Process Capability Analysis in Non Normal Linear Regression Profiles

When the distribution of a process characterized by a profile is non normal, process capability analysis using normal assumption often leads to erroneous interpretations of the process performance. Profile monitoring is a relatively new set of techniques in quality control that is used in situations where the state of product or process is represented by a function of two or more quality characteristics. Such profiles can be modeled using linear or nonlinear regression models. In some applications, it is assumed that the quality characteristics follow a normal distribution; however, in certain applications this assumption may fail to hold and may yield misleading results. In this article, we consider process capability analysis of non normal linear profiles. We investigate and compare five methods to estimate non normal process capability index (PCI) in profiles. In three of the methods, an estimation of the cumulative distribution function (cdf) of the process is required to analyze process capability in profiles. In order to estimate cdf of the process, we use a Burr XII distribution as well as empirical distributions. However, the resulted PCI with estimating cdf of the process is sometimes far from its true value. So, here we apply artificial neural network with supervised learning which allows the estimation of PCIs in profiles without the need to estimate cdf of the process. Box-Cox transformation technique is also developed to deal with non normal situations. Finally, a comparison study is performed through the simulation of Gamma, Weibull, Lognormal, Beta and student-t data.     

The Correlated Bivariate Noncentral F Distribution and Its Application

This article proposes the singly and doubly correlated bivariate noncentral F (BNCF) distributions. The probability density function (pdf) and the cumulative distribution function (cdf) of the distributions are derived for arbitrary values of the parameters. The pdf and cdf of the distributions for different arbitrary values of the parameters are computed, and their graphs are plotted by writing and implementing new R codes. An application of the correlated BNCF distribution is illustrated in the computations of the power function of the pre-test test for the multivariate simple regression model (MSRM).     

Comments on zacks' recent paper on one-sided cusum procedures

The goal of this paper is to propose approximations for the cdf and the inverse cdf of the normal sample median. The presented methodology, which seems to not have been investigated before, suggests to fit the normal sample median distribution with a symmetrical Johnson S_U: distribution having ap-proximatively the same second and fourth moments. The results obtained with this approach, compared with the normal approximation, are very impressive, especially for the inverse cdf. One important application of the inverse cdf approximation of the normal sample median is the computation of accurate α‐level median/range control limits for any value of α (and not only for the popular value α = 0.0027). This paper can be also viewed as an homage to Professor N.L. Johnson's works by making a link between two of his major papers.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

An entropic structure in capability indices

Abstract

Analysis capability indices for symmetric process in normal case is obtained via maximum entropy approach of distribution function of the data. In view of it, we have perused on production processes to be in statistical control. Generally a process is capable based on capability indices when its reasonable index was more than a known threshold value. Thus by conditioning on indices, the most general distribution is found out whose parameters can be approximated by using the data of process. Also analysis via Kullback-Leibler information measure based on the above arguments is obtained in the last part of the paper.     

Properties and Inference for a New Class of Skew-t Distributions

In this article we introduce a new generalization of skew-t distributions, which contains the standard skew-t distribution, as a special case. This new class of distributions is an adequate model for modeling some dataset rather than the standard skew-t distributions. This kind of distributions can be represented as a scale-shape mixture of the extended skew-normal distributions. The main properties of this family of distributions are studied and a recurrence relation for the cumulative distribution functions (cdf) of them is presented. We derive the distribution of the order statistics from the trivariate exchangeable t-distribution in terms of our distribution and then an exact expression for the cdf of order statistics is derived. Likelihood inference for this distribution is also examined. The method is illustrated with a numerical example via a simulation study.     

Combined Exponentially Weighted Moving Average Charts for the Mean and Variance Based on the Predictive Distribution

This article develops combined exponentially weighted moving average (EWMA) charts for the mean and variance of a normal distribution. A Bayesian approach is used to incorporate parameter uncertainty. We first use a Bayesian predictive distribution to construct the control chart, and we then use a sampling theory approach to evaluate it under various hypothetical specifications for the data generation model. Simulations are used to compare the proposed charts for different values of both the weighing constant for the exponentially weighted moving averages and for the size of the calibration sample that is used to estimate the in-statistical-control process parameters. We also examine the separate performance of the EWMA chart for the variance.     

On the generalized t(GT) distribution

McDonald and Newey [J.B. McDonald and W.K. Newey, Partially adaptive estimation of regression models via the generalized t distribution, Econ. Theor. 4 (1988), pp. 428–457.] introduced the generalized t(GT) distribution. In this paper, several explicit formulas for its cumulative distribution function (cdf) are derived. These formulas will be useful for future developments in the theory and applications of the distribution. One such situation is explained and an application is provided to rainfall data from Orlando, Florida.     

Confidence Bounds for Positive Ratios of Normal Random Variables

Some applications of ratios of normal random variables require both the numerator and denominator of the ratio to be positive if the ratio is to have a meaningful interpretation. In these applications, there may also be substantial likelihood that the variables will assume negative values. An example of such an application is when comparisons are made in which treatments may have either efficacious or deleterious effects on different trials. Classical theory on ratios of normal variables has focused on the distribution of the ratio and has not formally incorporated this practical consideration. When this issue has arisen, approximations have been used to address it. In this article, we provide an exact method for determining (1 ? α) confidence bounds for ratios of normal variables under the constraint that the ratio is composed of positive values and connect this theory to classical work in this area. We then illustrate several practical applications of this method.     

Three New Multivariate Process Capability Indices

Different multivariate process capability indices are developed by researchers to evaluate process capability when vectors of quality characteristics are considered in a study. This article presents three indices referred to as NCpM, MCpM, and NMC _PM in order to evaluate process capability in multivariate environment. The performance of the proposed indices is investigated numerically. Simulation results indicate that the proposed indices have descended estimation error and improved performance compared to the existing ones. These results can be important to researchers and practitioners who are interested in evaluating process capability in multivariate domain.     

An algorithm to compute the cdf of the product of two normal random variables

This paper provides a fortran algorithm that can be used to compute the cdf of the product of two normal distribution random variables. We also give references that provide mathematical properties, tables, and applications of this distribution     

Shannon Entropy and Mutual Information for Multivariate Skew‐Elliptical Distributions

Abstract. The entropy and mutual information index are important concepts developed by Shannon in the context of information theory. They have been widely studied in the case of the multivariate normal distribution. We first extend these tools to the full symmetric class of multivariate elliptical distributions and then to the more flexible families of multivariate skew‐elliptical distributions. We study in detail the cases of the multivariate skew‐normal and skew‐t distributions. We implement our findings to the application of the optimal design of an ozone monitoring station network in Santiago de Chile.     

An asymptotic confidence interval for the process capability index Cpm

Abstract

The use of indices as an estimation tool of process capability is long-established among the statistical quality professionals. Numerous capability indices have been proposed in last few years. C_pm constitutes one of the most widely used capability indices and its estimation has attracted much interest. In this paper, we propose a new method for constructing an approximate confidence interval for the index C_pm. The proposed method is based on the asymptotic distribution of the index C_pm obtained by the Delta Method. Under some regularity conditions, the distribution of an estimator of the process capability index C_pm is asymptotically normal.     

The Folded Logistic Distribution

ABSTRACT

Physical measurements like dimensions, including time, and angles in scientific experiments are frequently recorded without their algebraic sign. The directions of those physical quantities measured with respect to a frame of reference in most practical applications are considered to be unimportant and are ignored. As a consequence, the underlying distribution of measurements is replaced by a distribution of absolute measurements. When the underlying distribution is logistic, the resulting distribution is called the “folded logistic distribution”. Here, the properties of the folded logistic distribution will be presented and the techniques for estimating parameters will be given. The advantages of using this folded logistic distribution over the folded normal distribution will be discussed and some examples will be cited.     

Asymptotic Tail Properties of Student's t-Distribution

We investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of Student's t-distribution with ν > 0 degrees of freedom (t _ν for short) for ν tending to infinity when the argument x = x _ν of the pdf (cdf) depends on ν and tends to ± ∞ (?∞). To this end, we consider the ratio of the pdf's (cdf's) of the t _ν- and the standard normal distribution. Depending on the choice of the argument x _ν, the pdf-ratio (cdf-ratio) tends to 1, a fixed value greater than 1, or to ∞. As a byproduct, we obtain a result for Mill' ratio when x _ν → ?∞.     

The product <Emphasis Type="Italic">t</Emphasis> density distribution arising from the product of two Student’s <Emphasis Type="Italic">t</Emphasis> PDFs

The Student’s t distribution has become increasingly prominent and is considered as a competitor to the normal distribution. Motivated by real examples in Physics, decision sciences and Bayesian statistics, a new t distribution is introduced by taking the product of two Student’s t pdfs. Various structural properties of this distribution are derived, including its cdf, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Finally, an application to a Bayesian testing problem is illustrated.     

The inverse problem of multivariate and matrix-variate skew normal distributions

In this paper, we prove that the joint distribution of random vectors Z ₁ and Z ₂ and the distribution of Z ₂ are skew normal provided that Z ₁ is skew normally distributed and Z ₂ conditioning on Z ₁ is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.     

A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

A generalized two-piece skew normal distribution and some of its properties

In this paper, we develop a generalized version of the two-piece skew normal distribution of Kim [On a class of two-piece skew-normal distributions, Statistics 39(6) (2005), pp. 537–553] and derive explicit expressions for its distribution function and characteristic function and discuss some of its important properties. Further estimation of the parameters of the generalized distribution is carried out.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.