On estimation of the exponentiated Pareto distribution under different sample schemes

Bayes and classical estimators have been obtained for a two-parameter exponentiated Pareto distribution for when samples are available from complete, type I and type II censoring schemes. Bayes estimators have been developed under a squared error loss function as well as under a LINEX loss function using priors of non-informative type for the parameters. It has been seen that the estimators obtained are not available in nice closed forms, although they can be easily evaluated for a given sample by using suitable numerical methods. The performances of the proposed estimators have been compared on the basis of their simulated risks obtained under squared error as well as under LINEX loss functions.     

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 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.     

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.     

A Note on Estimation of a Distribution Function in a Nonparametric Set-up Using Stein’s Shrinkage Estimation Technique

Based on Stein’s famous shrinkage estimation of a multivariate normal distribution, we propose a new type of estimators of the distribution function of a random variable in a nonparametric setup. The proposed estimators are then compared with the empirical distribution function, which is the best equivariant estimator under a well-known loss function. Our extensive simulation study shows that our proposed estimators can perform better for moderate to large sample sizes.     

Comparison of normal variance estimators under multiple criteria and towards a compromise estimator

For estimating a normal variance under the squared error loss function it is well known that the best affine (location and scale) equivariant estimator, which is better than the maximum likelihood estimator as well as the unbiased estimator, is also inadmissible. The improved estimators, e.g., stein type, brown type and Brewster–Zidek type, are all scale equivariant but not location invariant. Lately, a good amount of research has been done to compare the improved estimators in terms of risk, but comparatively less attention had been paid to compare these estimators in terms of the Pitman nearness criterion (PNC) as well as the stochastic domination criterion (SDC). In this paper, we have undertaken a comprehensive study to compare various variance estimators in terms of the PNC and the SDC, which has been long overdue. Finally, using the results for risk, the PNC and the SDC, we propose a compromise estimator (sort of a robust estimator) which appears to work ‘well’ under all the criteria discussed above.     

Bayes Estimator of Inverse Gaussian Parameters Under General Entropy Loss Function Using Lindley's Approximation

In this article, we obtained Bayes estimators of parameters of Inverse Gaussian distributions under asymmetric loss function using Lindley's Approximation (L-Approximation). The proposed estimators have been compared with the corresponding estimators obtained under symmetric loss function and MLE for their risks. This comparison is illustrated using Monte-Carlo study of 2,000 simulated sample from the Inverse Gaussian distribution.     

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.     

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.     

Bayesian Estimation for Poisson-exponential Model under Progressive Type-II Censoring Data with Binomial Removal and Its Application to Ovarian Cancer Data

In this article, we propose Maximum likelihood estimators (MLEs) and Bayes estimators of parameters of Poisson-exponential distribution (PED) under General entropy loss function (GELF) and Squared error loss function (SELF) for Progressive type-II censored data with binomial removals (PT-II CBRs). The MLEs and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from PED. The proposed methodology is illustrated on a real dataset of ovarian cancer.     

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.     

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.     

Admissibility for linear estimators of regression coefficient in a general Gauss–Markoff model under balanced loss function

A generalization of Zellner's balanced loss function is proposed according to unified theory of least squares under a general Gauss–Markoff model. Admissibility of linear estimators is investigated under the balanced loss function. And necessary and sufficient conditions that linear estimators are admissible in a class of homogeneous and nonhomogeneous linear estimators are obtained, respectively.     

A generalization of the compound rayleigh distribution: using a bayesian method on cancer survival times

In this paper the generalized compound Rayleigh model, exhibiting flexible hazard rate, is high¬lighted. This makes it attractive for modelling survival times of patients showing characteristics of a random hazard rate. The Bayes estimators are derived for the parameters of this model and some survival time parameters from a right censored sample. This is done with respect to conjugate and discrete priors on the parameters of this model, under the squared error loss function, Varian's asymmetric linear-exponential (linex) loss function and a weighted linex loss function. The future survival time of a patient is estimated under these loss functions. A Monte Carlo simu¬lation procedure is used where closed form expressions of the estimators cannot be obtained. An example illustrates the proposed estimators for this model.     

Linearly admissible estimators of the common mean parameter under balanced loss function

Admissibility of linear estimators of the common mean parameter is investigated in the context of a linear model under balanced loss function. Sufficient and necessary conditions for linear estimators to be admissible in classes of homogeneous and non homogeneous linear estimators are obtained, respectively.     

Linearly admissible estimators of stochastic regression coefficient under balanced loss function

The admissibility of linear estimators in a linear model with stochastic regression coefficient is investigated under a balanced loss function. The sufficient and necessary conditions for linear estimators to be admissible in classes of homogeneous and non-homogeneous linear estimators are obtained, respectively.     

A weighted linear quantile regression

In this article, we introduce a new weighted quantile regression method. Traditionally, the estimation of the parameters involved in quantile regression is obtained by minimizing a loss function based on absolute distances with weights independent of explanatory variables. Specifically, we study a new estimation method using a weighted loss function with the weights associated with explanatory variables so that the performance of the resulting estimation can be improved. In full generality, we derive the asymptotic distribution of the weighted quantile regression estimators for any uniformly bounded positive weight function independent of the response. Two practical weighting schemes are proposed, each for a certain type of data. Monte Carlo simulations are carried out for comparing our proposed methods with the classical approaches. We also demonstrate the proposed methods using two real-life data sets from the literature. Both our simulation study and the results from these examples show that our proposed method outperforms the classical approaches when the relative efficiency is measured by the mean-squared errors of the estimators.     

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.     

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.     

Bayesian multivariate normal analysis under the extended reflected normal loss function

This paper considers the Bayesian analysis of the multivariate normal distribution under a new and bounded loss function, based on a reflection of the multivariate normal density function. The Bayes estimators of the mean vector can be derived for an arbitrary prior distribution of [d]. When the covariance matrix has an inverted Wishart prior density, a Bayes estimator of[d] is obtained under a bounded loss function, based on the entropy loss. Finally the admissibility of all linear estimators c[d]+ d for the mean vector is considered