Sudhansu S. Maiti

This work provides Double Sampling (DS) inspection plans considering the average lifetime as a quality characteristic which follows one parameter polynomial exponential (OPPE) family of distributions. Exponential, Lindley, Akash, Aradhana, Sujatha, length-biased Lindley, etc., are a few particular cases of the OPPE family. The quality of a lot is computed in this technique, by the lot average (μ) for first sample and, for the second sample we have taken combined mean to measure the lot quality. Also, we have estimated the optimum value of parameters of the proposed plan by non-linear optimization approaches considering acceptable quality level and rejectable quality level. A comparison part of the study is given, with respect to the sample size, between the Double Sampling (DS) plan and the Single Sampling (SS) plan for the variable. To describe the proposed work, we have also taken one example.

In this study, we propose a Bayesian Extended Exponentially Weighted Moving Average (BEEWMA) control chart. We have considered that the control chart statistic follows a normal distribution and the location parameter has an informative prior (normal). Average run length (ARL) has been used to evaluate the performance of the BEEWMA control chart. A simulation study has been performed to evaluate the new BEEWMA control chart’s performance for different smoothing constants values. The performance of the proposed control chart is also measured using a real-life application.

Estimation of generalized process capability index (GPCI), [ ] defined as the ratio of the proportion of specification conformance (or process yield) to the proportion of desired (or natural) conformance, is considered under study. The index covers normal as well as nonnormal and continuous as well as discrete quality characteristics. It can be assessed under either unilateral or bilateral specifications. Its expressions have been derived for one parameter polynomial exponential (OPPE), and natural discrete one parameter polynomial exponential (NDOPPE) distributed quality characteristic. Two estimators, maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU), have been derived. Theoretical comparison of these estimators is made with respect to mean squared error (MSE) through a simulation study since the closed-form expression of the maximum likelihood estimator (MLE) of the distributions’ parameter is not available. Three data sets are analyzed for illustration purposes; the estimates are compared in respect of estimated variances as MSEs cannot be computed.

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In this article, we have considered the estimation of the probability density function and cumulative distribution function of the one-parameter polynomial exponential family of distributions. A number of probability distributions like the exponential, Lindley, length-biased Lindley and Sujatha are particular cases. Two estimators—maximum likelihood and uniformly minimum variance unbiased estimators of the probability density function and cumulative distribution function of the family have been discussed. The estimation issues of the length-biased Lindley and Sujatha distribution have been considered in detail. The estimators have been compared in mean squared error sense. Monte Carlo simulations and real data analysis are performed to compare the performances of the proposed estimators.

In this paper, a new natural discrete version of the one parameter polynomial exponential family of distributions have been proposed and studied. The distribution is named as Natural Discrete One Parameter Polynomial Exponential (NDOPPE) distribution. Structural and reliability properties have been studied. Estimation procedure of the parameter of the distribution have been mentioned. Compound NDOPPE distribution in the context of collective risk model have been obtained in closed form. The new compound distribution has been compared with the classical compound Poisson, compound Negative binomial, compound discrete Lindley, compound xgamma-I and compound xgamma-II distributions regarding suitability of modelling extreme data with the help of some automobile claim.

In this article, we have considered one parameter polynomial exponential (OPPE) distribution. The exponential, Lindley, length-biased Lindley and Sujatha distribution are particular cases. Two estimators viz, MLE and UMVUE of the PDF and the CDF of the OPPE distribution have been discussed. The estimation issue of the length-biased Lindley and Sujatha distribution have been considered in detail. The estimators have been compared in MSE sense. Monte Carlo simulations and real data analysis are performed to compare the performances of the proposed methods of estimation.