A Simple Approximate Procedure for Constructing Binomial and Poisson Tolerance Intervals

The problems of constructing tolerance intervals for the binomial and Poisson distributions are considered. Closed-form approximate equal-tailed tolerance intervals (that control percentages in both tails) are proposed for both distributions. Exact coverage probabilities and expected widths are evaluated for the proposed equal-tailed tolerance intervals and the existing intervals. Furthermore, an adjustment to the nominal confidence level is suggested so that an equal-tailed tolerance interval can be used as a tolerance interval which includes a specified proportion of the population, but does not necessarily control percentages in both tails. Comparison of such coverage-adjusted tolerance intervals with respect to coverage probabilities and expected widths indicates that the closed-form approximate tolerance intervals are comparable with others, and less conservative, with minimum coverage probabilities close to the nominal level in most cases. The approximate tolerance intervals are simple and easy to compute using a calculator, and they can be recommended for practical applications. The methods are illustrated using two practical examples.     

An Extension of Economic Design of x-Bar Control Charts for Non Normally Distributed Data Under Weibull Shock Models

Three parameters—sample size, sampling intervals, and the control limits—must be determined when the x bar chart to monitor a manufacturing process. The constant sampling intervals were widely employed because of its administrative simplicity. However, the variable sampling interval (VSI) has recently been shown to give substantially faster detection of most process shifts than fixed-sampling-interval (FSI) for x-bar charts. In addition, these measurements in the subgroup are assumed to be normally distributed. That assumption may not be tenable. This investigation compares the economic design of x-bar control charts for non normal data under Weibull shock models with various sampling avenues.     

A Design of s Control Charts with a Combined Double Sampling and Variable Sampling Interval Scheme

In this paper, we propose new estimation techniques in connection with the system of S-distributions. Besides “exact” maximum likelihood (ML), we propose simulated ML and a characteristic function-based procedure. The “exact” and simulated likelihoods can be used to provide numerical, MCMC-based Bayesian inferences.     

Confidence Bounds for Multiplicative Comparisons

Statements that are inherently multiplicative have historically been justified using ratios of random variables. Although recent work on ratios has extended the classical theory to produce confidence bounds conditioned on a positive denominator, this current article offers a novel perspective that eliminates the need for such a condition. Although seemingly trivial, this new perspective leads to improved lower confidence bounds to support multiplicative statements. This perspective is also more satisfying as it allows comparisons that are inherently multiplicative in nature to be properly analyzed as such.     

On Sharp Jensen's Inequality and Some Unusual Applications

The purpose of this article is two-fold. First, we find it very interesting to explore a kind of notion of optimality of the customary Jensen-bound among all Jensen-type bounds. Without this result, the customary Jensen-bound stood alone simply as just another bound. The proposed notion and the associated optimality are important given that in some situations the Jensen's inequality does leave us empty handed.

When it comes to highlighting Jensen's inequality, unfortunately only a handful of nearly routine applications continues to recycle time after time. Such encounters rarely produce any excitement. This article may change that outlook given its second underlying purpose, which is to introduce a variety of unusual applications of Jensen's inequality. The collection of our important and useful applications and their derivations are new.     

Interval Estimation by Frequentist Model Averaging

An important contribution to the literature on frequentist model averaging (FMA) is the work of Hjort and Claeskens (2003 Hjort , N. L. , Claeskens , G. ( 2003 ). Frequestist model average estimators . J. Amer. Statist. Assoc. 98 : 879 – 899 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), who developed an asymptotic theory for frequentist model averaging in parametric models based on a local mis-specification framework. They also proposed a simple method for constructing confidence intervals of the unknown parameters. This article shows that the confidence intervals based on the FMA estimator suggested by Hjort and Claeskens (2003 Hjort , N. L. , Claeskens , G. ( 2003 ). Frequestist model average estimators . J. Amer. Statist. Assoc. 98 : 879 – 899 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are asymptotically equivalent to that obtained from the full model under both parametric and the varying-coefficient partially linear models. Thus, as long as interval estimation rather than point estimation is concerned, the confidence interval based on the full model already fulfills the objective and model averaging provides no additional useful information.     

Percentile-t Bootstrap Confidence Intervals for Period Using Periodogram

The magnitude of light intensity of many stars varies over time in a periodic way. Therefore, estimation of period and making inference about this parameter are of great interest in astronomy. The periodogram can be used to estimate period, properly. Bootstrap confidence intervals for period suggested here, are based on using the periodogram and constructed by percentile-t methods. We prove that the equal-tailed percentile-t bootstrap confidence intervals for period have an error of order n ^?1. We also show that the symmetric percentile-t bootstrap confidence intervals reduce the error to order n ^?2, and hence have a better performance. Finally, we assess the theoretical results by conducting a simulation study, compare the results with the coverages of percentile bootstrap confidence intervals for period and then analyze a real data set related to the eclipsing system R Canis Majoris collected by Shiraz Biruni Observatory.     

ML and Bayes estimation in a two-phase tandem queue with a second optional service and random feedback

ABSTRACT

In this article, we consider a two-phase tandem queueing model with a second optional service and random feedback. The first phase of service is essential for all customers and after the completion of the first phase of service, any customer receives the second phase of service with probability α, feedback to the tail of the first queue with probability β if the service is not successful and leaves the system with probability 1 ? α ? β. In this model, our main purpose is to estimate the parameters of the model, traffic intensity, and mean system size, in the steady state, via maximum likelihood and Bayesian methods. Furthermore, we find asymptotic confidence intervals for mean system size. Finally, by a simulation study, we compute the confidence levels and mean length for asymptotic confidence intervals of mean system size with a nominal level 0.95.     

Empirical bayes interval estimates involving uniform distributions

Bayesian and empirical Bayesian decision rules are exhibited for the interval estimation of the parameter 0 of a Uniform (0,θ) distribution. The estimate ?,δ>resulting in the interval [?,?+δ]suffers loss given by L(?,δ>,θ)=1-[?≦e≦?+δ]+c₁((?-θ)²+(?+δ?θ)²))+c₂δ. The solution is presented for prior distributions G which have bounded support, no point masses,∫θ^?mdG(θ)<∞ and for some integer m. An example is presented involving a particular parametric form for G and rates of risk convergence in the empirical Bayes problem for this example are calculated.     

Isotontic regression for interval censored survival data using an e-m algorithm

This paper develops a nonparametric model of the relationship between survival S and a dichotomous random variable X under the order constraint that P(X=1|S=s) is increasing (or decreasing) with s. The estimation procedure, called isotonic regression, has been studied in some depth for the case of uncensored data, but we give a methodology which is appropriate in the more general context of right, left, and interval censored data. An E-M Algorithm (Dempster et. al., 1977) is used for maximum likelihood estimation.