Two-parameter Rayleigh Distribution Research Articles

Statistical inference for a two-parameter Rayleigh distribution under generalized progressive hybrid censoring scheme

The two-parameter Rayleigh distribution, as an extended distribution of the Rayleigh distribution, has been widely applied in reliability analysis. With the introduction of the location parameter, two-parameter Rayleigh distribution becomes more flexible in fitting real-data. In this paper, based on generalized progressively hybrid censored(GPHC) sample from the two-parameter Rayleigh distribution, Classical and Bayesian inferences are discussed. The Newton-Raphson(NR) and Expectation-Maximization(EM) algorithms are used to compute the maximum likelihood estimates(MLEs). As well as the asymptotic confidence interval(ACI) estimation is obtained through the asymptotic distribution theory of maximum likelihood estimation(MLE) and computation of the observed Fisher information matrix. In Bayesian frame, the estimation of unknown parameters and prediction of future observable are taken into consideration. Due to Bayesian estimation is challenging to compute precisely and for the purpose of comparison, the Lindley's approximation, the Tierney-Kadane(TK) approximation and Markov chain Monte Carlo(MCMC) method are employed to obtain Bayesian estimates. Then, combining the MCMC algorithm mentioned in the article, the one- and two- samples Bayesian prediction are obtained. Finally, the simulation results are provided and a real-life data set is used for illustration purpose.

Bayesian Estimation in Rayleigh Distribution under a Distance Type Loss Function

Estimation of unknown parameters using different loss functions encompasses a major area in the decision theory. Specifically, distance loss functions are preferable as it measures the discrepancies between two probability density functions from the same family indexed by different parameters. In this article, Hellinger distance loss function is considered for scale parameter λ of two-parameter Rayleigh distribution. After simplifications, form of loss is obtained and that is meaningful if parameter is not large and Bayes estimate of λ is calculated under that loss function. So, the Bayes estimate may be termed as ‘Pseudo Bayes estimate’ with respect to the actual Hellinger distance loss function as it is obtained using approximations to actual loss. To compare the performance of the estimator under these loss functions, we also consider weighted squared error loss function (WSELF) which is usually used for the estimation of the scale parameter. An extensive simulation   is carried out to study the behaviour of the Bayes estimators under the three different loss functions, i.e. simplified, actual and WSE loss functions. From the numericalresults it is found that the estimators perform well under the Hellinger distance loss function in comparison with the traditionally used WSELF. Also, we demonstrate the methodology by analyzing two real-life datasets.

Comparison Different Estimation Method for Reliability Function of Rayleigh Distribution Based On Fuzzy Lifetime Data

    In this study, we present different methods of estimating fuzzy reliability of a two-parameter Rayleigh distribution via the maximum likelihood estimator, median first-order statistics estimator, quartile estimator, L-moment estimator, and mixed Thompson-type estimator. The mean-square error MSE as a measurement for comparing the considered methods using simulation through different values for the parameters and unalike sample sizes is used. The results of simulation show that the fuzziness values are better than the real values for all sample sizes, as well as  the fuzzy reliability at the estimation  of the Maximum likelihood Method, and Mixed Thompson Method perform better than the other methods in the sense of MSE, so that  we recommend to use this type of estimation.

Order restricted classical and Bayesian inference of a multiple step-stress model from two-parameter Rayleigh distribution under Type I censoring

This article considers a multiple step-stress model under two-parameter Rayleigh distribution with Type I censoring. Given that the lifetime of the experiment unit gradually decreases as the stress level grows step by step, the parameters of lifetime distribution with the increasing level have a natural order restriction. The reparametrization is applied to deal with this order limitation. Based on the proportional hazard model, the continuous cumulative distribution function and the corresponding likelihood function are inferred. The procedures of computing order restricted maximum likelihood estimations whether at least one failure exists in each stress level or not are introduced. Through transformation, the asymptotic confidence intervals of the original parameters are calculated based on the observed Fisher information matrix. Furthermore, taking the square error loss function, the Linex loss function, and the general entropy loss function into account, Bayesian estimations are discussed. With the importance sampling, the unknown parameters’ estimates are obtained and the associated highest posterior density credible intervals are built up. With simulations under different circumstances, the effectiveness of each method proposed is demonstrated. Finally, an analysis of a real dataset is provided.

Estimation of Stress-Strength Reliability for Multicomponent System with Rayleigh Data

Inference is investigated for a multicomponent stress-strength reliability (MSR) under Type-II censoring when the latent failure times follow two-parameter Rayleigh distribution. With a context that the lifetimes of the strength and stress variables have common location parameters, maximum likelihood estimator of MSR along with the existence and uniqueness is established. The associated approximate confidence interval is provided via the asymptotic distribution theory and delta method. Meanwhile, alternative generalized pivotal quantities-based point and confidence interval estimators are also constructed for MSR. More generally, when the lifetimes of strength and stress variables follow Rayleigh distributions with unequal location parameters, likelihood and generalized pivotal-based estimators are provided for MSR as well. In addition, to compare the equivalence of different strength and stress parameters, a likelihood ratio test is provided. Finally, simulation studies and a real data example are presented for illustration.

Inference based on partly interval censored data from a two-parameter Rayleigh distribution

In this paper, the maximum likelihood and Bayesian estimation of the parameters of location-scale Rayleigh distribution with partly interval censored data is considered. For computing the maximum likelihood estimators with partly interval censored data, three methods are used, namely, Newton-Raphson, Expectation-Maximization and Monte-Carlo Expectation-Maximization algorithms. The standard errors of the estimates are computed using the observed information matrix. Also, two types of confidence intervals are constructed using the Wald method and the nonparametric percentile bootstrap confidence intervals. For computing the Bayes estimators, three methods viz Lindley's approximation, Tierney-Kadane approximation and importance sampling methods are used. Highest posterior density (HPD) credible intervals of the two parameters are constructed using importance sampling technique. Monte-Carlo simulation experiments are conducted to investigate the performance of the proposed methods. Finally, the methods are illustrated by using two real data sets, one is related with diabetic patients data set and the other is related to HIV infection data set.

Estimating the Parameters of the Two-Parameter Rayleigh Distribution Based on Adaptive Type II Progressive Hybrid Censored Data with Competing Risks

This paper attempts to estimate the parameters for the two-parameter Rayleigh distribution based on adaptive Type II progressive hybrid censored data with competing risks. Firstly, the maximum likelihood function and the maximum likelihood estimators are derived before the existence and uniqueness of the latter are proven. Further, Bayesian estimators are considered under symmetric and asymmetric loss functions, that is the squared error loss function, the LINEXloss function, and the general entropy loss function. As the Bayesian estimators cannot be obtained explicitly, the Lindley method is applied to compute the approximate Bayesian estimates. Finally, a simulation study is conducted, and a real dataset is analyzed for illustrative purposes.

Confidence interval, prediction interval and tolerance limits for a two-parameter Rayleigh distribution

ABSTRACTThe problems of interval estimating the parameters and the mean of a two-parameter Rayleigh distribution are considered. We propose pivotal-based methods for constructing confidence intervals for the mean, quantiles, survival probability and for constructing prediction intervals for the mean of a future sample. Pivotal quantities based on the maximum likelihood estimates (MLEs), moment estimates (MEs) and the L-moments estimates (L-MEs) are proposed. Interval estimates based on them are compared via Monte Carlo simulation. Comparison studies indicate that the results based on the MEs and the L-MEs are very similar. The results based on the MLEs are slightly better than those based on the MEs and the L-MEs for small to moderate sample sizes. The methods are illustrated using an example involving lifetime data.

Estimation on a two-parameter Rayleigh distribution under the progressive Type-II censoring scheme: comparative study

Estimation on a two-parameter Rayleigh distribution under the progressive Type-II censoring scheme: comparative study

A Progressive Interval Type-I Censored Life Test Plan for Rayleigh Distribution

In this paper, we have considered the problem of optimal inspection times for the progressive interval type-I censoring scheme where uncertainty in the process is governed by the two-parameter Rayleigh distribution. Here, we also introduced some optimality criterion and determined the optimum inspection times, accordingly. The effect of the number of inspections and choice of optimally spaced inspection times based on the asymptotic relative efficiencies of the maximum likelihood estimates of the parameters are also investigated. Further, we have discussed the optimal progressive type-I interval censoring plan when the inspection times and the expected proportions of total failures in the experiment are under control.

Inference of R = P(Y < X) for two-parameter Rayleigh distribution based on progressively censored samples

ABSTRACTThe UMVUE and maximum likelihood estimator of for independent progressively Type-II censored samples from two-parameter Rayleigh distributions with different scale parameters are derived. Also the exact, asymptotic and bootstrap confidence intervals for R are evaluated. Using Gibbs sampling, the Bayes estimates and corresponding credible intervals for R are obtained. Applying Monte Carlo simulations, we compare the performances of the different estimation methods. Finally we use of two real data sets and show the competitive performance of the presented methods.

Confidence sets for the two-parameter Rayleigh distribution under progressive censoring

• Exact confidence sets for the two-parameter Rayleigh distribution are presented. • Constrained optimization problems are stated to determine the minimum-size sets. • Efficient procedures based on minimizing Lagrangian functions are proposed. • Three real data sets are analyzed. • Pointwise and simultaneous confidence bands and hypothesis tests can be derived. Exact confidence intervals and regions are proposed for the location and scale parameters of the Rayleigh distribution. These sets are valid for complete data, and also for standard and progressive failure censoring. Constrained optimization problems are solved to find the minimum-size confidence sets for the Rayleigh parameters with the required confidence level. The smallest-area confidence region is derived by simultaneously solving four nonlinear equations. Three numerical examples regarding remission times of leukemia patients, strength data and the titanium content in an aircraft-grade alloy, as well as a simulation study, are included for illustrative purposes. Further applications in hypothesis testing and the construction of pointwise and simultaneous confidence bands are also pointed out.

Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms

In this article, Maximum likelihood estimates for the shape and scale parameters of two-parameter Rayleigh distribution are obtained based on progressive type-II censored samples using the Newton-Raphson (NR) method and the Expectation-Maximization (EM) algorithm. A simple algorithm discussed in [2-3] is used for generating progressive type-II censored samples. Based on this censoring scheme, approximate asymptotic variances are derived and used to construct approximate confidence intervals of the parameters. The performance of these two maximum likelihood estimation algorithms is compared in terms of simulation results of root mean squared error (RMSE) and the coverage rates. Simulation results showed that in nearly all the combination of simulation conditions the estimators based on the EM algorithm have less root mean squared error (RMSE) and narrower widths of confidence intervals compared to those obtained using the NR algorithm. Finally, an illustrative example with real-life data sets is provided to illustrate how maximum likelihood estimation using the two algorithms works in practice.

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

ABSTRACTIn the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.

Exact Interval Inference for the Two-Parameter Rayleigh Distribution Based on the Upper Record Values

The maximum likelihood method is the most widely used estimation method. On the other hand, it can produce substantial bias, and an approximate confidence interval based on the maximum likelihood estimator cannot be valid when the sample size is small. Because the sizes of the record values are considerably smaller than the original sequence observed in the majority of cases, a method appropriate for this situation is required for precise inference. This paper provides the exact confidence intervals for unknown parameters and exact predictive intervals for the future upper record values by providing some pivotal quantities in the two-parameter Rayleigh distribution based on the upper record values. Finally, the validity of the proposed inference methods was examined from Monte Carlo simulations and real data.

On Progressively Type-II Censored Two-parameter Rayleigh Distribution

Recently, Rayleigh distribution has received considerable attention in the statistical literature. In this article, we consider the point and interval estimation of the functions of the unknown parameters of a two-parameter Rayleigh distribution. First, we obtain the maximum likelihood estimators (MLEs) of the unknown parameters. The MLEs cannot be obtained in explicit forms, and we propose to use the maximization of the profile log-likelihood function to compute the MLEs. We further consider the Bayesian inference of the unknown parameters. The Bayes’ estimates and the associated credible intervals cannot be obtained in closed forms. We use the importance sampling technique to approximate (compute) the Bayes’ estimates and the associated credible intervals. For comparison purposes, we have also used the exact method to compute the Bayes’ estimates and the corresponding credible intervals. Monte Carlo simulations are performed to compare the performances of the proposed method, and one dataset has been analyzed for illustrative purposes. We further consider the Bayes’ prediction problem based on the observed samples, and provide the appropriate predictive intervals. A data example has been provided for illustrative purposes.

Estimation of the reliability function for two-parameter exponentiated Rayleigh or Burr type X distribution

Problem Statement: The two-parameter exponentiated Rayleigh distribution has been widely used especially in the modelling of life time event data. It provides a statistical model which has a wide variety of application in many areas and the main advantage is its ability in the context of life time event among other distributions. The uniformly minimum variance unbiased and maximum likelihood estimation methods are the way to estimate the parameters of the distribution. In this study we explore and compare the performance of the uniformly minimum variance unbiased and maximum likelihood estimators of the reliability function R(t)=P(X&gt;t) and P=P(X&gt;Y) for the two-parameter exponentiated Rayleigh distribution. Approach: A new technique of obtaining these parametric functions is introduced in which major role is played by the powers of the parameter(s) and the functional forms of the parametric functions to be estimated are not needed. We explore the performance of these estimators numerically under varying conditions. Through the simulation study a comparison are made on the performance of these estimators with respect to the Biasness, Mean Square Error (MSE), 95% confidence length and corresponding coverage percentage. Conclusion: Based on the results of simulation study the UMVUES of R(t) and ‘P’ for the two-parameter exponentiated Rayleigh distribution found to be superior than MLES of R(t) and ‘P’.

Two-Parameter Rayleigh Distribution: Different Methods of Estimation

SYNOPTIC ABSTRACTIn this study we have considered different methods of estimation of the unknown parameters of a two-parameter Rayleigh distribution from both the frequentists' and the Bayesian view points. First, we briefly describe different frequentists' approaches: maximum likelihood estimators, method of moments estimators, L-moment estimators, percentile-based estimators, and least squares estimators, and we compare them using extensive numerical simulations. We have also considered Bayesian inferences of the unknown parameters. It is observed that the Bayes estimates and the associated credible intervals cannot be obtained in explicit forms, and we have suggested using an importance sampling technique to compute the Bayes estimates and the associated credible intervals. We analyze one dataset for illustrative purposes.

Moment Estimation of the Parameters in Rayleigh Distribution with Two Parameters

In this article, the moment estimation of the parameters in two-parameter Rayleigh distribution is studied. For given sample values, the necessary and sufficient conditions on the estimating equations which have unique solution are obtained, and it is also proved that these conditions are satisfied on asymptotic probability 1. Furthermore, the strong consistency and asymptotic normality are obtained. Finally, some simulation experiments are made to show the conclusions.