Bayesian estimation of mean and square of mean of normal distribution using linex loss function

In this paper, the Bayes estimators for mean and square of mean ol a normal distribution with mean μ and vaiiance σ r2 (known), relative to LINEX loss function are obtained Comparisons in terms of risk functions and Bayes risks of those under LINEX loss and squared error loss functions with their respective alternative estimators viz, UMVUE and Bayes estimators relative to squared error loss function, are made. It is found that Bayes estimators relative to LINEX loss function dominate the alternative estimators m terms of risk function snd Bayes risk. It is also found that if t2 is unknown the Bayes estimators are still preferable over alternative estimators.     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

Estimators for the Finite Mixture of Rayleigh Model Based on Progressively Censored Data

In this article, based on progressively Type-II censored samples from a heterogeneous population that can be represented by a finite mixture of two-component Rayleigh lifetime model, the problem of estimating the parameters and some lifetime parameters (reliability and hazard functions) are considered. Both Bayesian and maximum likelihood estimators are of interest. A class of natural conjugate prior densities is considered in the Bayesian setting. The Bayes estimators are obtained using both the symmetric (squared error) loss function, and the asymmetric (LINEX and General Entropy) loss functions. It has been seen that the estimators obtained can be easily evaluated for this type of censoring by using suitable numerical methods. Finally, the performance of the estimates have been compared on the basis of their simulated maximum square error via a Monte Carlo simulation study.     

Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring

The present article obtains the point estimators of the exponentiated-Weibull parameters when all the three parameters of the distribution are unknown. Maximum likelihood estimator generalized maximum likelihood estimator and Bayes estimators are proposed for three-parameter exponentiated-Weibull distribution when available sample is type-II censored. Independent non-informative types of priors are considered for the unknown parameters to develop generalized maximum likelihood estimator and Bayes estimators. Although the proposed estimators cannot be expressed in nice closed forms, these can be easily obtained through the use of appropriate numerical techniques. The performances of these estimators are studied on the basis of their risks, computed separately under LINEX loss and squared error loss functions through Monte-Carlo simulation technique. An example is also considered to illustrate the estimators.     

Comparison of linex and quadratic bayes estimators foe the rayleigh distribution

In this paper, the Bayes estimators for the parameter, the reliability function, and failure rate function of the Rayleigh distribution are obtained when based on complete or type II censored samples. Some types of the linex loss function are used. Comparieons in terms of risks of those under linex loss and squared error loss function with Bayes estimators relative to squared error loss function are made, Numerical example and simulation example are included.     

Estimation of parameters of Kumaraswamy-Exponential distribution under progressive type-II censoring

In this paper, the problem of estimating unknown parameters of a two-parameter Kumaraswamy-Exponential (Kw-E) distribution is considered based on progressively type-II censored sample. The maximum likelihood (ML) estimators of the parameters are obtained. Bayes estimates are also obtained using different loss functions such as squared error, LINEX and general entropy. Lindley's approximation method is used to evaluate these Bayes estimates. Monte Carlo simulation is used for numerical comparison between various estimates developed in this paper.     

Bayesian and Non Bayesian Estimations on the Exponentiated Modified Weibull Distribution for Progressive Censored Sample

The four-parameter Exponentiated Modified Weibull (EMW) is considered as an important lifetime distribution. Based on progressive Type-II censored sample, maximum likelihood and Bayesian estimators of the parameters, reliability function, and hazard rate function are derived. Two cases are considered: first, the case of one unknown exponent parameter of EMW and second, the case when two parameters of the EMW are both unknown. The Bayes estimators are studied under squared error and LINEX loss functions. The standard Bayes and importance sampling are considered for the estimation. Monte Carlo simulations are performed under different samples sizes and different censoring schemes for investigating and comparing the methods of estimation.     

Risk of a homoscedasticity pre-test estimator of the regression scale under LINEX loss

In this paper we consider the risk of an estimator of the error variance after a pre-test for homoscedasticity of the variances in the two-sample heteroscedastic linear regression model. This particular pre-test problem has been well investigated but always under the restrictive assumption of a squared error loss function. We consider an asymmetric loss function — the LINEX loss function — and derive the exact risks of various estimators of the error variance.     

Statistical analysis for Kumaraswamy’s distribution based on record data

In this paper we review some results that have been derived on record values for some well known probability density functions and based on m records from Kumaraswamy’s distribution we obtain estimators for the two parameters and the future sth record value. These estimates are derived using the maximum likelihood and Bayesian approaches. In the Bayesian approach, the two parameters are assumed to be random variables and estimators for the parameters and for the future sth record value are obtained, when we have observed m past record values, using the well known squared error loss (SEL) function and a linear exponential (LINEX) loss function. The findings are illustrated with actual and computer generated data.     

Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring

In this paper, we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution under type II progressive censored samples. Estimation of reliability and hazard functions is also considered. Maximum likelihood estimators are obtained using the Expectation–Maximization (EM) algorithm. Further, we obtain expected Fisher information matrix using the missing value principle. Bayes estimators are derived under squared error and linex loss functions. We have used Lindley, and Tiernery and Kadane methods to compute these estimates. In addition, Bayes estimators are computed using importance sampling scheme as well. Samples generated from this scheme are further utilized for constructing highest posterior density intervals for unknown parameters. For comparison purposes asymptotic intervals are also obtained. A numerical comparison is made between proposed estimators using simulations and observations are given. A real-life data set is analyzed for illustrative purposes.     

Statistical inference for two exponential populations under joint progressive type-I censored scheme

In this article, we introduce a new scheme called joint progressive type-I (JPC-I) censored and as a special case, joint type-I censored scheme. Bayesian and non Bayesian estimators have been obtained for two exponential populations under both JPC-I censored scheme and joint type-I censored. The maximum likelihood estimators of the parameters, the asymptotic variance covariance matrix, have been obtained. Bayes estimators have been developed under squared error loss function using independent gamma prior distributions. Moreover, approximate confidence region based on the asymptotic normality of the maximum likelihood estimators and credible confidence region from a Bayesian viewpoint are also discussed and compared with two Bootstrap confidence regions. A numerical illustration for these new results is given.     

Bayesian inference of the inverse Weibull mixture distribution using type-I censoring

A large number of models have been derived from the two-parameter Weibull distribution including the inverse Weibull (IW) model which is found suitable for modeling the complex failure data set. In this paper, we present the Bayesian inference for the mixture of two IW models. For this purpose, the Bayes estimates of the parameters of the mixture model along with their posterior risks using informative as well as the non-informative prior are obtained. These estimates have been attained considering two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the former case, Bayes estimates are obtained under three loss functions while for the latter case only the squared error loss function is used. Simulation study is carried out in order to explore numerical aspects of the proposed Bayes estimators. A real-life data set is also presented for both cases, and parameters obtained under case when shape parameter is known are tested through testing of hypothesis procedure.     

Bayesian estimation for the exponentiated Weibull model under Type II progressive censoring

Based on progressive Type II censored samples, we have derived the maximum likelihood and Bayes estimators for the two shape parameters and the reliability function of the exponentiated Weibull lifetime model. We obtained Bayes estimators using both the symmetric and asymmetric loss functions via squared error loss and linex loss functions. This was done with respect to the conjugate priors for two shape parameters. We used an approximation based on the Lindley (Trabajos de Stadistca 21, 223–237, 1980) method for obtaining Bayes estimates under these loss functions. We made comparisons between these estimators and the maximum likelihood estimators using a Monte Carlo simulation study.     

Minimax estimation of a probability of success under LINEX loss

Minimax estimation of a binomial probability under LINEX loss function is considered. It is shown that no equalizer estimator is available in the statistical decision problem under consideration. It is pointed out that the problem can be solved by determining the Bayes estimator with respect to a least favorable distribution having finite support. In this situation, the optimal estimator and the least favorable distribution can be determined only by using numerical methods. Some properties of the minimax estimators and the corresponding least favorable prior distributions are provided depending on the parameters of the loss function. The properties presented are exploited in computing the minimax estimators and the least favorable distributions. The results obtained can be applied to determine minimax estimators of a cumulative distribution function and minimax estimators of a survival function.     

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.     

Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.     

Classical and Bayesian estimation of P(X<Y) using upper record values from Kumaraswamy’s distribution

In this paper, maximum likelihood and Bayesian approaches have been used to obtain the estimation of \(P(X<Y)\) based on a set of upper record values from Kumaraswamy distribution. The existence and uniqueness of the maximum likelihood estimates of the Kumaraswamy distribution parameters are obtained. Confidence intervals, exact and approximate, as well as Bayesian credible intervals are constructed. Bayes estimators have been developed under symmetric (squared error) and asymmetric (LINEX) loss functions using the conjugate and non informative prior distributions. The approximation forms of Lindley (Trabajos de Estadistica 3:281–288, 1980) and Tierney and Kadane (J Am Stat Assoc 81:82–86, 1986) are used for the Bayesian cases. Monte Carlo simulations are performed to compare the different proposed methods.     

Bayesian estimation under a mixture of the Burr type XII distribution and its reciprocal

The Bayesian estimation for the parameters of the finite mixture of the Burr type XII distribution with its reciprocal are obtained based on the type-I censored data. The Bayes estimators are computed based on squared error and Linex loss functions and using the idea of Markov chain Monte Carlo algorithm. Based on the Monte Carlo simulation, Bayes estimators are compared with their corresponding maximum-likelihood estimators.     

Bayesian estimation of parameters of inverse Weibull distribution

The present paper describes the Bayes estimators of parameters of inverse Weibull distribution for complete, type I and type II censored samples under general entropy and squared error loss functions. The proposed estimators have been compared on the basis of their simulated risks (average loss over sample space). A real-life data set is used to illustrate the results.