Estimating the minimum and maximum location parameters for two ig-exponential scale mixtures

Suppose that we have two components, each having a two-parameter exponential distribution. Suppose further that these components are conditionally independent, sharing a common random hazard rate and possessing unequal, fixed, unknown location parameters. We develop estimators for the minimum and maximum of these location parameters when the random hazard rate has an inverse Gaussian distribution. Performance comparisons are made among the proposed estimators. Maximum likelihood estimators are shown to be inadmissible.     

Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data

In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.     

On the disribution of residuals form fitted parametric models

Results of a simulation study of the fit of data to an estimated parametric model are reported. Three particular models including the two-parameter normal and exponential distributions, and the simple linear regression model are considered. A number of scaled versions of the least squares residuals from the regression model and quantities that we call residuals from the other two models arc seen follow the parent distribution form loo well. i.e., to be supernormal and superexponential. A point of particular interest is that this tendency does not appear to decrease with increasing sample size, at least for the sample sizes considered here.     

Pareto analysis based on records

Estimation of the parameters of an exponential distribution based on record data has been treated by Samaniego and Whitaker [On estimating population characteristics from record-breaking observations, I. Parametric results, Naval Res. Logist. Q. 33 (1986), pp. 531–543] and Doostparast [A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. Recently, Doostparast and Balakrishnan [Optimal record-based statistical procedures for the two-parameter exponential distribution, J. Statist. Comput. Simul. 81(12) (2011), pp. 2003–2019] obtained optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning location and scale parameters based on record data from a two-parameter exponential model. In this paper, we derive optimal statistical procedures including point and interval estimation as well as most powerful tests based on record data from a two-parameter Pareto model. For illustrative purpose, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.     

Exact linear inference and prediction for exponential distributions based on general progressively type-ii censored samples

In this paper, we make use of an algorithm of Huffer and Lin (2001) in order to develop exact interval estimation for the location and scale parameters of an exponential distribution based on general progressively Type-II censored samples. The exact prediction intervals for failure times of the items censored at the last observation are also presented for one-parameter and two-parameter exponential distributions. Finally, we give two examples to illustrate the methods of inference developed here.     

Estimation of Pr(Y < X) for the two-parameter generalized exponential records

This article considers the maximum likelihood and Bayes estimation of the stress–strength reliability based on two-parameter generalized exponential records. Here, we extend the results of Baklizi [Computational Statistics and Data Analysis 52 (2008), 3468–3473] to explain a wide variety of real datasets. We also consider the estimation of R when the same shape parameter is known. The results for exponential distribution can be obtained as a special case with different scale parameters.     

Estimation for one- and two-parameter exponential distributions under multiple type-II censoring

In this paper, we derive explicit best linear unbiased estimators for one- and two-parameter exponential distributions when the available sample is multiply Type-II censored. Further, after noting that the maximum likelihood estimators do not exist explicitly, we propose some linear estimators by approximating the likelihood equations appropriately. Some illustrative examples from life-testing experiments are also presented.     

On the power of goodness-of-fit tests for the exponential distribution under progressive Type-II censoring

The aim of this article is to review existing goodness-of-fit tests for the exponential distribution under progressive Type-II censoring and to provide some new ideas and adjustments. In particular, we consider two-parameter exponentially distributed random variables and adapt the proposed test procedures to our scenario if necessary. Then, we compare their power by an extensive simulation study. Furthermore, we propose five new test procedures that provide reasonable alternatives to those already known.     

Exponential family distribution with unknown truncation parameter: two sample problem

The uniformly most powerful unbiased tests are formulated for two sample problem of a given continuous distribution belonging to the exponential family with unknown scale and truncation parameters. The two-parameter exponential and Paretc distributions are considered in examples.     

Maximum likelihood estimators of the location and scale parameters of the exponential distribution from a censored sample

In this note explicit expressions are given for the maximum likelihood estimators of the parameters of the two-parameter exponential distribution, when a doubly censored sample is available.     

Unbiased Estimation of the Distribution Function of a Two-Parameter Exponential Population Using Order Statistics

In this article, we consider the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population. We give necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics and suggest unbiased estimators in some situations where unbiased estimators exist. A few properties of the suggested estimators for some special cases have also been discussed.     

Estimation of quantiles of exponential and double exponential distributions based on two order statistics

Asymptotically best linear unbiased estimators (ABLUE) of quantiles, x^., in the two-parameter (location-scale) exponential and double exponential families are obtained as linear combinations of two suitably chosen order statistics. Exact formulae for the linear combinations are given as functions of £. The derived estimators in both cases compare favorably with the usual nonparametric estimator. Also, in the exponential case the derived estimator compares favorably with the Sarhan-Greenberg BLUE based on a complete sample     

On Unimodal Regression in the Exponential Family

In this article, we discuss statistical methods for curve-estimation under the assumption of unimodality for variables with distributions belonging to the two-parameter exponential family with known or constant dispersion parameter. An important special case is a one-parameter distribution. We suggest a nonparametric method based on monotonicity properties. The method is applied to Swedish data on laboratory verified diagnoses of influenza and data on inflation from an episode of hyperinflation in Bulgaria.     

An extension of the generalized exponential distribution

The two-parameter generalized exponential distribution has been used recently quite extensively to analyze lifetime data. In this paper the two-parameter generalized exponential distribution has been embedded in a larger class of distributions obtained by introducing another shape parameter. Because of the additional shape parameter, more flexibility has been introduced in the family. It is observed that the new family is positively skewed, and has increasing, decreasing, unimodal and bathtub shaped hazard functions. It can be observed as a proportional reversed hazard family of distributions. This new family of distributions is analytically quite tractable and it can be used quite effectively to analyze censored data also. Analyses of two data sets are performed and the results are quite satisfactory.     

Discriminating between the log-normal and generalized exponential distributions

The two-parameter generalized exponential distribution was recently introduced by Gupta and Kundu (Austral. New Zealand J. Statist. 40 (1999) 173). It is observed that the Generalized Exponential distribution can be used quite effectively to analyze skewed data set as an alternative to the more popular log-normal distribution. In this paper, we use the ratio of the maximized likelihoods in choosing between the log-normal and generalized exponential distributions. We obtain asymptotic distributions of the logarithm of the ratio of the maximized likelihoods and use them to determine the required sample size to discriminate between the two distributions for a user specified probability of correct selection and tolerance limit.     

Distributions of certain variables in samples from two-parameter exponential distribution

Cumulative distribution function of the variable Y=(U+c)/(Z/2ν)) is given. Here U and Z are independent random variables, U has the exponential distribution (1.1) with θ=0, σ=1, Z has the distribution χ² (2ν) and c is a real quantity. The variable Y with U and Z given by (2.2) and (2.3) is used for inference about the parametric functions ?=θ?kσ of a two-parameter exponential distribution (1.1) with k or ? known. Special cases of ? or k are: the parameter θ, the Pth quantile Xp, the mean θ+σ and the value of the cumulative distribution function or of the reliability function at given point a. Also one-sided tolerance limits for a two-parameter exponential distribution can be derived from the distribution of the variable Y. The results are also applied to the Pareto distribution.     

Estimation of the smallest scale parameter of two-parameter exponential distributions

Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values.     

The simultaneous confidence intervals for all distances from the extreme populations for two-parameter exponential populations based on the multiply type II censored samples

Among k independent two-parameter exponential distributions which have the common scale parameter, the lower extreme population (LEP) is the one with the smallest location parameter and the upper extreme population (UEP) is the one with the largest location parameter. Given a multiply type II censored sample from each of these k independent two-parameter exponential distributions, 14 estimators for the unknown location parameters and the common unknown scale parameter are considered. Fourteen simultaneous confidence intervals (SCIs) for all distances from the extreme populations (UEP and LEP) and from the UEP from these k independent exponential distributions under the multiply type II censoring are proposed. The critical values are obtained by the Monte Carlo method. The optimal SCIs among 14 methods are identified based on the criteria of minimum confidence length for various censoring schemes. The subset selection procedures of extreme populations are also proposed and two numerical examples are given for illustration.     

Robust Bayesian estimation of a two-parameter exponential distribution under generalized Type-I progressive hybrid censoring

A generalized Type-I progressive hybrid censoring scheme was proposed recently to overcome the limitations of the progressive hybrid censoring scheme. In this article, we provide a robust Bayesian method to estimate the unknown parameters of the two-parameter exponential distribution of a generalized Type-I progressive hybrid censored sample. For each parameter, we derive the marginal posterior density functions and the corresponding Bayesian estimators under the squared error loss function. To assess the proposed method, Monte Carlo simulations are performed using a real dataset.     

Nonnegative Unbiased Estimation of Scale Parameters and Associated Quantiles Based on a Ranked Set Sample

In this article, we are interested in estimating the scale parameter in location and scale families. It is well known that the best linear unbiased estimator (BLUE) of scale parameter based on a simple random sample (SRS) is nonnegative. However, the BLUE of scale parameter based on a ranked set sample (RSS) can assume negative values. We suggest various modifications of BLUE of scale parameter based on RSS so that the resulting estimators are unbiased as well as nonnegative. Their performances in terms of relative efficiencies are compared and some recommendations are made for normal, logistic, double exponential, two-parameter exponential and Weibull distributions. We also briefly discuss an application of the proposed nonnegative BLUE of scale parameter for quantile estimation for the above populations.