Volume 29, Issue 1

In this article , acceptance sampling plans are developed for the Linear Failure Rate Distribution percentiles when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the specified life percentile is obtained under a given consumerś risk. The operating characteristic values of the sampling plans as well as the producerś risk are presented.

In this paper we propose a conditional distribution of the waiting time given the queue length. We assume that the number of customers in a queue, that is the queue length, follows a geometric distribution and the waiting time of a newly arrived customer in a queue follows an exponential distribution. We model the conditional distribution of the waiting time given the queue length using Gumbel's bivariate exponential copula. Parameters are estimated using the likelihood approach with a two stage estimation procedure. A simulation study indicates the performance of the estimators. Asymptotic properties of the estimators of the parameters are studied. We apply the model to a real data set to illustrate the dependence between the waiting time and the queue length.

Control charts for the process average play an important role in process control. In the presence of prior information, using Bayesian approach, one can construct posterior control chart for process average ( X ̅ posterior control chart). The control limits of the proposed chart are derived under the assumption that the process average has a conjugate prior distribution. Both cases – variance is known and variance is unknown – are discussed. When the variance is unknown, the control limits are constructed using Unbiased Estimator (UE) and Maximum Likelihood Estimator (MLE) of the variance. The power and Average Run Length (ARL) of the proposed chart are obtained. The newly constructed X ̅ posterior control chart is compared with a few other X ̅ charts described in the literature.

Reliability sampling plans are used to take decisions on the disposition of lots based on life testing of products. Such plans are developed taking into the consideration of relevant probability distributions of the lifetimes of the products under testing. When the quality of products varies over lots, then a predictive distribution of the lifetime should be used to design sampling plans. In this paper, designing of reliability single sampling plan based on the predictive distribution of the lifetime is considered. It is assumed that sampling inspection is carried out through life testing of products with hybrid censoring. The predictive distribution is obtained assuming that the probability distribution of the lifetime of the product is Rayleigh and the process parameter has an inverse-Rayleigh prior. Plan parameters are determined using hypergeometric, binomial and Poisson probabilities, providing protection to both producer as well as consumer.

Effective management and the assessment of quality performance of products is important in modern enterprises. Often, the business performance is measured using the lifetime performance index C L to evaluate the potential of a process, where L is a lower specification limit. In this paper the maximum likelihood estimator (MLE) of C L is derived based on progressive Type II sampling and assuming the Lomax distribution. The MLE of C L is then utilized to develop a new hypothesis testing procedure for given value of L . Moreover, we develop the Bayes estimator of C L assuming the conjugate prior distribution and applying the squared-error loss function. The Bayes estimator of C L is then utilized to develop a credible interval again for given L . Finally, we propose a Bayesian test to assess the lifetime performance of products and give two examples and a Monte Carlo simulation to assess and compare the two ML-approach with the Bayes-approach with respect to the lifetime performance index C L .

This paper considers the problem of an acceptance sampling plan for a truncated life test when the lifetime follows the three parameter Kappa distribution. For different acceptance numbers, confidence levels and values of the ratio of the fixed experiment time to the specified lower bound of the mean lifetime, the minimum sample sizes necessary to ensure a specified mean lifetime are obtained. The operating characteristic function values of the suggested sampling plans and the corresponding producer's risk are stated. Tables are provided so that the plans can be easily applied in practice.

The natural exponential families (NEF) play an important role in many fields of statistics. Two subclasses of NEF have interesting mathematical properties and have therefore been investigated by many researchers. These two subfamilies contain the distributions with quadratic variance function (NEF-QVF) and with cubic variance function (NEF-CVF), respectively. In estimation theory, the variance of an unbiased estimator is of eminent importance whenever it is necessary to evaluate its accuracy. In many cases, however, it is difficult to calculate the variance because of its complicated form. In such cases one can use a lower bound of the variance as approximation. The best known lower bounds for unbiased estimators are the Cramér–Rao bound (by Rao, 1945 and Cramér, 1946), Bhattacharyya bound (by Bhattacharyya, 1946, 1947), Hammersley–Chapman–Robbins bound (by Hammersley, 1950 and Chapman and Robbins, 1951) and Kshirsagar bound (by Kshirsagar, 2000). The Bhattacharyya and Kshirsagar bounds, respectively, are generalizations of the Cramér–Rao and Hammersley–Chapman–Robbins bounds. Several authors such as Shanbhag (1972, 1979), Pommeret (1996), Mohtashami Borzadaran (2001), Tanka and Akahira (2003) and Tanaka (2006) derived some useful properties and characterizations of the Bhattacharrya bounds for NEF, NEF-QVF and NEF-CVF. This paper presents explicitly the Bhattacharyya and Kshirsagar bounds for some examples (normal, gamma, exponential, negative binomial, Abel and Takacs distributions) from NEF-QVF and NEF-CVF. The aim is twofold, first we want to illustrate by the examples that the bounds may well be used as approximation of unknown variances and secondly we want to compare the two bounds to show that sometimes the calculation of the Bhattacharyya bound is sufficient since the Kshirsagar bound is only marginally larger but much more difficult to compute.

In this paper we introduce the so-called McDonald Exponentiated Pareto distribution. We provide a number of mathematical properties of this distribution and derive among others expressions for its moment-generating function, the r th moment, and the Renyi entropy. Finally, we present the likelihood equations to support the application of the McDonald Exponentiated Pareto distribution. The aim of the paper is to introduce an interesting distribution which is distinguished by high flexibility with respect to skewness and tail behavior.