Aamir Saghir

The bivariate dispersion control charts for non-normal processes

This article study the performance of the bivariate dispersion control charts, namely the and the charts, for the bivariate normal, bivariate exponential, bivariate t(5), bivariate Logistic and bivariate Laplace distributions. The asymmetrical probability limits of these charts are proposed for the distributions under study. The estimated factors and quantile points used in the construction of these two charts are provided for the bivariate non-normal distributions. The effect of improper use of constants and quantile points in the construction of the and the charts are studied in terms of the associated false alarm rates. The performances of the proposed limits of the given charts are evaluated in terms of average run length for the various bivariate distributions and compared. The performances of these control charts are also compared with their respective 3σ limits.

Phase-I Design Scheme for -chart Based on Posterior Distribution

This article develops a Phase I design structure of -Chart, namely Bayesian -Chart, based on Bayesian (posterior distribution) framework assuming the normality of the quality characteristic to incorporate parameter uncertainty. Our approach consists of two stages: (i) construction of the control limits for -Chart based on posterior distribution of unknown mean μ and (ii) evaluation of the performance of the proposed design structure. The proposed design structure of -Chart is compared with the frequents design structure of - Chart in terms of (i) width of the control region and (ii) power of detecting a shift in the location parameter of the process. It has been observed that the proposed design structure of -Chart is performs better than the usual design structure to detecting shifts in the parameter of the process when the prior mean is close to the unknown target value.

The Phase I Dispersion Charts for Bivariate Process Monitoring

Multivariate control charts are usually implemented in statistical process control to monitor several correlated quality characteristics. Process dispersion charts are used to determine the stability of process variation (which is typically done before monitoring the process location/mean). A Phase-I study is generally used when population parameters are unknown. This article develops Phase-I |S| and |G| control charts, to monitor the dispersion of a bivariate normal process. The charting constants are determined to achieve the required nominal false alarm probability (FAP0). The performance of the proposed charts is evaluated in terms of (i) the attained false rate and (ii) the probability of signaling for out-of-control situations. The analysis shows that the proposed Phase-I bivariate charts correctly control the FAP (the false alarm probability) and detect a shift occurring in the bivariate dispersion matrix with adequate probability. An example is given to explain the practical implementation of these charts. Copyright

On the Performance of Phase-I Bivariate Dispersion Charts to Non-Normality

A phase-I study is generally used when population parameters are unknown. The performance of any phase-II chart depends on the preciseness of the control limits obtained from the phase-I analysis. The performance of phase-I bivariate dispersion charts has mainly been investigated for bivariate normal distribution. However, this assumption is seldom fulfilled in reality. The current work develops and studies the performance of phase-I |S| and |G| charts for monitoring the process dispersion of bivariate non-normal distributions. The necessary control charting constants are determined for the bivariate non-normal distributions at nominal false alarm probability (FAP0). The performance of these charts is evaluated and compared in a situation when samples are generated by bivariate logistic, bivariate Laplace, bivariate exponential, or bivariate t5 distribution. The analysis shows that the proper consideration to underlying bivariate distribution in the construction of phase-I bivariate dispersion charts is very important to give a real picture of in or out of control process status. Copyright

A control chart for COM-Poisson distribution using a modified EWMA statistic

ABSTRACT In this paper, a control chart has been developed for the Conway–Maxwell Poisson (COM-Poisson) distribution using the modified exponentially weighted moving average statistic. The proposed chart provides an efficient detection of smaller changes in the location parameter of the COM-Poisson distribution. The performance of the proposed control chart has been evaluated by the average and the standard deviation of the run length distribution for various parameters. Better detecting ability has also been compared with the existing control chart using EWMA statistic. Using simulation, we also showed the detecting ability over the traditional EWMA chart.

The length-biased weighted exponentiated inverted Weibull distribution

Abstract Length-Biased distributions are a special case of the more general form known as weighted distributions. We can exploit the conceptuality of Length-Biased distribution in the development of appropriate models for lifetime data. Its method is adjusting the original probability density function from real data and the expectation of those data. This modification can lead to correct conclusions of the models. Therefore, we introduced the Length-Biased version of the weighted Exponentiated inverted Weibull distribution in this paper. Various properties and the expressions for moments, coefficient of skewness, coefficient of kurtosis, moment generating function, hazard rate function, etc. are derived. The maximum likelihood estimates of the parameters of the proposed distribution are determined. The study results suggest that this distribution is an efficacious model in life time data analysis and other related fields.

A EWMA control chart based on an auxiliary variable and repetitive sampling for monitoring process location

ABSTRACT This article develops an exponentially weighted moving average (EWMA) control chart using an auxiliary variable and repetitive sampling for efficient detection of small to moderate shifts in location. A EWMA statistic of a product estimator of the average (which utilities the information of auxiliary variables as well as repetitive sampling) is plotted on the proposed chart. The control chart coefficients of the proposed EWMA chart are determined for two strategic limits known as outer and inner control limits for the target in-control average run length. The performance of the proposed EWMA chart is studied using average run length when a shift occurs in the process average. The efficiency of the developed chart is compared with the competitive existing control charts. The results of the study revealed that proposed EWMA chart is more efficient than others to detect small changes in process mean.

The Maxwell length-biased distribution: Properties and estimation

The concept of length-biased distribution can be employed in the development of proper models for the lifetime data. Length-biased distribution is a special case of the more general form known as the weighted distribution. This article introduces a new class of Maxwell length-biased distribution. The statistical properties of the proposed distribution are determined, including shape, kth moment, reliability function, hazard function, reverse hazard function, and so on. The maximum likelihood estimate (MLE), method of moments (MM) estimate, and Bayesian estimate of the unknown parameter are derived. An illustrative example demonstrates the application of the proposed distribution in real life.

Phase-I design structure of Bayesian variance chart

Abstract This article develops a new design structure for S2-Chart, namely Bayesian variance chart, in Phase-I analysis assuming the normality of the quality characteristic to incorporate the parameter uncertainty. Our approach consists of two stages: (i) construction of the control limits for S2-Chart and (ii) performance evaluation of the proposed control limits. The comparison of the proposed design structure with the frequentist design structure of S2-Chart is examined in terms of (i) width of control region and (ii) OC curves when the process variance goes out of control. It is observed that the proposed Phase-I S2-Chart is more efficient than the frequentist S2-Chart in discriminatory power of detecting a shift in the process dispersion. When the process variance is in-control (after implementation of Bayesian variance chart), then the control limits for -Chart using in-control standard deviation are also given here for monitoring unknown mean under unknown standard deviation case.

Designing of Gini-chart for Exponential, t, Logistic and Laplace Distributions

This article extended the work of Saghir and Lin (2012) to develop the probability limits of Gini chart, a process dispersion chart based on ginis mean difference proposed by Riaz and Saghirr (2007), for the exponential, t(5), Logistic and Laplace distributions. The asymmetrical control limits of the Gini chart are proposed for the distributions under study. The estimated factors and quantile points used in the construction of the Gini chart are provided for the exponential, t(5), Logistic and Laplace distributions. The effect of improper use of constants and quantile points in the construction of Gini chart is studied in terms of the associated false alarm rates. The performance of the asymmetrical control limits of the given chart is evaluated in terms of average run length (ARL) for the exponential, t(5), Logistic and Laplace distributions and compared with the 3σ limits proposed by Saghir and Lin (2012). The ARL performance of the proposed probability limits is compared with the existing limits of R and S charts for the under study parent distributions. Finally, the design scheme of chart associated with Gini chart is developed.