain Monte Carlo (MCMC) approximation is needed to solve the ompliated integrations in the ase of Bayesian tehniques. Numerial results using generation data sets are presented. AMS 1 Subjet Classifiation: Primary 6G3; Seondary 6F15. Keywords: Type-I Hybrid Progressive Censoring Sheme; Bayesian Non-Bayesian Estimations; Gibbs Metropolis Sampler; Extended Burr Type-XII distribution. INTRODUCTION The Burr-XII distribution was first introdued by Burr (194), has gained great attention due to the importane of using it in many pratial situations. Inferenes on the Burr-XII distribution have been studied by many authors. Wingo (1993), Moore Papadopoulos (), Mousa Jaheen (), Wu Yu (5), Soliman (5), Xiuhun, et al. (7), Jaheen Okasha (11) Jong-Wuu Wu et al. (14). Gharib et al. (1) have derived a new distribution alled extended Burr-XII distribution (EBXIID) by using the Marshall-Olkin tehnique. They have disussed some properties for the proposed distribution shown that the extended Burr-XII distribution may be obtained as a ompound distribution with mixing exponential distribution. Censored sampling have great interest due to the ost time. In many ases, the experimenter may not always obtain omplete information on failure times for all experimental units. Lifetime distributions under ensored sampling have been attrating great interest in the reent years due to their wide appliations. The most ommon ensoring shemes are Type-I Type-II ensoring, in whih the experiment will be terminated at a predetermined time T in the ase of Type-I ensoring or m th failure in the ase of Type-II ensoring. A new model of ensoring sheme alled Type-I hybrid progressive ensoring sheme. Its a mixture of Type- progressive hybrid ensoring sheme (For more details, see Balakrishnan Aggarwala () Childs et al. (3)). Its onsidered n independent idential items whih plaed on a lifetest. In the beginning, the T value of m are fixed beforeh. The experiment is terminated at a rom
time T = min(x (m:m:n), T ). Suppose, the failure times are denoted by X i, i = 1,,..., m, m<n. Let {R 1,..., R m } are presribed ensoring shemes number of units removed from the experiment satisfying m R i+m = n.when the first failure ours X (1:m:n), the R 1 of the remaining units are romly removed from the experiment. Similarly, when the seond failure ours X (:m:n), the rest of units R are romly moved from the test so on. There are two ases of the end of the experiment, If the m th failure happen before a presribed time T, the experiment is finished at the time point X (m:m:n). Furthermore, the test quit at time T satisfying X (j:m:n) <T <X (j+1:m:n) all the remaining live test units Rj = n R 1 R j j are removed. The observed sample might be one of the following two ases: Case I : {X (1:m:n), X (:m:n),, X (m:m:n) }, if X (m:m:n) <T, Case II : {X (1:m:n), X (:m:n),, X (j:m:n) }, if X (j:m:n) <T <X (j+1:m:n). Now, If the failure times of the n items originally on the test are from an extended Burr Type-XII population with df F (x) pdf f (x) as follow: f (x) = αβx 1 (1 + x ) β 1, ((1 + x ) β ᾱ) α, β,, x>, α 1, ᾱ = 1 α (1) F (x) = 1 α (1 + x ) β ᾱ. () Also, the hazard rate funtion reliability funtion are; respetively: Remarks 1. H (x) = βx 1 (1 + x ) β 1 (1 + x ) β ᾱ S(x) = (3) α (1 + x ) β ᾱ. (4) 1. For α = 1, (1) redue to the ase of Burr-XII distribution.. For α = 1, = 1, (1) redue to the ase of stard Lomax distribution. 3. For β = 1, (1) redue to the ase of log-logisti distribution. MAXIMUM LIKELIHOOD ESTIMATION (MLE) This setion deals with obtaining the point estimation for unknown parameters β using maximum likelihood tehnique. The joint probability density funtion an written from as follows: f 1,,,m:m:n (x 1, x,..., x m ) m f (x i:m:n) [1 F (x i:m:n )] R i, Case I: X m <T; j f (x i:m:n) [1 F (x i:m:n )] R i [1 F (T )] R j, Case II: T<X m. (5) where R j = n R 1 R j j <x i:m:n, T<. The maximum likelihood for EBXIID an be written from (5), as follow: l d β d where d [( x 1 i:m:n (1 + x i:m:n )β 1) ( (1 + x i:m:n )β ᾱ ) ( (1 + x i:m:n ) β ᾱ ) R i ] ( (1 + T ) β ᾱ ) dr j (6) d = Taking logarithm of l gives: { j, T<Xm ; m, X m <T. L = log l d log + d log β d + ( 1) log x i:m:n + (β 1) d log ((1 + xi:m:n )β ᾱ) d R i log ((1 + xi:m:n )β ᾱ) R j log ((1 + T ) β ᾱ) d log (1 + xi:m:n ) The point estimation for β an obtained by differentiating (7) to β, equating the new equations to zero solving to obtain ĉ ˆβ. The analytial solution (7) 4491
may be very diffiult to find. So, Matematia 8 are used to solve the two equations. Also, the orresponding MLE of S(t) H (t) an be written as: xĉ) ˆβ 1 H ˆ (x) = ĉ ˆβxĉ 1 (1 + (1 + xĉ) ˆβ ᾱ α Ŝ(x) = (1 + xĉ) ˆβ ᾱ ASYMPTOTIC CONFIDENCE INTERVALS In this setion, the approximate onfidene interval for, β, S(t) H (t) are obtained., β was approximately bivariate normal with mean (ĉ, ˆβ) varianeovariane matrix I 1. see Greene (), Agresti (). Where I (, β) = L L β L β L β (8) (9). (1) Inverse of the matrix in (1), then the variane-ovariane matrix will be: [ ] I 1 varĉ ovar(ĉ, ˆβ) (, β) =. (11) ovar(ĉ, ˆβ) var ˆβ The (1-α)% approximate onfidene interval for, β, S(t) H (t) ; respetively: ĉ ± Zη varĉ, ˆβ ± Zη H (t) ± Zη var ˆβ, Ŝ(t) ± Zη varŝ(t), ˆ varh ˆ (t), where, var() var(β) are the varianes of the unknown parameters β whih obtained from inverse fisher information matrix Z η is the perentile of the two tail probability η alulated from stard normal distribution. The varh ˆ (t) vars(t)ˆ an be alulated from the next equations: varh ˆ (t) [ G T 1 I 1 G 1 ](ĉ, ˆβ), (1) where ( H(t) G 1 = varŝ(t) [ G T I 1 G ](ĉ, ˆβ), (13) H(t) β ) ( S(t), G = S(t) β I 1 is the variane-ovariane matrix defined in (11). BAYES ESTIMATION Bayesian estimation for unknown parameters are studied in this setion. First, let the parameters β are independent have gamma distributions as prior funtions in the form; respetively: π 1 () = π (β) = ba a 1 e b, >, (14) Ɣ(a) ξ γ β γ 1 e ξβ, β>. (15) Ɣ(γ ) The joint prior distribution of β an be given as: g(, β) = π 1 ()π (β). (16) The joint posterior density funtion an be written as: π(, β x) = l(, β x)π 1 ()π (β) l(x, β)π 1 ()π (β)ddβ. (17) The Bayes estimator under squared error loss funtion of any funtion g is the posterior mean has been given by: ĝ(, β x) = E(g(, β x)) = g(, β)l((, β x))π 1 ()π (β)ddβ. l(, β x)π 1 ()π (β)ddβ ) (18) 449
In general, eq. (18) an t redue to losed form the exat solution may be very hard. In this ase, the MCMC method to generate sample of (17) maybe the best proedure to ompute the double integrations in (18). Next setion shows the MCMC approah inluding Metropolis-Hastings within Gibbs Sampling tehnique. BAYESIAN ESTIMATION USING MCMC APPROACH Now, onsidered MCMC approah to generate samples from the posterior. Its lear that, the solution of (18) is very hard to alulate. So, the MCMC maybe suitable approximation to solve (18) hene estimate ĉ ˆβ. The Metropolis-Hastings algorithm is a very general MCMC method developed by Metropolis et al. (1953) extended by Hastings (197). It an be used to obtain rom samples from any arbitrarily ompliated target distribution of any dimension that is known up to a normalizing onstant. In fat, Gibbs Sampler is a speial ase of a Monte Carlo Markov hain algorithm. It generates a sequene of samples from the full onditional probability distributions of two or more rom variables. Gibbs sampling requires deomposing the joint posterior distribution into full onditional distributions for eah parameter then sampling from them. In order to use the method of MCMC for estimating the parameters of the EBXIID, let us onsider independent priors (14 15), respetively, for the parameters β. The joint posterior density funtion an be obtained up to proportionality by multiplying the likelihood with the prior this an be written as: π(, β x) β d+γ 1 d+a 1 e b ξβ d [ (x 1 i:m:n (1 + x i:m:n )β 1)( (1 + xi:m:n )β ᾱ ) ] d [ ((1 + x i:m:n ) β ᾱ ) ] R i ((1 + T ) β ᾱ ) drj (19) From (19), the posterior density funtion of given β β given are, respetively: g 1 ( β, x) d+a 1 e b d [ (x 1 i:m:n (1 + x i:m:n )β 1)( (1 + xi:m:n )β ᾱ ) ] d [ ((1 + x i:m:n ) β ᾱ ) ] R i ((1 + T ) β ᾱ ) dr j, g (β, x) β d+γ 1 e ξβ d [ ((1 + x i:m:n ) β 1)( (1 + xi:m:n )β ᾱ ) ] () d [ ((1 + x i:m:n ) β ᾱ ) ] R i ((1 + T ) β ᾱ ) drj (1) Its lear that the posterior density funtion of β given in ( 1) annot be redued analytially to well known distributions therefore it is not possible to sample diretly by stard methods, but the plot of it shows that it is similar to normal distribution. So, to generate rom numbers from this distribution, the Metropolis- Hastings method with normal proposal distribution will be used. The proposal sheme to generate β from the posterior density funtions in turn obtain the Bayes estimates the orresponding redible intervals an be written as: 1. Start with (). Set ν = 1 3. Generate ν from g 1 with the normal N( ν 1, var()), where var() is the variane of parameter is obtained from (11). 4. Also, generate β ν from g with the normal N(β ν 1, var(β), where var(β) is the variane of parameter β is obtained from (11). 5. Compute ν β ν 6. Set ν = ν + 1 7. Repeat 3-6 steps N times 8. Rearrange the values i β i, i = Q + 1, Q +,, N. 9. Obtain the Bayes estimates of β with respet to the squared error loss funtion as: E( data) = E(β data) = where Q is burn-in. 1 N Q 1 N Q N i i=q+1 N i=q+1 β i, 4493
Table 1. Censoring Shemes Censoring Shemes R CS1 R 1 ={4,, 4, 3,, 4,, 4, 3, } CS R ={1,, 1, 3,, 1,, 1, 3,, 1,, 1, 1, } CS3 R 3 ={1,, 1,,, 4,,1,,1,,,1,,1,,} CS4 R 4 ={4, 3, 4, 3, 5, 4, 5, 4, 3, 5} CS5 R 5 ={4,, 4, 3,, 1,, 1, 3,, 1,, 3, 3, } CS6 R 6 ={4,,, 3,,,,, 3,,,,, 3,,,,, 3, } CS7 R 7 ={ 8,, 6,8,3, 3,5, 8,} CS8 R 8 ={4,, 4, 3,, 4,,, 3,,,,, 3,,,,, 3, } CS9 R 9 ={1,, 1,,, 1,, 1,,, 1,, 1,,, 1,, 1,,, 1,, 1,,, 1,, 1,, } where r means that repeated r times. Table. Mean, mse, LL, UL, IL ov for β using maximum likelihood method ĉ CS n m mean mse LL UL I L Cov CS1 4 1 3.1883.799 1.7554 4.613.8659.998 CS 15 3.365.549.473 4.458.3785 1 CS3 3.465.496.1846 4.384.138.96 CS4 5 1 3.178.3164 1.6978 4.658.964.988 CS5 15 3.188.4 1.9738 4.839.311.99 CS6 3.48.336.14 4.63.389.99 CS7 3 3.5.466.5855 4.4149 1.893.98 CS8 6 3.11.184.5 4.37 1.983.98 CS9 3 3.3399.555.4541 4.57 1.7716.968 ˆβ CS1 4 1.1563.5735.548 4.8373 5.361 1 CS 15.3397.774.187 4.495 4.355 1 CS3.468.7187.646 4.31 3.6763 1 CS4 5 1.378 1.7815 1.11 5.8539 6.9661.998 CS5 15.1618.619.939 4.98 4.1359.988 CS6.35.58.578 4.71 3.449 1 CS7 3 3.717 1.915 1.1867 4.9566 3.7699 1 CS8 6.41.574.55 3.9768 3.4716 1 CS9 3.5566.83.981 4.133 3.1513 1 1. (1 η)% redible intervals of β an be omputed as ( η ), 1 η (N Q) β( η ), β 1 η (N Q) (N Q) (N Q). NUMERICAL RESULTS Confidene interval for, β, S H using maximum likelihood Bayes inlude MCMC tehnique are alulated numerially. (ns = 5) samples generated from Type-I hybrid progressive samples from the EBXIID with different ensoring shemes R ontains N = 11 values with disard the first Q = 1 values in the ase of MCMC as burn-in. The performanes of ML Bayesian (with joint gamma priors) methods are ompared via mean squared errors tehnique. In Tables ( 5) below, Average point estimation (mean), mean squared error (mse), interval estimation lower limit (LL), interval estimation upper limit (UL), interval length (IL) overage probability (ov) 4494
Table 3. Mean, mse, LL, UL, IL ov for S H using maximum likelihood method Ŝ CS n m mean mse LL UL I L Cov CS1 4 1.876.547.6877.974.398.956 CS 15.81.558.688.9139.57.96 CS3.79.59.687.937.3.94 CS4 5 1.7999.57.6839.9159.3.96 CS5 15.81.5.695.973.11.97 CS6.81.51.74.939.35.94 CS7 3.7915.537.6957.8874.1918.98 CS8 6.845.5.789.91.191.96 CS9 3.7998.479.717.8888.178.936 Hˆ CS1 4 1 1.5311.35.4861.5761.91.996 CS 15 1.6.398.758.491 1.7384.99 CS3 1.755.3959.9138.4973 1.5835 1 CS4 5 1 1.698.4763.41.873.3951.988 CS5 15 1.5661.3435.74.4119 1.6915.99 CS6 1.6144.3484.8678.361 1.4931.98 CS7 3 1.8584.53 1.148.5741 1.4314.916 CS8 6 1.5843.3489.858.3158 1.463.986 CS9 3 1.6987.379 1.546.349 1.883.966 Table 4. Mean, mse, LL, UL, IL ov for β using Bayesian method under mm approximation ĉ CS n m mean mse LL UL I L Cov CS1 4 1.5773.81 1.5873 3.6847.974.93 CS 15.7538.161 1.8649 3.747 1.884.96 CS3.856.135.6 3.7848 1.76.96 CS4 5 1.551.3 1.5588 3.645.456.93 CS5 15.6896.116 1.88 3.6413 1.8185.938 CS6.876.136.691 3.7644 1.6954.966 CS7 3 3.1418.181.3857 3.9547 1.5691.978 CS8 6.8574.13.65 3.733 1.658.956 CS9 3 3.361.15.3 3.839 1.587.98 ˆβ CS1 4 1 1.3416.533.474.9453.4749.99 CS 15 1.6466.11.739 3.1478.438.994 CS3 1.98.986.9349 3.347.413.98 CS4 5 1 1.95.319.476.9418.514.974 CS5 15 1.5416.1586.6597.9993.3396.964 CS6 1.7931.848.8871 3.1676.85.994 CS7 3.4161.476 1.3533 3.99.5696.99 CS8 6 1.7199.181.838 3.598.19.986 CS9 3.88.1835 1.1691 3.39.9 1 for, β, S H are omputed. All results are obtained at = 3; β = 1.8; T =.9; t =.4; α =.5; a =.5; b = 1; γ =.8; ξ = 1; N = 11; Q = 1; η =.5 different ensoring shemes (see Table 1). Remarks: From Tables -5,observe that: When n m inrease, the mean squared error (MSE) derease. 4495
Table 5. Mean, mse, LL, UL, IL, interval length ov for S H using Bayesian method under mm approximation Ŝ CS n m mean mse LL UL I L Cov CS1 4 1.7938.48.671.9155.434.998 CS 15.7853.533.679.8997.88.99 CS3.7756.574.666.8886.6.98 CS4 5 1.7937.463.677.915.335.99 CS5 15.7887.478.6817.8957.14.99 CS6.789.51.6863.8918.55.956 CS7 3.7774.566.68.8747.1947.93 CS8 6.7943.477.6983.893.1919.96 CS9 3.7875.49.6974.8776.18.948 Hˆ CS1 4 1 1.3195.46.3795.595 1.88.994 CS 15 1.4856.356.691.8 1.5893.994 CS3 1.617.9.8814.361 1.487 1 CS4 5 1 1.96.734.714.36.49.988 CS5 15 1.45.33.6434.7 1.5636.986 CS6 1.563.461.88.46 1.3966.984 CS7 3 1.7839.418 1.1137.4541 1.344.98 CS8 6 1.4777.395.797.1581 1.369.988 CS9 3 1.6376.955 1.65.488 1.3.996 In most ases, mean squared errors interval lengths alulated for Bayesian proedure are smaller than alulated for maximum likelihood, so Bayesian proedure is better than the maximum likelihood as expeted. Coverage probabilities in the two methods nearly so lose. CONCLUSION In this paper, the maximum likelihood Bayes methods are used to make interval estimation for the unknown parameters of EBXIID. Markov Chain Monte Carlo approximation is used in Bayesian proedure to solve the hard integrations. Some numerial omputations omparisons are presented to illustrate the methods of inferene developed here. REFERENCES [1] Agresti, A. (). Categorial Data Analysis. nd Edition. New York: Wiley. [] Ali Mousa, M. A. M. Jaheen, Z. F. (). Statistial inferene for the Burr model based on progressively ensored data. Computers & Mathematis with Appliation, Vol. 43, No. 1 11, pp 1441 1449. [3] Balakrishnan, N. Aggarwala R. (). Progressive Censoring: Theory, Methods, Appliations, Birkhauser, Boston, Berlin. [4] Childs A., Chrasekar B., Balakrishnan N., Kundu D. (3). Exat Likelihood inferene based on type-i type-ii hybrid ensored samples from the exponential distribution, Ann. Insti. Statistial Mathematis 55, pp: 319 33. [5] Gharib, M., Mohie El - Din, M.M. Mohammed, B.E. (1). An extended Burr (XII) distribution its appliation to ensored data; Journal of Advaned Researh in Statistis Probability; Vol: ; Issue 1; 37 5: [6] Greene, W.H. (). Eonometri Analysis, 4th Ed. New York: Prentie Hall. [7] Hastings, W.K. (197). Monte Carlo sampling methods using Markovhains their appliations. Biometrika 57, 97 19. [8] Jaheen, Z. F. Okasha, H. M. (11). E-Bayesian estimation for the Burr Type-XII model based on type- ensoring. Applied Mathematial Modelling, 35, 473 4737. [9] Jong-Wuu Wu, Wen-Chuan Lee, Ching-Wen Hong Sung-Yu Yeh. (14). Implementing Lifetime Performane Index of Burr-XII Produts With Progressively 4496
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