On the Bivariate Kummer-Beta Type IV Distribution

In this article, the non central bivariate Kummer-beta Type IV distribution is introduced and derived via the Laplace transform of the non central bivariate beta distribution by Gupta et al. (2011 ). We focus on and discuss the central bivariate Kummer-beta Type IV distribution; this distribution is a special case of the non central bivariate Kummer-beta Type IV distribution and extends the popular Jones’ bivariate beta distribution. The probability density functions of the product and the ratio of the components of the central bivariate Kummer-beta Type IV distribution are also derived and we provide tabulations of the associated lower percentage points as well as some upper percentage points that are useful in reliability

Shewhart-type control charts for variation in phase I data analysis

Control charts for variation play a key role in the overall statistical process control (SPC) regime. We study the popular Shewhart-type S2, S and R control charts when the mean and the variance of a normally distributed process are both unknown and are estimated from m independent samples (subgroups) each of size n. This is the Phase I setting. Current uses of these charts do not recognize that in this setting the signalling events are statistically dependent and that m comparisons are made with the same control limits simultaneously. These are important issues because they affect the design and the performance of the control charts. The proposed methodology addresses these issues (which leads to working with the joint distribution of a set of dependent random variables) by calculating the correct control limits, so that the false alarm probability (FAP), defined as the probability of at least one false alarm, is at most equal to some given nominal value FAP0. To aid practical implementation, tables are provided for the charting constants for each Phase I chart, for an FAP0 of 0.01 and 0.05, respectively. An illustrative example is given.

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Nonparametric control charts can provide a robust alternative in practice to the data analyst when there is a lack of knowledge about the underlying distribution. A nonparametric exponentially weighted moving average (NPEWMA) control chart combines the advantages of a nonparametric control chart with the better shift detection properties of a traditional EWMA chart. A NPEWMA chart for the median of a symmetric continuous distribution was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see Gibbons and Chakraborti, 2003). This is called the nonparametric exponentially weighted moving average Signed-Rank (NPEWMA-SR) chart. However, important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this paper we address these issues with a more in-depth study of the two-sided NPEWMA-SR chart. A Markov chain approach is used to compute the run-length distribution and the associated performance characteristics. Detailed guidelines and recommendations for selecting the chart's design parameters for practical implementation are provided along with illustrative examples. An extensive simulation study is done on the performance of the chart including a detailed comparison with a number of existing control charts, including the traditional EWMA chart for subgroup averages and some nonparametric charts i.e. runs-rules enhanced Shewhart-type SR charts and the NPEWMA chart based on signs. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. A summary and some concluding remarks are given.Contaminated normal Distribution-free Markov chain Median Outlier Quality control Robust Run-length Search Algorithm Simulation