The comparative studies on reliability for Rayleigh models

Transcription

1 Journal of the Korean Data & Information Science Society 018, 9, 한국데이터정보과학회지 The comparative studies on reliability for Rayleigh models Ji Eun Oh 1 Joong Kweon Sohn 1 Department of Statistics, Kyungpook National University Received 1 January 018, revised 1 March 018, accepted 13 March 018 Abstract In this paper, several methods to estimate the reliability R of a Rayleigh model as a function of given time t are proposed and studied. Many research have been done by providing some informations on parameters of a Raleigh model rather R itself. Here R is a probability to endure a certain level of stress. Thus for the Bayesian method, several priors are given to R directly. Also a bootstrap method and a maximum likelihood method are examined and compared with a Bayesian method in term of mean square errors. we find that Bayesian methods perform better than other methods. Keywords: Bayes estimator, bootstrap estimator, maximum likelihood estimator, rayleigh distribution, reliability. 1. Introduction Reliability is the probability that a product or system will work properly for a specified period of time under the design operating conditions without failure. So that the reliability depends on specific time as a random variable and its function from the chosen distribution can be estimated and controled for system maintenance. The study of reliability is important to develop the appropriate future plans to improve the quality and performance of the system recently. Many authors had been studied to estimate the reliability from their own statistical distribution function that they assumed. Among all the statistical life time model, the Rayleigh distribution is the most important models and the widely used in large-scale tests in life and reliability. In this study, we assume that the life times of units follow a Rayleigh distribution, which has been widely used to describe life time reliability analyses. The study of maximum likelihood estimation MLE of reliability with different distribution have been discussed by several authors. The MLE is used to derive point and asymptotic confidence estimates of the unknown parameters. First Harter and Moore 1965 showed an explicit form for the MLE of based on type II censored data. Thoman et al studied with maximum likelihood ML estimator and exact confidence interval for reliability and 1 Graduate student, Department of Statistics, Kyungpook National University, Daegu 41566, Korea. Corresponding author: Professor, Department of Statistics, Kyungpook National University, Daegu 41566, Korea. jsohn@knu.ac.kr

2 534 Ji Eun Oh Joong Kweon Sohn tolerance limits in the Weibull distribution. Dyer and Whisenand 1973 considered the best linear unbiased estimator of the parameter of the Rayleigh distribution. Smith et al considered MLE and exact confidence intervals for reliability in the Weibull Distribution. Kang and Kim 1994 considered the approximate MLE of scale parameter of weibull distribution with type II censoring. Lee et al. 014 also studied reliability estimation and ratio distribution in a general exponential distribution. Kim and Cha 016 considered the bivariate reliability models with multiple dynamic competing risks. Inferences for the Rayleigh distribution have been discussed by several authors. Sinha and Howlader 1983 showed the Credible and highest predictive density HPD intervals of the parameter and reliability of Rayleigh distribution. The approximate MLE is proofed as efficient as the best linear unbiased estimator of Rayleigh distribution by Balakrishnan Bayesian estimation and prediction problems for the Rayleigh distribution based on Type II censored sample have been considered by Fernandez 000. Wu et al. 006 considered the Bayesian estimator and prediction intervals for future observations based on progressively type II censored samples. Kim and Han 009 considered ML estimator, approximate ML estimator and Bayes estimation procedures for the scale parameter based on a multiply type II censored sample. Dey 009 compared the Bayes estimates of reliability function with different loss function. Recently, Lee et all 010 considered the reliability of right truncated Rayleigh distribution. Pak et al. and Kweon et al. 014 considered the reliability estimation with Rayleigh distribution on fuzzy lifetime data and type I hybrid censored sample, respectively. Rasheed and Aaref 016 studied reliability estimation in inverse Rayleigh distribution using precautionary loss function with MLE method and Bayesian method. Most of studies deal with scale parameter of Rayleigh or inverse Rayleigh distribution so far according to literature reviews. We focus on estimating reliability of Rayleigh distribution and derive three estimator of reliability that not focus on the distribution parameter but on reliability parameter directly. We use the following notation that the Rayleigh distribution with the scale parameter σ and the corresponding density function fx; σ for σ > 0 is given by and reliability function as follow. fx; σ = x x e σ σ, x > 0, σ > θ = Rt = P X > t = e t σ, x > 0, t > 0, σ > This study has two objectives. First, it aims to develop three estimators of reliability based on the MLE method, bootstrap method and Bayesian method. We also focus on reliability function to estimate not the scale parameter of Rayleigh distribution. The second objective is to compute and compare the confidence intervals of reliability using the estimators from the previous three method. In Section, we calculate the maximum likelihood ML estimator of the reliability function θ with MLE method and derive the asymptotic distribution and the confidence interval of θ. In order to the confidence limits for parameter, we calculate the Fisher information matrix to obtain the asymptotic variances and covariance of the ML estimator of the parameter.

3 The comparative studies on reliability for Rayleigh models 535 In Section 3, we generate the bootstrap samples using the ML estimator from section and obtain the bootstrap estimator and confidence interval of bootstrap sample with N replications. In Section 4, we derive the Bayes estimator of reliability with the inverse gamma prior on scale parameter and HPD confidence interval. In Section 5, we show some results based on Monte-Carlo simulation to compare the mean squared error MSE of each estimator using MLE method and bootstrap method and Bayesian method in different sample size and the scale parameter σ. Also we showed some results of three kinds of confidence interval of reliability and compare the coverage probability and interval length of each estimator.. Maximum likelihood estimation MLE method The method of maximum likelihood estimation by Harter and Moore 1965 is a commonly used procedure because it has very desirable properties. The Rayleigh distribution applies to a non-limited positive continuous variable, the probability density function and the cumulative distribution function is given by: fx, σ = x x e σ σ, x > 0, σ > 0,.1 F x, σ =1 e x σ, x > 0, σ > 0,. where x is a continuous random variable defined over 0, and σ is the scale parameter. The reliability function on specific time t which is Rt, θ = Rt = P X > t = t 0 F x, σdx = e t σ, x > 0, t > 0, σ > 0. We recalled Rt is a function of θ, θ = e t σ, t > 0, σ > 0 and σ = t logθ. To find the estimate of ML estimator of reliability function Rt, we first have to find the ML estimator of σ which is a scale parameter of the Rayleigh distribution. Suppose x = x 1, x,, x n be an observed random sample of size n from the Rayleigh distribution, where σ is a scale parameter. The likelihood function of the observed sample is Lσ x = 1 n σ n x i exp i=1 The log-likelihood function may then be written as n i=1 x i σ..3 lσ x = loglσ x = n logx i nlogσ i=1 n i=1 x i σ..4 The ML estimator of σ can be obtained by first derivative of.4 with respect to σ,

4 536 Ji Eun Oh Joong Kweon Sohn ˆσ = i n. The likelihood function of θ, which can be obtained by replace the σ = t given by Lθ x = logθ n n [ x t x i exp i logθ ] t i=1 logθ to.3 is.5 and log-likelihood function of θ may then be written as lθ x = nloglogθ nlogt + x logx i + i t logθ..6 Calculating the first partial derivatives of.6 with respect to θ and equating to zero, we obtain the likelihood equation Hence, the ML estimator of θ lθ x θ = n θlogθ + i 1 t θ = 0. ˆθ MLE = exp t n..7 i We can get the ˆθ MLE with invariance properties of MLE. ˆθ MLE = ˆRt = exp t ˆσ = exp t n..8 i.1. Asymptotic distribution and confidence interval of reliability In this section, first we need to find the asymptotic distribution of ˆσ and then we drive the asymptotic distribution of ˆθ. Base on the asymptotic distribution of ˆθ, we can get the asymptotic confidence interval of θ. In order to the confidence limits for parameter, we calculate the Fisher information matrix to obtain the asymptotic variances and covariance of the ML estimator of the parameter. From the log-likelihood function in.6, we have lθ x n x θ = 1 + logθ i 1 t θ..9

5 The comparative studies on reliability for Rayleigh models 537 The Fisher information Iθ is then obtained by taking expectation of minus.9. In practice, we usually estimate the Iθ by I 0 ˆθ that obtained by following [ ] lθ x Iθ = E, θ [ I 0 ˆθ = E n 1 + logθ ] x i 1 t θ = nσ 1 + logθ nθ t θ t, 1 + logθ V arˆθ = 1 I 0 ˆθ = θ t 1 + logθ n ˆσ 1 + logθ nθ t. Under some mild regularity conditions ˆθ is approximately normally distributed with mean θ and variance V arˆθ. ˆθ Nθ, V arˆθ. Thus, the 1001 α% approximate confidence interval for θ is ˆθ z α V arˆθ, ˆθ + z α V arˆθ, θ V t 1 + logθ ARˆθ = n ˆσ 1 + logθ nθ t, θ t 1 + logθ = i 1 + logθ nθ t, where z α is the percentile of the standard normal distribution with right-tail probability α and ˆσ = i n. 3. Bootstrap method The bootstrap method introduced in Efron 198 is a very general resampling procedure for estimating the distributions of statistics based on independent observations. The bootstrap method is shown to be successful in many situations, which is being accepted as an alternative to the asymptotic methods. For the above example, we can easily calculate its bootstrap distribution. We can easily imagine that the above computation becomes too complicated to compute directly if n is

6 538 Ji Eun Oh Joong Kweon Sohn large. Therefore, simple random sampling was proposed to generate bootstrap distribution. In the bootstrap literature, a variety alternatives are suggested other than simple random sampling. The following steps are followed to obtain bootstrap sample from Rayleigh distribution with parameter. Step 1 : Generate random samples x 1, x,, x n is a data sample drawn from Rayleigh distribution with the scale parameter σ and compute the ML estimator of σ, ˆσ. Step : Generate a bootstrap sample x 1, x,, x n from Rayleigh distribution with ˆσ. Based on bootstrap sample, compute the new estimator of scale parameter, ˆσ and bootstrap estimator of reliability using equation.8, say ˆθ Boot. Step 3 : Repeat Step, NBOOT times. NBOOT is 000 replications The bootstrap confidence interval of reliability In this section, we propose to use the percentile bootstrap method based on the idea of Efron 198. We follow the next few step to get the bootstrap interval of estimator using bootstrap method. To generate the confidence intervals by bootstrapping, the sampling distribution of the parameter θ is simulated by sampling over and over from the current data and recomputing parameter estimates θ from each bootstrapped sample. The variability shown by the many θ values gives us a hint about the variability of the one estimate θ we got from our data. Step 1 : Generate random samples x 1, x,, x n is a data sample drawn from Rayleigh distribution with the scale parameter σ and compute the ML estimator of σ, ˆσ. Step : Using ˆσ, generate a bootstrap sample x 1, x,, x n from Rayleigh distribution. Based on bootstrap sample, compute bootstrap estimate of reliability using ˆσ and equation.8, say ˆθ Boot. Step 3 : Repeat Step-3 NBOOT times based on N different bootstrap samples. Step 4 : Generate the confidence interval of which has asymptotic normal distribution with mean θ and variance V arˆθ. 4. Bayesian method Bayesian analysis is an important approach to statistics, which formally seeks use of prior information and Bayes Theorem provides the formal basis for using this information. An important pre-requisite in Bayesian estimation is the appropriate choice of prior for the parameters. However, Bayesian analysts also hard to choose which prior is better than the other. Very often, priors are chosen according to ones subjective knowledge and beliefs. In this paper, we derive the Bayes estimator with the inverse gamma prior on scale parameter and comparing with non-informative prior at simulation study. First we develop the Bayesian estimation procedure for the estimation of reliability form Rayleigh distribution assuming independent inverse gamma prior for the unknown model parameters. Thus, we consider the natural conjugate prior for given by

7 The comparative studies on reliability for Rayleigh models 539 πσ = βα Γα σ α+1 exp 1 σ β σ α+1 exp β σ, σ > 0, α > 0, β > 0, 4.1 where shape parameter α > 0 and scale parameter β > 0. This density is known as the square-root inverted gamma distribution. By combining.6 and 4.1, the posterior density of σ is given by πσ x = n+α x i +β Γ n+α σ exp i + β n+α+1 σ, 4. i +β. where it is the inverse gamma density, IGα, β with α = n+α, β = It is well known that the Bayes estimator of σ under squared loss function. To estimate the reliability function θ, θ = e t σ, t > 0, σ > 0 and substituting σ = t logθ into 4.. The posterior probability density function of θ is given by πθ x = = i +β Γ n+α i +β Γ n+α n+α n+α exp i +β t logθ n+α+ t logθ t logθ 1 θ, θ n+α 1 logθ n+α 1, 0 < θ < The Bayes estimator of reliability function θ with the squared error loss function which is Lˆθ, θ = ˆθ θ, ˆθ Bayes is by Dey 009, ˆθ Bayes =Eθ x = ˆθ Bayes = t i + β θpiθ xdθ n+α. 4.4 If α = 0, β = 0, we get a non-informative prior and also if α = 1, β = 1, we get a inverse gamma prior The Bayesian confidence interval of relability In order to derive the confidence interval for Bayes estimator, we use the posterior density function of θ. We already have the posterior probability density function from previous chapter and given by

8 540 Ji Eun Oh Joong Kweon Sohn πθ x = i +β Γ n+α n+α θ i +β 1 logθ n+α 1, 0 < θ < 1. From the posterior probability density function of θ, we know that as followed, Rt = θ = exp t σ, i + β t log 1 θ n + α χ. By the logarithm of posterior probability density function and some transformation, log 1 θ is chi-squared distribution with degree of freedom n+α. The 1001 α% x i +β t HPD credible set for θ is the interval c L, c U which satisfying c U c L πθ xdθ = 1 α. L 0 πθ xdθ = α and πθ xdθ = α. Therefore 1 α equal-tail credible limit c L and c U for the reliability θ are the solution of as followed. i Since σ χ n, 1001 α% equal tail interval limits c L and c U for the reliability function are given by U i + β t 1 log = χ 1 α c U t n + α and i + β t 1 log = χ α c L t n + α. The HPD limits H L t and H U t are calculated by Sinha and Howlader 1983, x P i + β 1 n + α t log < χ < H U i + β 5. Simulation study t 1 log = 1 α. H L In this section, a simulation study is conducted to investigate and compare the performance of ML estimator and Bootstrap estimators and Bayes estimators under squared error loss function of reliability and also derived the confidence intervals for three estimators. We mainly compute MSE and average estimates of the ML estimator, Bootstrap estimator and Bayes estimator. To compare the performance of the confidence interval of reliability, we compute the coverage probability CP and interval lengths IL of each estimators. We compared the results of estimates of reliability using MLE method, bootstrap method and the Bayesian method with different sample sizes and values of the scale parameter : n, σ

9 The comparative studies on reliability for Rayleigh models 541 = 15, 1, 30,1, 50, 1, 100, 1 and 15, 30,, 50,, 100, using Monte-Carlo simulation. Through this simulation study, we find the MSE and mean of each estimates so that we compare the estimates with true value of. In bootstrap sampling, we replicate bootstrap sampling process for 000 times. We considered two Bayes estimators with different prior that one is the inverse gamma prior and the other is the non-informative prior for the reliability function of Rayleigh distribution. We assume that the parameters of inverse gamma distribution as a prior distribution with α = 1, β = 1 and consider as a non-informative prior distribution with α = 0, β = 0 for Bayes estimators. We compared the confidence interval length and coverage probability of the estimates of reliability. The confidence level taken is 1 α = Table 5.1 MSE and average values AV θ = 0.61, σ = 1, t = 1 n ˆθML ˆθBoot ˆθBayes ˆθBayes α = 1, β = 1 α = 0, β = 0 MSE AV MSE AV MSE AV MSE AV θ = 0.88, σ =, t = 1 n ˆθML ˆθBoot ˆθBayes ˆθBayes α = 1, β = 1 α = 0, β = 0 MSE AV MSE AV MSE AV MSE AV * ML : Maximum likelihood estimates * Boot : Bootstrap estimates * Bayes : Bayes estimates with inverse gamma prior * Bayes : Bayes estimates with non-informative prior Tables 5.1 shows that the MSE and average values of the three estimators. As the sample size increases, the Bayes estimator performs better than MLE overall but they all have their MSE values decreasing as sample size increase. At small sample like n = 15, MSE values of Bayes estimates corresponds to a larger than other estimators but at sample size n = 30, 50 and 100, MSE values of Bayes estimators were smaller than others for σ = 1. When the σ is, Bayes estimator has the smaller MSE than the MLE estimator as the sample size increases. Table 5. summaries that the coverage probability and interval length of each estimators with different sample size and σ. The mean of interval length decrease as sample size increase and the coverage probability of bootstrap and Bayes estimators are higher than ML estimators. The Bayes estimator is the shortest interval length.

10 54 Ji Eun Oh Joong Kweon Sohn Table 5. Coverage probability and interval length θ = 0.61, σ = 1, t = 1 n ˆθML ˆθBoot ˆθBayes ˆθBayes α = 1, β = 1 α = 0, β = θ = 0.88, σ =, t = 1 n ˆθML ˆθBoot ˆθBayes ˆθBayes α = 1, β = 1 α = 0, β = Table 5.3 Average interval estimation θ = 0.61, σ = 1, t = 1 n MLE Bootstrap Bayes Method Method Method c L, c U c L, c U c L, c U , , , , , , , , , , , , θ = 0.88, σ =, t = 1 n MLE Bootstrap Bayes Method Method Method c L, c U c L, c U c L, c U , , , , , , , , , , , , Table 5.3 summaries that the average confidence interval of three estimators of reliability. Figure 5.1 shows below that the asymptotic distribution of each estimators at different sample size and σ.

11 The comparative studies on reliability for Rayleigh models 543 a MLE estimates σ = 1 b MLE estimates σ = c Bootstrap estimates σ = 1 d Bootstrap estimates σ = e Bayes estimates σ = 1 f Bayes estimates σ = Figure 5.1 Box plots and histograms of each estimates

12 544 Ji Eun Oh Joong Kweon Sohn 6. Conclusion and discussion A simulation studies were conducted to examine the performance of the different estimators how close to the true value of reliability and the coverage probability of confidence interval of each estimator was compared. It is shown that the asymptotic distribution of reliability depends only on time and sample size. The ML estimator close to the true value and has the small MSE when sample size is large. So the ML estimator of reliability was accurate results of simulation when the sample size is large. Its clear that the MSE of Bayes estimator is smaller than the other two estimators except the sample size is 15. The interval length of three estimators are quite similar and the coverage probability of ML estimator is the lowest. Among three estimators, the Bayes estimators interval length is the smallest. Based on the above discussion, we can conclude that the Bayes estimator has the smallest MSE and the shortest interval length as the sample size increases even if MSE of ML estimator performed good at small sample size. We consider that Bayes estimator is good estimator of this study. References Balakrishnan, N Approximate MLE of the scale parameter of the Rayleigh distribution with censoring. IEEE Transactions on Reliability, 38, Dey, S Comparison of Bayes estimators of the parameter and reliability function of Rayleigh distribution under different loss functions. Malaysian Journal of Mathematical Sciences, 3, Dyer, D. D. and Whisenand, C. W Best linear estimator of the parameter of the Rayleigh distributionpart I: Small sample theory for censored order statistics. IEEE Transactions on Reliability,, Efron, B The Jackknife, the bootstrap and other resampling plans. CBMS-NSF Regional Conference Series in Applied Mathematics, 38. Fernandez, A. J Bayesian inference from type II censored Rayleigh data. Statistics and Probability Letters, 48, Harter, H. L. and Moore, A. H Maximum-likelihood estimation of the parameters of gamma and Weibull populations from complete and from censored samples. Technometrics, 7, Kim, J. Y. and Cha, J. H Bivariate reliability models with multiple dynamic competing risks. Journal of the Korean Data & Information Science Society, 5, Kang, S. B. and Kim, M. H Approximate MLE for the scale parameter of the Weibull distribution with type- censoring. Journal of the Korean Data & Information Science Society, 5, Kim, C. S. and Han, K. H Estimation of the scale parameter of the Rayleigh distribution with multiply type-ii censored sample. Journal of Statistical Computation and Simulation, 79, Kwon, B. W., Lee, K. J. and Cho, Y. S Estimation for the Rayleigh distribution based on type I hybrid censored sample. Journal of the Korean Data & Information Science Society, 5, Lee, C. S. and Moon, Y. G Reliability estimation and ratio distribution in a general exponential distribution. Journal of the Korean Data & Information Science Society, 5, Lee, J. C. and Lee, C. S Reliability and ratio in a right truncated Rayleigh distribution. Journal of the Korean Data & Information Science Society, 1, Pak, A., Parham, G. A. and Saraj, M Reliability estimation in Rayleigh distribution based on fuzzy lifetime data. International Journal of System Assurance Engineering and Management, 5, Rasheed, A. and Aref, R. K Bayesian approach in estimation of scale parameter of inverse Rayleigh distribution. Mathematics and Statistics Journal,, Sinha, S. K. and Howlader, H. A Credible and HPD intervals of the parameter and Reliability of Rayleigh distribution. IEEE Transactions on Reliability, 3, Smith, R. L. and Naylor, J. C A comparison of maximum likelihood and Bayesian estimators for the three- parameter Weibull distribution. Journal of the Royal Statistical Society, 36, Thoman, D. R., Lee, J. and Antler, C. E Maximum likelihood estimation, exact confidence intervals forrreliability and tolerance limits in the Weibull distribution. Technometrics, 1,

13 The comparative studies on reliability for Rayleigh models 545 Wu, S. J., Chen, D. H. and Chen, S. T Bayesian inference for Rayleigh distribution under progressive censored sample. Applied Stochastic Models in Business and Industry,,