Statistical analysis of type I multiply left-censored samples from exponential distribution

Left-censored data with one or more detection limits (DLs) often arise in environmental contexts. The computational procedure for the calculation of maximum likelihood estimators of the parameter for Type I multiply left-censored data from underlying exponential distribution is suggested and used considering various numbers of DLs. The expected Fisher information measure (FIM) is analytically determined and its performance is compared with sample (observed) FIM using simulations. Simulations are focused primarily on the properties of estimators for small sample sizes. Moreover, the simulations follow the possible applications of the results in the statistical analysis of real chemical data.     

Analysis of left censored data from the generalized exponential distribution

The generalized exponential distribution proposed by Gupta and Kundu [Gupta, R.D and Kundu, D., 1999, Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173–188.] is an important lifetime distribution in survival analysis. In this paper, we consider the maximum likelihood estimation procedure of the parameters of the generalized exponential distribution when the data are left censored. We obtain the maximum likelihood estimators of the unknown para-meters and the Fisher information matrix. Simulation studies are carried out to observe the performance of the estimators in small sample.     

Optimal step-stress testing for progressively Type-I censored data from exponential distribution

In this paper, a k  -step-stress accelerated life-testing is considered with an equal step duration τ$\tau$. For small to moderate sample sizes, a practical modification is made to the model previously considered by Gouno et al. [2004. Optimal step-stress test under progressive Type-I censoring. IEEE Trans. Reliability 53, 383–393] in order to guarantee a feasible k  -step-stress test under progressive Type-I censoring, and the optimal τ$\tau$ is determined under this model. Next, we discuss the determination of optimal τ$\tau$ under the condition that the step-stress test proceeds to the k  -th stress level, and the efficiency of this conditional inference is compared to that of the previous case. In all cases considered, censoring is allowed at each point of stress change (viz., iτ$i\tau$, i=1,2,…,k$i=1,2,\dots ,k$). The determination of optimal τ$\tau$ is discussed under C-optimality, D-optimality, and A-optimality criteria. We investigate in detail the case of progressively Type-I right censored data from an exponential distribution with a single stress variable.     

Optimal step-stress test under Type-I censoring for multivariate exponential distribution

In this paper, we study a k-step-stress accelerated life test under Type-I censoring. The lifetime of the items follows the multivariate exponential distribution and a cumulative exposure model is considered. We derive the maximum likelihood estimators of the model parameters and establish the asymptotic properties of them. The problem of choosing the optimal time is addressed by using V-optimality as well as D-optimality criteria. Finally, some numerical studies are discussed to illustrate the proposed procedures.     

Conditional maxima and inferences for the truncated exponential distribution

This paper derives the conditional distribution of the maximum given the sample total for a random sample from the truncated exponential distribution. Based on that result, the paper develops tests or associated confidence intervals for the truncation parameter θ with another parameter θ assumed unknown.     

Improved estimators for the location of double exponential distribution

In this paper, attention is focused on estimation of the location parameter in the double exponential case using a weighted linear combination of the sample median and pairs of order statistics, with symmetric distance to both sides from the sample median. Minimizing with respect to weights and distances we get smaller asymptotic variance in the second order. If the number of pairs is taken as infinite and the distances as null we attain the least asymptotic variance in this class of estimators. The Pitman estimator is also noted. Similarly improved estimators are scanned over their probability of concentration to investigate its bound. Numerical comparison of the estimators is shown.     

Optimum reliability acceptance sampling plans under progressive type-I interval censoring with random removal using a cost model

A reliability acceptance sampling plan (RASP) is a variable sampling plan, which is used for lot sentencing based on the lifetime of the product under consideration. If a good lot is rejected then there is a loss of sales, whereas if a bad lot is accepted then the post sale cost increases and the brand image of the product is affected. Since cost is an important decision-making factor, adopting an economically optimal RASP is indispensable. This work considers the determination of an asymptotically optimum RASP under progressive type-I interval censoring scheme with random removal (PICR-I). We formulate a decision model for lot sentencing and a cost function is proposed that quantifies the losses. The cost function includes the cost of conducting the life test and warranty cost when the lot is accepted, and the cost of batch disposition when it is rejected. The asymptotically optimal RASP is obtained by minimizing the Bayes risk in a set of decision rules based on the maximum likelihood estimator of the mean lifetime of the items in the lot. For numerical illustration, we consider that lifetimes follow exponential or Weibull distributions.     

Weighted exponential distribution: properties and different methods of estimation

In this article, we investigate various properties and methods of estimation of the Weighted Exponential distribution. Although, our main focus is on estimation (from both frequentist and Bayesian point of view) yet, the stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics are derived first time for the said distribution. Different types of loss functions are considered for Bayesian estimation. Furthermore, the Bayes estimators and their respective posterior risks are computed and compared using Gibbs sampling. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and two real data sets have been analysed for illustrative purposes.     

Change-point analysis with bathtub shape for the exponential distribution

Likelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

Bayesian sample size determination for the bounded linex loss function

In this paper, we discuss optimum sample size determination for a bounded linex loss function (blinex). The linex loss function is often used when losses are asymmetric, but it has the disadvantage that it can only be used if the moment-generating function of the posterior distribution exists. Blinex loss has the advantage that it can always be calculated. Also many authors have argued that a bounded loss function is to be preferred. We have obtained the optimum sample size for a number of distributions when the cost of sampling is linear. The form of the posterior risk function does not allow an analytical solution so simulation is necessary.     

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.     

Bayesian analysis for the two-parameter Pareto distribution based on record values and times

Doostparast and Balakrishnan (Pareto record-based analysis, Statistics, under review) recently developed optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning shape and scale parameters, for the two-parameter Pareto distribution based on record data. In this paper, on the basis of record values and inter-record times from the two-parameter Pareto distribution, maximum-likelihood and Bayes estimators as well as credible regions are developed for the two parameters of the Pareto distribution. For illustrative purposes, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.     

A new method for generating distributions with an application to exponential distribution

A new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility. A special case has been considered in detail, namely one-parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter exponential distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.     

Assessment of the Gould-Shih procedure for sample size re-estimation

The power of a clinical trial is partly dependent upon its sample size. With continuous data, the sample size needed to attain a desired power is a function of the within-group standard deviation. An estimate of this standard deviation can be obtained during the trial itself based upon interim data; the estimate is then used to re-estimate the sample size. Gould and Shih proposed a method, based on the EM algorithm, which they claim produces a maximum likelihood estimate of the within-group standard deviation while preserving the blind, and that the estimate is quite satisfactory. However, others have claimed that the method can produce non-unique and/or severe underestimates of the true within-group standard deviation. Here the method is thoroughly examined to resolve the conflicting claims and, via simulation, to assess its validity and the properties of its estimates. The results show that the apparent non-uniqueness of the method's estimate is due to an apparently innocuous alteration that Gould and Shih made to the EM algorithm. When this alteration is removed, the method is valid in that it produces the maximum likelihood estimate of the within-group standard deviation (and also of the within-group means). However, the estimate is negatively biased and has a large standard deviation. The simulations show that with a standardized difference of 1 or less, which is typical in most clinical trials, the standard deviation from the combined samples ignoring the groups is a better estimator, despite its obvious positive bias.     

Optimal sample size allocation for accelerated life test under progressive type-II censoring with competing risks

In this article, we study the optimization problem of sample size allocation when the competing risks data are from a progressive type-II censoring in a constant-stress accelerated life test with multiple levels. The failure times of the individual causes are assumed to be statistically independent and exponentially distributed with different parameters. We obtain the estimates of the unknown parameters through a maximum likelihood method, and also derive the Fisher information matrix. We propose three optimization criteria and two search scenarios to obtain the sample size allocation at each stress level. Some numerical results are studied to illustrate the usage of the proposed methods.     

Continuous-time capture-recapture models with time variation and behavioural response

This paper develops a likelihood-based inference procedure for continuous-time capture-recapture models. The first-capture and recapture intensities are assumed to be in constant proportion but may otherwise vary arbitrarily through time. The full likelihood is partitioned into two factors, one of which is analogous to the likelihood in a special type of multiplicative intensity model arising in failure time analysis. The remaining factor is free of the non-parametric nuisance parameter and is easily maximized. This factor provides an estimator of population size and an asymptotic variance under a counting process framework. The resulting estimation procedure is shown to be equivalent to that derived from a martingale-based estimating function approach. Simulation results are presented to examine the performance of the proposed estimators.     

Optimal record-based statistical procedures for the two-parameter exponential distribution

There are a number of situations in which the experimental data observed are record statistics. In this paper, optimal confidence intervals as well as uniformly most powerful (MP) tests for one-sided alternatives are developed. Since a uniformly MP test for a two-sided alternative does not exist, generalized likelihood ratio and uniformly unbiased and invariant tests are derived for the two parameters of the exponential distribution based on record data. For illustrative purposes, a data set on the times between consecutive telephone calls to a company's switchboard is analysed using the proposed procedures. Finally, some open problems in this direction are pointed out.     

Reliability estimation in generalized inverted exponential distribution with progressively type II censored sample

In this paper, a generalization of inverted exponential distribution is considered as a lifetime model [A.M. Abouammoh and A.M. Alshingiti, Reliability estimation of generalized inverted exponential distribution, J. Statist. Comput. Simul. 79(11) (2009), pp. 1301–1315]. Its reliability characteristics and important distributional properties are discussed. Maximum likelihood estimation of the two parameters involved along with reliability and failure rate functions are derived. The method of least square estimation of parameters is also studied here. In view of cost and time constraints, type II progressively right censored sampling scheme has been used. For illustration of the performance of the estimates, a Monte Carlo simulation study is carried out. Finally, a real data example is given to show the practical applications of the paper.