COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH *

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1 Jordan Journal of Mathematics and Statistics JJMS 4,, pp COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH * Sanku Dey ABSTRACT: Based on complete as well as type II censored samples, the Bayes estimators for the parameter and reliability function of Rayleigh distribution are obtained. These estimators are obtained on the basis of squared error loss function and LINEX loss function. Comparisons in terms of risks of those under squared error loss and LINEX loss functions with Bayes estimators relative to squared error loss function have been made. Finally, Monte Carlo simulations are performed to compare the performances of the Bayes estimates under different situations.. INTRODUCTION Rayleigh distribution, which is a special case of Weibull distribution, has wide applications in lifetime data analysis especially in reliability theory and survival analysis. Polovko 968 and Dyer and Whisenand 973 demonstrated the importance of this distribution in electro vacuum devices and communication engineering. The origin and other aspects of this distribution can be found in Siddiqui 96, Hirano 986. Ariyawansa and Templeton 984 have also discussed some of its applications. Howlader and Hossian 995 obtained Bayes estimators and highest posterior density intervals for the scale parameter and the reliability function in case of type-ii censored sampling by using Hartigan prior. Singh et al 5 compared maximum likelihood estimators, generalized maximum likelihood estimators and Bayes estimators under type-ii censored sampling of an exponentiated Weibull distribution. They considered independent noninformative types of priors to obtain Bayes estimators. The performances of the Bayes estimators are studied by their simulated risks. Abd Elfattah et al 6a studied the efficiency of maximum likelihood estimates of the parameter of Rayleigh distribution under three cases, Mathematics Subject Classification: 6F; 6F5; 6N5; 6N. Keywords: Bayes Estimator, LINEX Loss Function, Reliability Function, Risk Function, Squared Error Loss Function. Copyright Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan. Received on: July 3, Accepted on: March 3, 6

2 6 Sanku Dey type-i, type-ii and progressive type-ii censored sampling schemes. Abd Elfattah et al 6b have also obtained Bayes risk under squared error loss function and maximum likelihood risk functions for complete and type II censored sampling schemes. Hendi et al 7 obtained Bayes estimators of the scale parameter, reliability function and failure rate by using noninformative prior and Hartigan prior based on upper record values. The probability distribution function pdf and the reliability function of the Rayleigh distribution are respectively given by: x f x = exp x ; x, > and Rt = Ft = P X > t = exp - t ; t, > In the estimation of reliability function, use of symmetric loss function may be inappropriate as recognized by Canfield 97. Varian 975 proposed an asymmetric loss function known as LINEX loss function which has been found to be appropriate in the situation where overestimation is more serious than underestimation or vice-versa. The LINEX loss function for a parameter is given by a L = [ e -a -], a 3 where, = ˆ -, and ˆ is an estimate of. The sign and magnitude of a represents the direction and degree of asymmetry respectively. The positive value of a is used when over estimation is more serious than underestimation while negative value of a is used in the reverse situations. If a is close to zero, this loss function is approximately squared error loss and therefore almost symmetric. Several authors including Basu and Ebrahimi99, Rojo987, Soliman and Zellner986 have used this loss function in various estimation and prediction problems.

3 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 63 If we take = ˆ -, then L is equivalent to the loss function used by Varian 975 and Zellner 986.Again if we take = ˆ - then L is equivalent to the loss function used by Soliman. Here, we consider the natural conjugate family of priors: g β exp α +, α, β > 4 If β =, α =, we get a non-informative prior. Also, if β =, α =, we get the asymptotically invariant prior, proposed by Hartigan 964. The plan of the article is as follows: In section of the present paper, we have obtained the Bayes estimators of taking g as a prior distribution using the squared error ˆ loss function and LINEX loss function L, where = -. The risk of estimators has also been obtained. Comparisons in terms of risk with the estimators of under squared error loss and LINEX loss functions have been made. In section 3, a numerical example has been given to compare the results. In section 4, we obtain Bayes estimator of Rt when the LINEX loss function L, where = ˆ -, is used and compared with those corresponding to the squared error loss function and a simulation study is performed to endorse our estimation techniques in section 5.. BAYES ESTIMATE OF A group of n components have lifetimes which follow a Rayleigh distribution. Due to the cost and time considerations, the failure times are recorded as they occur until a fixed number r n of components have failed. It is quite common in life testing

4 64 Sanku Dey situations that only the first r lifetimes in a sample of n components can be obtained Type II censoring. Let x = x: n, x: n,..., xr: n are i.i.d. random variables, where x in : is the time of the ith component to fail. Since the remaining n-r components have not yet failed and thus have lifetimes greater than x rn :, the likelihood function can be written as n! r T r Lx = exp Π x n r! i : n i = ; > 5 Where, r T= x j: n + n r+ x r: n j= r = n j+ x x j= jn : j : n X j It is straightforward to show that is distributed exponential with mean which implies that T is gamma with shape parameter r and scale parameter or a Chi-square distribution with r degree of freedom giving the probability density function of T as t r h t = exp t T r r Γ r ; t > 6.. Bayes estimator of based on squared error loss function Using Bayes theorem, the posterior pdf of is π x = L x g L x g d

5 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 65 = r + α exp r + α r α + + Γ,, α, β > 7 Considering the squared error loss L ˆ, = ˆ, the Bayes estimator of denoted by ˆ for the above prior, given by the posterior mean of is ˆ = π x d r + α Γ = r + α Γ / 8.. Bayes estimator of based on LINEX Loss Function Under the LINEX loss function 3, the posterior expectation of the loss function L with respect to π x in 7 is E[L ] = ˆ a[ ] ˆ { a[ ] } e π xd 9 ˆ = e -a ˆ E [exp{a }] a E[ -] The value of ˆ that minimizes the posterior expectation of the loss function L, denoted by ˆ is obtained by solving the equation: E L ˆ [ ] = a ˆ ˆ ˆ E [ e exp a ] E =

6 66 Sanku Dey that is, ˆ is the solution of the equation ˆ ˆ ˆ [ exp a E a ] = e E provided that all expectation exists and are finite. ˆ ˆ exp a ˆ = e a exp r + α r + α + r + α Γ r + α exp d r + α r α + + Γ d On simplification, we get the optimal estimate of relative to L is ˆ = a [ { exp }] / a r + α The risk efficiency of ˆ with respect to ˆ under LINEX Loss L The risk functions of estimators These risk functions are denoted by RL ˆ and ˆ relative to L ˆ and R L ˆ are of interest., where subscript L denotes risk relative to L and are given by using h t in 6 as follows: R L ˆ = E L = ˆ a[ ] ˆ { e a[ ] } h T t dt T

7 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 67 [ D] a{ a } a r T e e = T exp dt r Γ r a{ [ D] a T r exp - } T dt r Γ r ar β β = exp[ + D a] r + D+ a r+ α+ 4 a where D = [ exp ] r + α + In the same manner, we get R L ˆ = E L = ˆ a[ ] ˆ { e a[ ] } h t dt T T A = exp[ r T e a a ] T exp dt r Γ r T A r T [ a ] T exp dt r Γ r a a β r β = e [exp A aa ] aa r + + a 5 where, r + α Γ A = r + α Γ

8 68 Sanku Dey The risk efficiency of defined as follows: RE L ˆ, ˆ = ˆ with respect to ˆ under LINEX Loss L may be R ˆ L R ˆ L 6.4. The risk efficiency of estimators ˆ with respect to ˆ under squared error loss The risk functions of the estimators ˆ and ˆ under squared error loss are denoted by RS RS ˆ and R ˆ = S ˆ and are given by : ˆ h t dt T = T r ˆ ˆ + exp T dt r Γ r T r = D T +β exp T a r Γ r dt D a On simplification, we get, / T r exp T r r Γ dt + β exp / Rs φβ + r φ φ Γ r ζ r β r r [ ζ + β ζ 3...] + 3/! 5 / 7 where,

9 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 69 a φ = [ exp ] a r+ α+ β ζ = Γ { r +, } β ζ =Γ{ r, } Again, ζ 3 3 β = Γ{ r, } S ˆ = ˆ T R h t dt = r T ˆ ˆ + T exp dt r Γ r [{ } { } ] / / = A A + T r exp T dt r Γ r On simplification, we get, RS ˆ = β exp A + r + β A Γ r ζ r β r r [ ζ + β ζ 3...] 3/! 5/ 8 The efficiency of RES ˆ with respect to ˆ under squared error loss is defined as: ˆ, ˆ = R ˆ S R ˆ S 9

10 7 Sanku Dey 3. NUMERICAL EXAMPLE To compare the proposed estimator ˆ with the estimator ˆ, the risk functions are computed so as to see whether L and how ˆ performs as compared to ˆ out performs ˆ under LINEX loss ˆ when true loss is squared error. A comparison of this type may be needed to check whether an estimator is inadmissible under some loss function. If it is so, the estimator would not be used for the losses specified by that loss function. For this purpose the risk efficiency have been computed. A random sample of size n = were generated from with =, and ordered as follows:.56,.483,.4864,.59,.6947,.746,.773,.87,.39,.59,.75,.8,.33,.588,.979,.9,.9659,.4,.878, Three sets of values:,,, 5,, were taken for n, r, to provide complete sample, 5% and 5% censored observations, respectively. We compute and report the estimates and their corresponding risk efficiencies for. Tables 4 show Bayes LINEX estimator loss ˆ, Bayes squared error estimator loss ˆ, risk efficiency REL efficiency RES ˆ, ˆ of ˆ with respect to ˆ under LINEX loss and risk ˆ ˆ, From tables -4, we note: under squared error loss. i The risk efficiency REL ˆ, ˆ is greater than one for all values of a, a = ±.5, ±, ±, this conclusion is valid for both complete and censored samples, which indicates that the proposed estimator ˆ is preferable to ˆ see Table-. ii The risk efficiency REL Thus the proposed estimator with proper choice of a. ˆ, ˆ is greater than one for a.5 see table. ˆ performs better than the Bayes estimator ˆ

11 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 7 iii For Hartigan prior, we see that except for a =, under squared error loss, the proposed estimator ˆ has smaller risk than ˆ see Table-. iv The values of the risk efficiencies RE L and RE S are very sensitive for variation in a. v In general, the values of the risk efficiencies RE L are very sensitive for variation in α and β. TABLE : The estimatorsˆ, ˆ, the risk efficiencies REL ˆ, ˆ and RES ˆ, ˆ under the prior g for α = β = a Complete sample 5% Censoring 5% Censoring ˆ =.575 ˆ =.947 ˆ =.68 ˆ RE L RE S ˆ RE L RE S ˆ RE L RE S TABLE : The estimatorsˆ, ˆ, the risk efficiencies REL ˆ, ˆ and RES ˆ, ˆ under the prior g for α =, β = a Complete sample 5% Censoring 5% Censoring ˆ =.3 ˆ =.583 ˆ =.9565 ˆ RE L RE S ˆ RE L RE S ˆ RE L RE S

12 7 Sanku Dey TABLE 3: The estimatorsˆ, ˆ, the risk efficiencies REL ˆ, ˆ and RES ˆ, ˆ under the prior g for α =, β = a Complete sample 5% Censoring 5% Censoring ˆ =.56 ˆ =.97 ˆ =.65 ˆ RE L RE S ˆ RE L RE S ˆ RE L RE S TABLE 4: The estimatorsˆ, ˆ, the risk efficiencies REL ˆ, ˆ and RES ˆ, ˆ under the prior g for α =, β = a Complete sample 5% Censoring 5% Censoring ˆ =.88 ˆ =.63 ˆ =.59 ˆ RE L RE S ˆ RE L RE S ˆ RE L RE S BAYES ESTIMATOR OF F t Let γ = Ft be the probability that a system will survive a specified mission time t. t By substituting = in 7, we obtain the posterior p.d.f. of γ as : logγ

13 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 73 r + α exp t log γ t π γ x = log γ r + α r + α + Γ γ t logγ = r+ α r+ α r+ α γ log γ r + α Γ t, <γ < Using the convex loss function L, = ˆ γ γ, it is seen that this loss function is quite asymmetric when a = with overestimation being more serious than underestimation. Also, when a <, L rises almost exponentially and almost linearly when < and almost linearly when >. For small values of a the loss function L is approximately squared error loss and therefore almost symmetric. For more details about L see Zellner 986. The posterior expectation of the LINEX loss function L is: E γ L ˆ γ γ = [ e a a ˆ γ γ - ] π γ x dγ ˆ γ aγ = e a E + γ ˆ γ e E γ a aγ For a minimum to exist at = E γ [ L ] ˆ = ae aγ E e aγ a ˆ γ γ = aγ E γ e = e a ˆ γ

14 74 Sanku Dey The Bayes estimator of γ relative to L simple algebra, denoted by and is given by after = - j = j r + α α a jt log[ + ] a j! Under the squared error loss function L ˆ γ, γ = ˆ γ γ, the Bayes estimator of γ, denoted by is given by ˆ γ = x d γπ γ γ = γ γ log γ r + α Γ t r+ α r+ α r+ α After simple algebra, the Bayes estimator of γ is obtained as: d γ ˆ γ = + r+ α t It can be seen that the risk functions relative to loss function L do not exist. Let us consider the set of data generated in example 3, let t =, we compute and, the results with the corresponding values of a are given in table-5. It can be seen from table-5 that, the Bayes estimates of reliability function relative to L for positive value of a are lower than the Bayes estimates under squared error loss function, this conclusion is valid for both complete and censored samples.

15 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 75 TABLE 5: Bayes estimation of reliability function based on LINEX loss and squared error loss for the variations in a, α and β t=, the true value γ t= =.67 a Complete Sample 5% Censoring 5% Censoring =.63 =.68 =.63 =.645 =.649 =.63 =.648 =.664 =.595 =.565 =.596 =.65 α=, β = α=, β = α=, β = α=, β = α=, β= α=, β = α=, β = α=, β = α=, β = α=, β = α=, β = α=, β= SIMULATION RESULTS One sample does not tell us much. We generated N = sample of sizes n =,, 3 from with =. Define the Monte Carlo estimator of a parameter θ by N = i i = / N and the root- mean-square -error RMSE = N i o / N, o = true value of i= Compute and at t = for the variations in a, α and β. The results are in Table-6.The entries within parentheses represents the corresponding RMSE. From Table-6, we see that the Bayes estimators of reliability function based on LINEX loss have smaller RMSE than that of squared error loss when a <.5 irrespective of sample sizes. Therefore, we conclude that in situations involving reliability estimation, asymmetric loss function is more appropriate than squared error loss function when a<.5.

16 76 Sanku Dey TABLE 6: Simulation estimates of the mean value of Bayes estimators of reliability function based on LINEX loss and squared error loss for the variations in a, α and β t=, the true value γ t= =.67. a α =, β = α =, β = α =, β = α =, β = n = = = = a =.67 =.5868 =.65 = a =.65 =.5997 =.66 =

17 COMPARISON OF RELATIVE RISK FUNCTIONS OF THE RAYLEIGH DISTRIBUTION UNDER TYPE-II CENSORED SAMPLES: BAYESIAN APPROACH 77 Acknowledgement: Author is highly grateful to the referee/s and the Editor for their constructive suggestions, which led to improvement in the quality of the paper. REFERENCES [] A. Ahmed Soliman, Comparison of Linex and Quadratic Bayes estimators for the Rayleigh Distribution. Communications in Statistics.-Theory and Methods, 9, [] A. M. Abd Elfattah, Amal S. Hassan and D.M. Ziedan, Efficiency of Maximum Likelihood Estimators under Different Censored Sampling Schemes for Rayleigh Distribution. Interstat. March 6a [3] A. M. Abd Elfattah, Amal S. Hassan and D.M. Ziedan. Efficiency of Bayes estimators for Rayleigh distribution. Interstat, July 6b [4] A. Zellner, Bayesian Estimation and Prediction using Asymmetric Loss Functions. Journal of American Statistical Association. 8986, [5] A.M. Polovko, Fundamental of Reliability Theory. Academic Press, New York, 968. [6] A.P. Basu and N. Ebrahimi, Bayesian Approach to life Testing and Reliability Estimation Using Asymmetric Loss Function. Journal of Statistical Planning and Inference, 999, -3. [7] D.D. Dyer and C.W. Whisenand, Best Linear Unbiased Estimator of the Parameter of the Rayleigh Distribution. IEEE Transactions on Reliability, R-973, [8] E.T. Lee, Statistical Methods for survival Data Analysis. Lifetime, Learning Publications, Inc., Belmount, 98. [9] H.A. Howlader and A. Hossain, On Bayesian Estimation and Prediction from Rayleigh based on Type-II Censored Data. Communications in Statistics.-Theory and Methods, , [] H.R. Varian, A Bayesian Approach to Real Estate Assessment. Amsterdam, North Holland 975, [] J. A. Hartigan, Invariant Prior Distribution. Annals of.mathematical Statistics,

18 78 Sanku Dey [] J. Rojo, On the admissibility of CX+d with respect to the LINEX Loss Function. Communications in.statistics-theory and Methods, 6987, [3] K. A. Ariyawansa and J.G.C. Templeton, Structural inference on the parameter of the Rayleigh distribution from doubly censored samples. Statistical Hefte, 5984, [4] K. Hirano, Rayleigh Distributions, New York, Wiley, 986. [5] M. I. Hendi, S.E. Abu-Youssef and A.A. Alraddadi, A Bayesian analysis of record statistics from the Rayleigh model. International Mathematical Forum, 9 7, [6] M.M. Siddiqui, Some Problems Connected with Rayleigh Distributions. Journal of Research, National Bureau of Standards, 6D 96, [7] R.A. Canfield, A Bayesian Approach to Reliability Estimation Using a Loss Function. IEEE Transactions on Reliability, R-9 97, 3-6. [8] U. Singh, P. Gupta and S. Upadhyay, S, Estimation of three-parameter exponentiated- Weibull distribution under type-ii censoring. Journal of Statistical Planning and Inference, 34 5, Sanku Dey Department of Statistics, St. Anthony s College, Shillong-793, Meghalaya, India. address: sanku_deyk3@yahoo.co.in