On minimum distance estimation of location based on the kolmogorov statistic

The asymptotic distribution is derived for the minimum distance estimator of a location parameter based on the Kolmogorov goodness of fit statistic. The distribution is expressed in terms of the distribution of a functional of a Brownian bridge. An upper bound is obtained for the length of the confidence interval based on the Kolmogorov statistic. A simulation study with sample sizes 10 and 20 compares the length of the interval based on the Kolmogorov statistic to the length of the interval based on the maximum likelihood estimator. Another simulation shows the effect of model misspecification on the coverage probabilities of the interval based on the Kolmogorov statistic.     

Interval estimation for the generalized inverted exponential distribution under progressive first failure censoring

In this paper, we consider the inferential procedures for the generalized inverted exponential distribution under progressive first failure censoring. The exact confidence interval for the scale parameter is derived. The generalized confidence intervals (GCIs) for the shape parameter and some commonly used reliability metrics such as the quantile and the reliability function are explored. Then the proposed procedure is extended to the prediction interval for the future measurement. The GCIs for the reliability of the stress-strength model are discussed under both equal scale and unequal scale scenarios. Extensive simulations are used to demonstrate the performance of the proposed GCIs and prediction interval. Finally, an example is used to illustrate the proposed methods.     

Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data

In this paper, inference for the scale parameter of lifetime distribution of a k-unit parallel system is provided. Lifetime distribution of each unit of the system is assumed to be a member of a scale family of distributions. Maximum likelihood estimator (MLE) and confidence intervals for the scale parameter based on progressively Type-II censored sample are obtained. A β-expectation tolerance interval for the lifetime of the system is obtained. As a member of the scale family, half-logistic distribution is considered and the performance of the MLE, confidence intervals and tolerance intervals are studied using simulation.     

A higher order approximation to a percentage point of the non–central t-distribution

A higher order approximation formula for a percentage point of the noncentral t–distribution with v degrees of freedom is given up to the order o(v^-3), using the Cornish-Fisher expansion for the statistic based on a lin-ear combination of a normal random variable and a chi-random variable. The upper confidence limit and the confidence interval for the non–centrality parameter are given. Numerical results are also obtained.     

Reliability inference for a general lower-truncated family of distributions under records

Based on record values, point and interval estimators are proposed in this paper for the parameters of a general lower-truncated family of distributions. Maximum likelihood and bias-corrected estimators are obtained for unknown model parameters. Based on a sufficient and complete statistic, the bias-corrected estimator is also shown to be uniformly minimum variance unbiased estimator. Different exact confidence intervals and exact confidence regions are constructed for the both model and truncated parameters, and other confidence interval estimates based on asymptotic distribution theory and bootstrap approaches are obtained as well. Finally, two real-life examples and a numerical study are presented to illustrate the performance of our methods.     

A TEST FOR THE NONPARAMETRIC TWO SAMPLE SCALE PROBLEM

We propose a distribution-free test for the nonparametric two sample scale problem. Unlike the other tests for this problem, we do not assume that the two distribution functions have a common median. We assume that they have a common quantile of order a (not necessarily 1/2). The test statistic is a modification of the Sukhatme statistic for the scale problem and the Wilcoxon-Mann-Whitney statistic for stochastic dominance. It is shown that the new test is uniformly more efficient (in the Pitman sense) than the Sukhatme test and has very good efficiency when compared to the Mood test.     

Fixed-width sequential confidence interval for the correlation coefficient of a bivariate normal distribution

A sequential confidence interval of fixed width 2d d > 0, is constructed for the correlation coefficient of a bivariate normal distribution. It is shown that the coverage probability is approximately equal to a preassigned number γ, 0 < γ < as d → 0.     

Confidence intervals for the mean of a normal distribution with restricted parameter space

For a normal distribution with known variance, the standard confidence interval of the location parameter is derived from the classical Neyman procedure. When the parameter space is known to be restricted, the standard confidence interval is arguably unsatisfactory. Recent articles have addressed this problem and proposed confidence intervals for the mean of a normal distribution where the parameter space is not less than zero. In this article, we propose a new confidence interval, rp interval, and derive the Bayesian credible interval and likelihood ratio interval for general restricted parameter space. We compare these intervals with the standard interval and the minimax interval. Simulation studies are undertaken to assess the performances of these confidence intervals.     

Confidence Intervals for a Binomial Parameter after Observing No Successes

The easily computed, one-sided confidence interval for the binomial parameter provides the basis for an interesting classroom example of scientific thinking and its relationship to confidence intervals. The upper limit can be represented as the sample proportion from a number of “successes” in a future experiment of the same sample size. The upper limit reported by most people corresponds closely to that producing a 95 percent classical confidence interval and has a Bayesian interpretation.     

Shrinkage confidence intervals for the normal mean: Using a guess for greater efficiency

If the unknown mean of a univariate population is sufficiently close to the value of an initial guess then an appropriate shrinkage estimator has smaller average squared error than the sample mean. This principle has been known for some time, but it does not appear to have found extension to problems of interval estimation. The author presents valid two‐sided 95% and 99% “shrinkage” confidence intervals for the mean of a normal distribution. These intervals are narrower than the usual interval based on the Student distribution when the population mean lies in such an “effective interval.” A reduction of 20% in the mean width of the interval is possible when the population mean is sufficiently close to the value of the guess. The author also describes a modification to existing shrinkage point estimators of the general univariate mean that enables the effective interval to be enlarged.     

Exact Non-parametric Confidence Intervals for Quantiles with Progressive Type-II Censoring

It is shown how various exact non-parametric inferences based on order statistics in one or two random samples can be generalized to situations with progressive type-II censoring, which is a kind of evolutionary right censoring. Ordinary type-II right censoring is a special case of such progressive censoring. These inferences include confidence intervals for a given parent quantile, prediction intervals for a given order statistic of a future sample, and related two-sample inferences based on exceedance probabilities. The proposed inferences are valid for any parent distribution with continuous distribution function. The key result is that each observable uncensored order statistic that becomes available with progressive type-II censoring can be represented as a mixture with known weights of underlying ordinary order statistics. The importance of this mixture representation lies in that various properties of such observable order statistics can be deduced immediately from well-known properties of ordinary order statistics.     

Estimating Process Capability Index C PM Using a Bootstrap Sequential Sampling Procedure

Construction of a confidence interval for process capability index C _PM is often based on a normal approximation with fixed sample size. In this article, we describe a different approach in constructing a fixed-width confidence interval for process capability index C _PM with a preassigned accuracy by using a combination of bootstrap and sequential sampling schemes. The optimal sample size required to achieve a preassigned confidence level is obtained using both two-stage and modified two-stage sequential procedures. The procedure developed is also validated using an extensive simulation study.     

Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application

In this paper, the estimation of parameters for a generalized inverted exponential distribution based on the progressively first-failure type-II right-censored sample is studied. An expectation–maximization (EM) algorithm is developed to obtain maximum likelihood estimates of unknown parameters as well as reliability and hazard functions. Using the missing value principle, the Fisher information matrix has been obtained for constructing asymptotic confidence intervals. An exact interval and an exact confidence region for the parameters are also constructed. Bayesian procedures based on Markov Chain Monte Carlo methods have been developed to approximate the posterior distribution of the parameters of interest and in addition to deduce the corresponding credible intervals. The performances of the maximum likelihood and Bayes estimators are compared in terms of their mean-squared errors through the simulation study. Furthermore, Bayes two-sample point and interval predictors are obtained when the future sample is ordinary order statistics. The squared error, linear-exponential and general entropy loss functions have been considered for obtaining the Bayes estimators and predictors. To illustrate the discussed procedures, a set of real data is analyzed.     

Comparison of two independent life tests subject to type II censoring

The problem of comparing two life distributions is considered. It Is assumed that each life test Is stopped after a predetermined number of failures without regard to the number of failures which have occurred in the other life test. A generalized Wilcoxon-rlann-Whitney test is proposed. Small and large sample distributional results are obtained for the test statistic. Under the assumption that the life distributions differ by at most a scale parameter an exact confidence interval for that parameter is obtained. An algorithm for computing the interval is given.     

An integrated dominance analysis and dynamic programing approach for measuring predictor importance for customer satisfaction

Abstract

Dominance analysis is a procedure for measuring the importance of predictors in multiple regression analysis. We show that dominance analysis can be enhanced using a dynamic programing approach for the rank-ordering of predictors. Using customer satisfaction data from a call center operation, we demonstrate how the integration of dominance analysis with dynamic programing can provide a better understanding of predictor importance. As a cautionary note, we recommend careful reflection on the relationship between predictor importance and variable subset selection. We observed that slight changes in the selected predictor subset can have an impact on the importance rankings produced by a dominance analysis.     

Interval shrinkage estimators

This paper considers estimation of an unknown distribution parameter in situations where we believe that the parameter belongs to a finite interval. We propose for such situations an interval shrinkage approach which combines in a coherent way an unbiased conventional estimator and non-sample information about the range of plausible parameter values. The approach is based on an infeasible interval shrinkage estimator which uniformly dominates the underlying conventional estimator with respect to the mean square error criterion. This infeasible estimator allows us to obtain useful feasible counterparts. The properties of these feasible interval shrinkage estimators are illustrated both in a simulation study and in empirical examples.     

Conditional interval estimation of the exponential location parameter following rejection of a pre-test

The conditional confidence interval for the location parameter of an exponential distribution following a preliminary test is investigated. The conditional confidence interval (CCI) may be shorter than the unconditional confidence interval (UCI) in contrast to the findings for the mean of a normal distribution by Meeks and D'Agostino (1983). The conditional coverage probability of the UCI is obtained by computing the coverage probability under the conditional probability density function. It is shown that the conditional coverage probability of the UCI is not uniformly greater than or less than the nominal level.     

Some Small-Sample Properties of the Student t Confidence Intervals

For the two-sided Student t confidence interval for the mean of a normal distribution there is, for any sample size, a sufficiently large confidence level that ensures that the interval covers all the observations; there are also sufficiently small confidence levels guaranteeing, respectively, that (a) the interval does not cover all the observations and (b) the interval lies within the extreme observations. Necessary and sufficient conditions are also obtained for the width of the confidence interval to always exceed the sample range, as well as for the reverse inequality. Some implications of the results are discussed.     

Jackknife method for intermediate quantiles

Quantile function plays an important role in statistical inference, and intermediate quantile is useful in risk management. It is known that Jackknife method fails for estimating the variance of a sample quantile. By assuming that the underlying distribution satisfies some extreme value conditions, we show that Jackknife variance estimator is inconsistent for an intermediate order statistic. Further we derive the asymptotic limit of the Jackknife-Studentized intermediate order statistic so that a confidence interval for an intermediate quantile can be obtained. A simulation study is conducted to compare this new confidence interval with other existing ones in terms of coverage accuracy.     

Empirical likelihood for generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.