Sudhansu S. Maiti

On Generalizing Process Capability Indices

Abstract A generalized process capability index, defined as the ratio of proportion of specification conformance (or, process yield) to proportion of desired (or, natural) conformance, has been developed. Almost all the process capabilities defined in the literature are directly or indirectly associated with this generalized index. Normal as well as non-normal and continuous as well as discrete random variables could be covered by this new index. It can also be assessed under either unilateral or bilateral specifications. We deal with the proposed index in case of normal, exponential and Poisson processes. Under each distributional assumption, point estimators for the proposed index are suggested and compared through simulation study. Two real-world applications have been discussed using the proposed index.

Tilted Normal Distribution and Its Survival Properties

To analyze skewed data, skew normal distribution is proposed by Azzalini (1985). For practical problems of estimating the skewness parame- ter of this distribution, Gupta and Gupta (2008) suggested power normal dis- tribution as an alternative. We search for another alternative, named tilted normal distribution following the approach of Marshall and Olkin (1997) to add a positive parameter to a general survival function and taking survival function is of normal form. We have found out dierent properties of this distribution. Maximum likelihood estimate of parameters of this distribu- tion have been found out. Comparison of tilted normal distribution with skew normal and power normal distribution have been made.

Bayesian Estimation of the Parameter of Maxwell Distribution under Different Loss Functions

In this paper, Bayes estimators of parameter of Maxwell distribution have been derived by considering non-informative as well as conjugate priors under different scale invariant loss functions, namely, Quadratic Loss Function, Squared-Log Error Loss Function and Modified Linear Exponential Loss Function. The risk functions of these estimators have been studied.

Bayesian Estimation of Generalized Process Capability Indices

Process capability indices (PCIs) aim to quantify the capability of a process of quality characteristic (X) to meet some specifications that are related to a measurable characteristic of its produced items. One such quality characteristic is life time of items. The specifications are determined through the lower specification limit (L), the upper specification limit (U), and the target value (T). Maiti et al. (2010) have proposed a generalized process capability index that is the ratio of proportion of specification conformance to proportion of desired conformance. Bayesian estimation of the index has been considered under squared error loss function. Normal, exponential (nonnormal), and Poisson (discrete) processes have been taken into account. Bayes estimates of the index have been compared with the frequentist counterparts. Data sets have been analyzed.

A Loglikelihood-Based Shape Measure of Past Lifetime Distribution

Generalized past information measure based on Rényi entropy for a continuous random variable useful in life testing is considered and its connection to the loglikelihood is established. From this relation, a loglikelihood-based past life distribution measure S ¯ f ( t ) has been proposed. From the gradient of the spectrum of generalized past information, the S ¯ f ( t ) measure has been calculated for some important parametric life distributions. Serving as a measure of the shape of a past life (inactivity time) distribution, S ¯ f ( t ) can be used to compare the tails and shapes of various frequently used densities, where the traditional kurtosis measure is not applicable.

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

ABSTRACT One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.

Parametric and non-parametric bootstrap confidence intervals of CNpk for exponential power distribution

Abstract For enhancing the quality and productivity, the use of process capability index (PCI) has become significant in statistical process control, designed to quantify the relation between the actual performance of the process and its specified requirements. Confidence interval is an important part of PCI because it is an estimated value and provides much more information about the population characteristic of interest than does a point estimate. In this article, bootstrap confidence intervals of non-normal PCI , is studied through simulation when the underlying distribution is exponential power distribution. Maximum likelihood method is used to estimate the parameters of the model. Four (parametric as well as non-parametric) bootstrap confidence intervals, namely standard bootstrap (s-boot), percentile bootstrap (p-boot), Student’s t bootstrap (t-boot), and bias-corrected percentile bootstrap (-boot), are considered for obtaining confidence intervals of . A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Simulation results showed that among these (s-boot, p-boot, t-boot, and -boot) confidence intervals, the performances of the s-boot confidence intervals is the best in terms of overage probabilities. Finally, three real data-sets are analyzed for illustrative purposes.

On estimation of the PDF and CDF of the Lindley distribution

ABSTRACT This article addresses two methods of estimation of the probability density function (PDF) and cumulative distribution function (CDF) for the Lindley distribution. Following estimation methods are considered: uniformly minimum variance unbiased estimator (UMVUE) and maximum likelihood estimator (MLE). Since the Lindley distribution is more flexible than the exponential distribution, the same estimators have been found out for the exponential distribution and compared. Monte Carlo simulations and a real data analysis are performed to compare the performances of the proposed methods of estimation.

Reliability from Damaged Stress and Strength Data

In case of stress-strength reliability R = P(X > Y), inference is made under various assumptions regarding the variables X and Y. In reality instead of observing X and Y one observes U and V which imply stress and strength subject to some sort of errors. In this article, procedures have been Indicated to estimate true reliability using U and V values under the assumption of exponentiality. Over and/or under-reporting has been treated generally as damage.

On Generalized Quality Capability Index

Abtsrcat A Generalized Process Capability Index, defined as the ratio of proportion of specification conformance (or, process yield) to proportion of desired (or, natural) conformance, has been developed by Maiti et al.(2010). Almost all the process capabilities defined in the literature are directly or indirectly associated with this generalized index. In this article, a loss function has been attached to the product quality characteristic for deviation from its target to define the index named as generalized quality capability index. Symmetric as well as asymmetric loss functions have been considered. We deal with the index in case of normal, non-normal and discrete processes. An algorithm for determining optimum process center for an off-centered process has been prescribed. A process data set has been analyzed.