Confidence intervals for a bounded mean in exponential families

Setting confidence bounds or intervals for a parameter in a restricted parameter space is an important issue in applications and is widely discussed in the recent literature. In this article, we focus on the distributions in the exponential families, and propose general forms of the truncated Pratt interval and rp interval for the means. We take the Poisson distribution as an example to illustrate the method and compare it with the other existing intervals. Besides possessing the merits from the theoretical inferences, the proposed intervals are also shown to be competitive approaches from simulation and real-data application studies.     

Shortest Confidence Intervals for Families of Distributions Involving Truncation Parameters

Statistical inferences for probability distributions involving truncation parameters have received recent attention in the literature. One aspect of these inferences is the question of shortest confidence intervals for parameters or parametric functions of these models. The topic is a classical one, and the approach follows the usual theory. In all literature treatments the authors consider specific models and derive confidence intervals (not necessarily shortest). All of these models can, however, be considered as special cases of a more general one. The use of this model enables one to obtain easily shortest confidence intervals and unify the different approaches. In addition, it provides a useful technique for classroom presentation of the topic.     

Generalized confidence interval for the slope in linear measurement error model

A generalized confidence interval for the slope parameter in linear measurement error model is proposed in this article, which is based on the relation between the slope of classical regression model and the measurement error model. The performance of the confidence interval estimation procedure is studied numerically through Monte Carlo simulation in terms of coverage probability and expected length.     

On lower confidence bounds for pcs in truncated location parameter models

We are concerned with deriving lower confidence bounds for the probability of a correct selection in truncated location-parameter models. Two cases are considered according to whether the scale parameter is known or unknown. For each case, a lower confidence bound for the difference between the best and the second best is obtained. These lower confidence bounds are used to construct lower confidence bounds for the probability of a correct selection. The results are then applied to the problem of seleting the best exponential populationhaving the largest truncated location-parameter. Useful tables are provided for implementing the proposed methods.     

Comparison of conditional and unconditional confidence intervals for the double exponential distribution

Conditional confidence intervals for the location parameter of the double exponential distribution based on maximum likelihood estimators conditioned on a set of ancillary statistics and the corresponding unconditional confidence intervals based on the maximum likelihood estimators alone are compared in two ways. Monte Carlo techniques are used and the conditional approach appears to give slightly better results although agreement as n becomes larger is noted     

Higher order approximations by a two-stage procedure for a negative exponential distribution

This paper deals with the problem of fixed-width confidence interval estimation of the location μ of a negative exponential distribution with unknown scale σ. Suppose we have information on the scale parameter σ such that σ>σ_L where σ_L(>0) is known to the experimenter from past experiences. We propose a two-stage procedure and provide higher order asymptotic expansions of the expected sample size and the coverage probability.     

A generalized score confidence interval for a binomial proportion

Constructing a confidence interval for a binomial proportion is one of the most basic problems in statistics. The score interval as well as the Wilson interval with some modified forms have been broadly investigated and suggested by many statisticians. In this paper, a generalized score interval CI_G(a) is proposed by replacing the coefficient 1/4 in the score interval with parameter a. Based on analyzing and comparing various confidence intervals, we recommend the generalized score interval CI_G(0.3) for the nominal confidence levels 0.90, 0.95 and 0.99, which improves the spike phenomenon of the score interval and behaves better and computes more easily than most of other approximate intervals such as the Agresti-Coull interval and the Jeffreys interval to estimate a binomial proportion.     

On classical and Bayesian order restricted inference for multiple exponential step stress model

In this article, we consider the multiple step stress model based on the cumulative exposure model assumption. Here, it is assumed that for a given stress level, the lifetime of the experimental units follows exponential distribution and the expected lifetime decreases as the stress level increases. We mainly focus on the order restricted inference of the unknown parameters of the lifetime distributions. First we consider the order restricted maximum likelihood estimators (MLEs) of the model parameters. It is well known that the order restricted MLEs cannot be obtained in explicit forms. We propose an algorithm that stops in finite number of steps and it provides the MLEs. We further consider the Bayes estimates and the associated credible intervals under the squared error loss function. Due to the absence of explicit form of the Bayes estimates, we propose to use the importance sampling technique to compute Bayes estimates. We provide an extensive simulation study in case of three stress levels mainly to see the performance of the proposed methods. Finally the analysis of one real data set has been provided for illustrative purposes.     

On accelerating sequential procedures for estimating exponential populations

In this study, we accelerate the purely sequential procedure due to Anscombe(1953), Chow and Robbins(1965) to reduce the number of sampling operations required to carry out the estimation process. The method is proposed while estimating the location parameter(s) of the exponential distribution(s). We also develop theory for the asymptotic characteristic of the associated stopping variables. Our findings are applicable to both point as well as confidence interval estimation problems. Other interesting results are also given.     

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.     

Interval Estimations for the Mean of the Generalized Poisson and Generalized Negative Binomial Distributions

In this article, we extend the Wald, score, skewness-corrected score, likelihood ratio, and mid-P intervals for the means of the generalized Poisson and generalized negative binomial distributions. These distributions are the members of the discrete version of the natural exponential family (NEF) with cubic variance function (CVF). Also, the coverage probabilities, the distal and mesial noncoverage probabilities, and the lengths of the proposed confidence intervals are estimated by means of a Monte Carlo simulation study. Finally, some practical examples are provided to show the applicability of the proposed intervals in applied studies.     

On optimal tests for general interval-hypotheses

Optimal statistical tests, using the normality assumptions for general interval hypotheses including equivalence testing and testing for nonzero difference (or for non-unit) are presented. These tests are based on the decision theory for Polya Type distributions and are compared with usual confidence tests and with ’two one-sided tests’- procedures. A formal relationship between some optimal tests and the Anderson and Hauck procedure as well as a procedure recommended by Patel and Gupta is given. A new procedure for a generalisation of Student's test as well as for equivalence testing for thet-statistics is shown.