Two-parameter Exponential Distribution Research Articles

On the estimation of the probabilities of a double linear inequality in the case of uniform and two-parameter exponential distributions

On the estimation of the probabilities of a double linear inequality in the case of uniform and two-parameter exponential distributions

Sequential Estimation of the Guarantee Times for Systems of Several Exponential Components

SYNOPTIC ABSTRACTThe minimum guaranteed lifetimes of series and parallel systems of components are the minimum and maximum of the individual guaranteed lifetimes, respectively. If the individual lifetimes independently follow two-parameter exponential distributions with a common scale parameter but possibly differing location parameters, then the minimum guaranteed lifetime of a series and parallel system of such components are the minimum and maximum of the location parameters, respectively. In this paper, we discuss sequential estimation of the minimum and maximum of several two-parameter exponential location parameters. Sequential stopping rules are developed under both the minimax risk and bounded maximum risk criteria. Asymptotic properties are provided for each stopping rule.

Distributions and moments for estimators of Gini index in an exponential distribution

In this paper, we propose several estimators of Gini index of the two-parameter exponential distribution and obtain distributions and moments of the proposed estimators. The proposed estimators are shown to cosistency and will be compared in terms of the mean squared error (MSE) through Monte Carlo method.

Estimation of the parameters of pareto distribution and the reliability function usin accelerated life testing with censoring

This paper addresses the situation where the time to failure is assumed to have a three-parameter Pareto distribution. This distribution is a mixture of two-parameter exponential distribution with scale parameter distributed as gamma distribution. The experiment is subject to type II censoring. A certain condition (stress) is assumed to affect the shape parameter of Pareto distribution. Maximum likelihood estimators (MLE) of an accelerated life‐test (ALT) model are derived. Prediction of the shape parameter under usual conditions is obtained. The reliability function at a certain mission time is also predicted. A numerical example is given

Estimation of a linear function of the parameters of an exponential distribution from doubly censored samples

In this paper improved estimators of the scale and its reciprocal, θ(0 −1), and the location, μ, of a two-parameter exponential distribution are given based on a doubly censored sample. Also the problem of estimating the linear function μ + zθ is considered, where z is a given constant. It is shown that the improved estimators are better than the best affine equivariant estimators.

The UMVUE ofP(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions

Based on two independent samples from Weinman multivariate exponential distributions with unknown scale parameters, uniformly minimum variance unbiased estimators ofP(X<Y) are obtained for both, unknown and known common location parameter. The samples are permitted to be Type-II censored with possibly different numbers of observations. Since sampling from two-parameter exponential distributions is contained in the model as a particular case, known results for complete and censored samples are generalized. In the case of an unknown common location parameter with a certain restriction of the model, the UMVUE is shown to have a Gauss hypergeometric distribution, which is further examined. Moreover, explicit expressions for the variances of the estimators are derived and used to calculate the relative efficiency.

Estimating guaranteed lifetimes of systems in a random environment

Suppose 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, based on samples taken from the mixed population. Several estimators are proposed and compared. Maximum likelihood estimators are shown to be inadmissible. These results are invariant to the distribution of the random hazard.

Unbiased estimators of p (X&lt;Y) and their variances in the case of uniform and two-parameter exponential distributions

In this paper, we consider the two-parameter exponential distribution with unknown shift and known scale parameters as well as the uniform distribution on a segment with one end known. For these cases unbiased estimators for (1) are found. We also obtain unbiased estimators for the variances of these unbiased estimators.

Parametric estimation of two-parameter exponential model in the presence of unidentified outliers

Abstract Parametric estimators of two-parameter exponential distribution in the presence of unidentified outliers are proposed and the means and variances of the estimators are obtained in exact forms.

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.

Tests for upper outliers in the two-parameter exponential distribution

The problem of detecting upper outliers in a sample from two parameter exponential distribution with unknown location and scale parameters is discussed. The block and inside-out sequential test procedures are investigated under a particular life testing situation. Two test statistics, one based on the prediction of future order statistics and the other based on score function are considered. The exact null distribution of one of the test statistic is quite involved and the simulated critical values are provided. The power performances of the proposed tests are investigated using a Monte Carlo study. The application of the above procedures for the case of Pareto sample is also indicated.

On the Shapiro-Wilk Test and Darling's Test for Exponentiality

Tests for exponentiality are widely used in studying time-structured phenomena. Especially in the analysis of behavioural data, however, hazard rates are constant only after a dead time during which no events can occur. To take this into account, Shapiro and Wilk (1972, Teclhnoinetrics 14, 355-370) developed a test for the two-parameter exponential distribution with unknown origin. They did not, however, consider the asymptotic distribution of the test statistic or its power properties. Although it has as yet been unnoticed, it is an elementary exercise to show that a transformed version of the Shapiro-Wilk test statistic is equal to Darling's (1953, Annals of Mathematical Statistics 24, 239-253) test statistic for the one-parameter exponential distribution. For this test, no small-sample critical values were known, but the asymptotic null distribution of the statistic is known to be normal, and the right-sided version of the test is locally most powerful against mixtures of exponentials. The two test statistics have the same distribution under the null hypothesis of exponentiality with unknown origin as well as under the alternative of a mixture of two-parameter exponentials with the same unknown origin. Since simulation results indicate that the convergence toward normality is rather slow, it is advised to use small-sample results for both test statistics. To this end we extend the table given by Shapiro and Wilk (1972) to values of n up to 500.

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

Suppose we have two components, each having a two-parameter exponential distribution. Suppose further that the components are conditionally independent with a random hazard rate and unequal, unknown, fixed location parameters. The objective of this paper is to estimate the minimum and maximum of the location parameters from conditionally independent samples taken from the mixed population. Several estimators are proposed and their properties in terms of MSE and absolute bias will be studied and compared. Interestingly, the relationships between these estimators always hold regardless of the mixing parameters assumed in the model.

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

In this article, there methods are proposed to give the lower confidence bounds for the parameters in one-and two-parameter exponential distributions with multiply Type II censored data. The accuracy of the approximate methods are compared both theoretically and numerically. The problem of significance testing for the threshold parameter is also addressed.

Estimation of parameters in a two-parameter exponential distribution using ranked set sample

In situations where the experimental or sampling units in a study can be easily ranked than quantified, McIntyre (1952,Aust. J. Agric. Res.,3, 385–390) proposed that the mean ofn units based on aranked set sample (RSS) be used to estimate the population mean, and observed that it provides an unbiased estimator with a smaller variance compared to a simple random sample (SRS) of the same sizen. McIntyre's concept ofRSS is essentially nonparametric in nature in that the underlying population distribution is assumed to be completely unknown. In this paper we further explore the concept ofRSS when the population is partially known and the parameter of interest is not necessarily the mean. To be specific, we address the problem of estimation of the parameters of a two-parameter exponential distribution. It turns out that the use ofRSS and its suitable modifications results in much improved estimators compared to the use of aSRS.

Bayesian prediction for the range based on a two-parameter exponential distribution with a random sample size

This paper is concerned with the problem of predicting the range of a future sample when the underlying distribution is a two-parameter exponential distribution. A Bayesian prediction distribution of the range will be derived under the assumption that the old sample is a censored type II and the sample size of the future sample is a random variable. A numerical illustration is provided.

Admissibility of a conditionally specified test procedure for time censored life data

Abstract A test procedure may have a sufficiently large power and size under control, but its use is not advisable unless it is admissible. In other words, the admissibility of a test procedure is a desirable property in addition to having size and power. In this paper, the necessary and sufficient conditions for the admissibility of a conditionally-specified test procedure, for testing the scale parameter called ‘average life’ in a two-parameter exponential distribution under time censoring, have been derived.

Optimal single variable sampling plans

A single variable sampling plan with a polynomial loss function is studied. The variable measured from each item of a 2 batch follows a normal distribution N(m,t2 ) where m has a normal prior distribution, and t2 has either a uniform prior distribution or a two-parameter exponential prior distribution. An algorithm for the determination of an optimal sampling planis given. The algorithm isfinite, i.e. an optimal solution can be obtained after a finite number of steps of searching. 371 Copyright 1993 by Marcel Dekker, Inc

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.

Bayesian reliability analysis for the two-parameter exponential distribution

This paper discusses the Bayesian reliability analysis for the two-parameter exponential distribution on the basis of the life tests that are terminated after a pre assigned number of failures, when the prior distribution of the location parameter, given the scale, is assumed to be exponential, while the scale parameter is assumed to have an inverse Gaussian prior distribution. The usual assumption of the squared error loss function is also made. A situation dealing with attribute life testing is also considered.