Shovan Chowdhury

A New Distribution-free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions

While the assumption of normality is required for the validity of most of the available control charts for joint monitoring of unknown location and scale parameters, we propose and study a distribution-free Shewhart-type chart based on the Cucconi statistic, called the Shewhart-Cucconi (SC) chart. We also propose a follow-up diagnostic procedure useful to determine the type of shift the process may have undergone when the chart signals an out-of-control process. Control limits for the SC chart are tabulated for some typical nominal in-control (IC) average run length (ARL) values; a large sample approximation to the control limit is provided which can be useful in practice. Performance of the SC chart is examined in a simulation study on the basis of the ARL, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparisons with a competing distribution-free chart, known as the Shewhart-Lepage chart (see Mukherjee and Chakraborti) show that the SC chart performs just as well or better. The effect of estimation of parameters on the IC performance of the SC chart is studied by examining the influence of the size of the reference (Phase-I) sample. A numerical example is given for illustration. Summary and conclusions are offered.

Distribution‐free Phase II CUSUM Control Chart for Joint Monitoring of Location and Scale

A single distribution-free (nonparametric) Shewhart-type chart on the basis of the Lepage statistic is well known in literature for simultaneously monitoring both the location and the scale parameters of a continuous distribution when both of these parameters are unknown. In the present work, we consider a single distribution-free cumulative sum chart, on the basis of the Lepage statistic, referred to as the cumulative sum-Lepage (CL) chart. The proposed chart is distribution-free (nonparametric), and therefore, the in-control properties of the chart remain invariant and known for all continuous distributions. Control limits are tabulated for implementation of the proposed chart in practice. The in-control and out-of-control performance properties of the cumulative sum-Lepage (CL) chart are investigated through simulation studies in terms of the average, the standard deviation, the median, and some percentiles of the run length distribution. Detailed comparison with a competing Shewhart-type chart is presented. Several existing cumulative sum charts are also considered in the performance comparison. The proposed CL chart is found to perform very well in the location-scale models. We also examine the effect of the choice of the reference value (k) on the performance of the CL chart. The proposed chart is illustrated with a real data set. Summary and conclusions are presented. Copyright

Some results on majorization and their applications

Majorization is a key concept in studying the Schur-convex property of a function, which is very useful in the study of stochastic orders. In this paper, some results on Schur-convexity have been developed. We have studied the conditions under which a function ? defined by ? ( x ) = ? i = 1 n u i g ( x i ) will be Schur-convex. This fills some gap in the theory of majorization. The results so developed have been used in the case of generalized exponential and gamma distributions. During this, we have also developed some stochastic properties of order statistics. Some useful results on majorization are developed.This enriches the theory of majorization.As applications, some distributions have been studied.

Estimation of Traffic Intensity Based on Queue Length in a Single M/M/1 Queue

In this article, maximum likelihood estimator (MLE) as well as Bayes estimator of traffic intensity (ρ) in an M/M/1/∞ queueing model in equilibrium based on number of customers present in the queue at successive departure epochs have been worked out. Estimates of some functions of ρ which provide measures of effectiveness of the queue have also been derived. A comprehensive simulation study starting with the transition probability matrix has been carried out in the last section.

Stochastic comparison of parallel systems with log-Lindley distributed components

In this paper, we study stochastic comparisons of parallel systems having log-Lindley distributed components. These comparisons are carried out with respect to reversed hazard rate and likelihood ratio ordering.

Bayes estimation in M/M/1 queues with bivariate prior

Abstract Bayes estimator of different queueing performance measures are derived in steady state by recording system size from each of n iid M/M/1 queues. The Bayes estimators are obtained under both squared error loss function and precautionary loss function with a bivariate distribution beta stacy as prior, with natural restriction 0 < λ < µ where λ and µ are arrival rate and service rate respectively. A comprehensive simulation results are also shown at the last section.

Bayes Estimation of Measures of Effectiveness in an M/M/1 Queueing Model

Abtsrcat In this paper, Bayes estimators of trafic intensity (ρ) and other measures of effectiveness are worked out under squared error loss function (SELF) and entropy loss function (ELF) with Beta distribution as prior. Number of customers present in each of n iid M/M/1 queueing systems are considered as data. A comprehensive simulation study has been carried out in the last section.

Ordering properties of sample minimum from Kumaraswamy-G random variables

ABSTRACT In this paper we compare the minimums of two independent and heterogeneous samples each following Kumaraswamy (Kw)-G distribution with the same and the different parent distribution functions. The comparisons are carried out with respect to usual stochastic ordering and hazard rate ordering with majorized shape parameters of the distributions. The likelihood ratio ordering between the minimum order statistics is established for heterogeneous multiple-outlier Kw-G random variables with the same parent distribution function.

On compounded geometric distributions and their applications

ABSTRACT Here, we introduce two-parameter compounded geometric distributions with monotone failure rates. These distributions are derived by compounding geometric distribution and zero-truncated Poisson distribution. Some statistical and reliability properties of the distributions are investigated. Parameters of the proposed distributions are estimated by the maximum likelihood method as well as through the minimum distance method of estimation. Performance of the estimates by both the methods of estimation is compared based on Monte Carlo simulations. An illustration with Air Crash casualties demonstrates that the distributions can be considered as a suitable model under several real situations.

Compounded inverse Weibull distributions: Properties, inference and applications

ABSTRACT In this paper two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions. The distributions can be used to model lifetime of series system where the lifetimes follow inverse Weibull distribution and the subgroup size being random follows either geometric or zero-truncated Poisson distribution. Some of the important statistical and reliability properties of each of the distributions are derived. The distributions are found to exhibit both monotone and non-monotone failure rates. The parameters of the distributions are estimated using the expectation-maximization algorithm and the method of minimum distance estimation. The potentials of the distributions are explored through three real life data sets and are compared with similar compounded distributions, viz. Weibull-geometric, Weibull-Poisson, exponential-geometric and exponential-Poisson distributions.