M. Hafidz Omar

Stabilizing the Performance of Kurtosis Estimator of Multivariate Data

The estimation of the kurtosis parameter of the underlying distribution plays a central role in many statistical applications. The central theme of the article is to improve the estimation of the kurtosis parameter using a priori information. More specifically, we consider the problem of estimating kurtosis parameter of a multivariate population when some prior information regarding the the parameter is available. The rationale is that the sample estimator of the kurtosis parameter has a large estimation error. In this situation we consider shrinkage and pretest estimation methodologies and reappraise their statistical properties. The estimation based on these strategies yield relatively smaller estimation error in comparison with the sample estimator in the candidate subspace. A large sample theory of the suggested estimators are developed and compared. The results demonstrate that suggested estimators outperform the estimator based on the sample data only in the candidate subspace. In an effort to appreciate the relative behavior of the estimators in a finite sample scenario, a Monte-carlo simulation study is planned and performed. The result of simulation study strongly corroborates the asymptotic result. To illustrate the application of the estimators, some example are showcased based on recently published data.

On some characteristics of bivariate chi-square distribution

Product moments of bivariate chi-square distribution have been derived in closed forms. Finite expressions have been derived for product moments of integer orders. Marginal and conditional distributions, conditional moments, coefficient of skewness and kurtosis of conditional distribution have also been discussed. Shannon entropy of the distribution is also derived. We also discuss the Bayesian estimation of a parameter of the distribution. Results match with the independent case when the variables are uncorrelated.

New V control chart for the Maxwell distribution

ABSTRACT Control charts have been popularly used as a user-friendly yet technically sophisticated tool to monitor whether a process is in statistical control or not. These charts are basically constructed under the normality assumption. But in many practical situations in real life this normality assumption may be violated. One such non-normal situation is to monitor the process variability from a skewed parent distribution where we propose the use of a Maxwell control chart. We introduce a pivotal quantity for the scale parameter of the Maxwell distribution which follows a gamma distribution. Probability limits and L-sigma limits are studied along with performance measure based on average run length and power curve. To avoid the complexity of future calculations for practitioners, factors for constructing control chart for monitoring the Maxwell parameter are given for different sample sizes and for different false alarm rate. We also provide simulated data to illustrate the Maxwell control chart. Finally, a real life example has been given to show the importance of such a control chart.

Estimation of mixture Maxwell parameters and its possible industrial application

Mixture Maxwell distribution has been proposed along with control chart application.The EM algorithm has been used for parameter estimation.Proposed chart performs better than control chart for regular Maxwell distribution.To monitor a process with subpopulations, this distribution could be used. Problem statementThe conventional methods of monitoring a process sometimes provide misleading results when the population consists of two or more subpopulations. Mixture distribution may provide better performance in this circumstance. ObjectiveTo establish a control chart named Mixture Maxwell Cumulative Quantity (MMCQ) control chart for two components Maxwell mixture distribution. This chart may be implemented to monitor non-conforming items in this process. MethodFor estimating the parameters, the Expectation-Maximization (EM) algorithm has been used. To measure performance and for comparison, the average run length (ARL) has been used. ResultsWe compared this MMCQ control chart with MCQ control chart where MMCQ chart performs better as compared to MCQ chart. Performance of the chart was measured using run length and detection properties. ConclusionAs one of non-normal skewed distributions, Maxwell distribution is studied to model control charts. To monitor time between events for processes with Maxwell subpopulations, the behavior of the control chart based on two components mixture Maxwell distribution were examined.

On a Correlated Variance Ratio Distribution and Its Industrial Application

The distribution of correlated variance ratio arises if variables in the parent population are correlated. One such case arises if sample observations follow independent bivariate normal distributions. We study its cumulative distribution function, raw moments, mean centered moments, coefficient of skewness and kurtosis, median and reliability. The density function is also graphed. We address the issue of the invariance of the distribution of correlated variance ratio, and testing equality of variances under correlation. Finally we exhibit an application of the said distribution in quality control problems for monitoring process outputs using control charts.