Estimation of common location parameter of two exponential populations based on records

Consider the problem of estimating the common location parameter of two exponential populations using record data when the scale parameters are unknown. We derive the maximum likelihood estimator (MLE), the modified maximum likelihood estimator (MMLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the common location parameter. Further, we derive a general result for inadmissibility of an equivariant estimator under the scaled-squared error loss function. Using this result, we conclude that the MLE and the UMVUE are inadmissible and better estimators are provided. A simulation study is conducted for comparing the performances of various competing estimators.     

Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

Improved estimation of common location of two exponential populations with order restricted scale parameters using censored samples

ABSTRACT

Estimation of common location parameter of two exponential populations is considered when the scale parameters are ordered using type-II censored samples. A general inadmissibility result is proved which helps in deriving improved estimators. Further, a class of estimators dominating the MLE has been derived by an application of integrated expression of risk difference (IERD) approach of Kubokawa. A discussion regarding extending the results to a general k( ? 2) populations has been done. Finally, all the proposed estimators are compared through simulation.     

Stress-strength reliability estimation for exponentially distributed system with common minimum guarantee time

Abstract

In this article, we study the problem of estimating the stress-strength reliability, where the stress and strength variables follow independent exponential distributions with a common location parameter but different scale parameters. All parameters are assumed to be unknown. We derive the MLE, the UMVUE of the reliability parameter. We also derive the Bayes estimators considering conjugate prior distributions for the scale parameters and a dependent prior for the common location parameter. Monte Carlo simulations have been carried out to compare among the proposed estimators with respect to different loss functions.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

On estimation of R=P(Y<X) for exponential distribution under progressive type-II censoring

This paper deals with the estimation of the stress–strength parameter R=P(Y<X), when X and Y are independent exponential random variables, and the data obtained from both distributions are progressively type-II censored. The uniformly minimum variance unbiased estimator and the maximum-likelihood estimator (MLE) are obtained for the stress–strength parameter. Based on the exact distribution of the MLE of R, an exact confidence interval of R has been obtained. Bayes estimate of R and the associated credible interval are also obtained under the assumption of independent inverse gamma priors. An extensive computer simulation is used to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose.     

A note on the common location parameter of several exponential populations

The problem of estimation of an unknown common location parameter of several exponential populations with unknown and possibly unequal scale parameters is considered. A wide class of estimators, including both a modified maximum likelihood estimator (MLE), and the uniformly minimum variance unbiased estimator (Umvue) proposed by ghosh and razmpour(1984), is obtained under a class of convex loss functions.     

Estimating the common hazard rate of several exponential distributions

Abstract

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.     

Estimating the negative binomial dispersion parameter with highly stratified surveys

We investigate several estimators of the negative binomial (NB) dispersion parameter for highly stratified count data for which the statistical model has a separate mean parameter for each stratum. If the number of samples per stratum is small then the model is highly parameterized and the maximum likelihood estimator (MLE) of the NB dispersion parameter can be biased and inefficient. Some of the estimators we investigate include adjustments for the number of mean parameters to reduce bias. We extend other estimators that were developed for the iid case, to reduce bias when there are many mean parameters. We demonstrate using simulations that an adjusted double extended quasi-likelihood estimator we proposed gives much improved estimates compared to the MLE. Adjusted extended quasi-likelihood and adjusted maximum likelihood estimators also give much-improved results. We illustrate the various estimators with stratified random bottom trawl survey data for cod (Gadus morhua) off the south coast of Newfoundland, Canada.     

Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

Estimation of Pr( x<y ) in the exponential case with common location parameter

This paper considers the problem of estimating the probability P = Pr(X < Y) when X and Y are independent exponential random variables with unequal scale parameters and a common location parameter. Uniformly minimum variance unbiased estimator of P is obtained. The asymptotic distribution of the maximum likelihood estimator is obtained and then the asymptotic equivalence of the two estimators is established. Performance of the two estimators for moderate sample sizes is studied by Monte Carlo simulation. An approximate interval estimator is also obtained.     

Inference of R=P[X<Y] for generalized logistic distribution

This paper deals with the estimation of R=P[X<Y] when X and Y come from two independent generalized logistic distributions with different parameters. The maximum-likelihood estimator (MLE) and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Assuming that the common scale parameter is known, the MLE, uniformly minimum variance unbiased estimator, Bayes estimation and confidence interval of R are obtained. The MLE of R, asymptotic distribution of R in the general case, is also discussed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.     

LOSS OF INFORMATION ASSOCIATED WITH THE ORDER STATISTICS AND RELATED ESTIMATORS IN THE DOUBLE EXPONENTIAL DISTRIBUTION CASE

Fisher (1934) derived the loss of information of the maximum likelihood estimator (MLE) of the location parameter in the case of the double exponential distribution. Takeuchi & Akahira (1976) showed that the MLE is not second order asymptotically efficient. This paper extends these results by obtaining the (asymptotic) losses of information of order statistics and related estimators, and by comparing them via their asymptotic distributions up to the second order.     

On improved estimation of a gamma shape parameter

This paper deals with improved estimation of a gamma shape parameter from a decision-theoretic point of view. First we study the second-order properties of three estimators – (i) the maximum-likelihood estimator (MLE), (ii) a bias corrected version of the MLE, and (iii) an improved version (in terms of mean squared error) of the MLE. It is shown that all the three estimators mentioned above are second-order inadmissible. Next, we obtain superior estimators which are second order better than the above three estimators. Simulation results are provided to study the relative risk improvement of each improved estimator over the MLE.     

On moments of dual generalized order statistics from Topp-Leone distribution

In this paper, we have derived exact and explicit expressions for the ratio and inverse moments of dual generalized order statistics from Topp-Leone distribution. This result includes the single and product moments of order statistics and lower records . Further, based on n dual generalized order statistics, we have deduced the expression for Maximum likelihood estimator (MLE) and Uniformly minimum variance unbiased estimator (UMVUE) for the shape parameter of Topp-Leone distribution. Finally, based on order statistics and lower records, a simulation study is being carried out to check the efficiency of these estimators.     

Estimation of p{y p> max(y1, y2, …, yp-1)}in theexponential case

This paper deals with the derivation of (i) the MLE (ii) the MVUE (iii) a Bayes estimator of the probability in the title, for the case p = 2. Simulation studies are carried out to compare these estimators. The results suggest that the MLE and the Bayes estimator are biased and the Bayes estimator have the smallest MSE. In the general case, explicit expression for the probability in the title is derived and the MLE and Bayes estimator are obtained. A general method of deriving the MVUE is pointed out. Because of the simulation studies for p = 2 it is recommended that the Bayes or predictive estimator should be used.     

Simultaneous robust estimation of location and scale parameters: A minimum-distance approach

Simultaneous robust estimates of location and scale parameters are derived from minimizing a minimum-distance criterion function. The criterion function measures the squared distance between the pth power (p > 0) of the empirical distribution function and the pth power of the imperfectly determined model distribution function over the real line. We show that the estimator is uniquely defined, is asymptotically bivariate normal and for p > 0.3 has positive breakdown. If the scale parameter is known, when p = 0.9 the asymptotic variance (1.0436) of the location estimator for the normal model is smaller than the asymptotic variance of the Hodges-Lehmann (HL)estimator (1.0472). Efficiencies with respect to HL and maximum-likelihood estimators (MLE) are 1.0034 and 0.9582, respectively. Similarly, if the location parameter is known, when p = 0.97 the asymptotic variance (0.6158) of the scale estimator is minimum. The efficiency with respect to the MLE is 0.8119. We show that the estimator can tolerate more corrupted observations at oo than at – for p < 1, and vice versa for p > 1.     

Ratio and product estimators in stratified random sampling

Khoshnevisan et al. [2007. A general family of estimators for estimating population mean using known value of some population parameter(s). Far East Journal of Theoretical Statistics 22, 181–191] have introduced a family of estimators using auxiliary information in simple random sampling. They have showed that these estimators are more efficient than the classical ratio estimator and that the minimum value of the mean square error (MSE) of this family is equal to the value of MSE of regression estimator. In this article, we adapt the estimators in this family to the stratified random sampling and motivated by the estimator in Searls [1964. Utilization of known coefficient of kurtosis in the estimation procedure of variance. Journal of the American Statistical Association 59, 1225–1226], we also propose a new family of estimators for the stratified random sampling. The expressions of bias and MSE of the adapted and proposed families are derived in a general form. Besides, considering the minimum cases of these MSE equations, the efficient conditions between the adapted and proposed families are obtained. Moreover, these theoretical findings are supported by a numerical example with original data.     

Estimation of the smaller and larger of two exponential location parameters

Assume independent random samples are drawn from two populations which are exponentially distributed with unknown location parameters and a common known scale parameter. We want to estimate the maximum and the minimum of the unknowo location paremeters. In this paper several estimators are proposed which are better than the natural estimations in terms of absolute bias and /or meaqn squared error.