Two zs are Better than One

Given only a random sample of observations, the usual estimator for the population mean is the sample mean. If additional information is provided it might be possible in some situations to obtain a better estimator. The situation considered here is when the variable whose mean is sought is composed of factors that are themselves observable. In the basic case, the variable can be expressed as the product of two, independent, more basic variables, but we also consider the case of more than two, the effect of correlation, and when there are observation costs.     

Asymptotic random extremal ratio and product based on generalized order statistics and its dual

The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results.     

The distribution of the product of two independent generalized trapezoidal random variables

ABSTRACT

In this article, we derive the probability density function (pdf) of the product of two independent generalized trapezoidal random variables having different supports, in closed form, by considering all possible cases. We also show that the results for the product of two triangular and uniform random variables follow as special cases of our main result. As an illustration, we obtain pdf of product for a suitably constrained set of parameters and plot some graphs using MATLAB, which express variation in pdf with change in different parameters of the generalized trapezoidal distribution.     

OPTIMAL DESIGN OF PLANS FOR ACCEPTANCE SAMPLING BY VARIABLES WITH INVERSE GAUSSIAN DISTRIBUTION

ABSTRACT

A vast majority of the literature on the design of sampling plans by variables assumes that the distribution of the quality characteristic variable is normal, and that only its mean varies while its variance is known and remains constant. But, for many processes, the quality variable is nonnormal, and also either one or both of the mean and the variance of the variable can vary randomly. In this paper, an optimal economic approach is developed for design of plans for acceptance sampling by variables having Inverse Gaussian (IG) distributions. The advantage of developing an IG distribution based model is that it can be used for diverse quality variables ranging from highly skewed to almost symmetrical. We assume that the process has two independent assignable causes, one of which shifts the mean of the quality characteristic variable of a product and the other shifts the variance. Since a product quality variable may be affected by any one or both of the assignable causes, three different likely cases of shift (mean shift only, variance shift only, and both mean and variance shift) have been considered in the modeling process. For all of these likely scenarios, mathematical models giving the cost of using a variable acceptance sampling plan are developed. The cost models are optimized in selecting the optimal sampling plan parameters, such as the sample size, and the upper and lower acceptance limits. A large set of numerical example problems is solved for all the cases. Some of these numerical examples are also used in depicting the consequences of: 1) using the assumption that the quality variable is normally distributed when the true distribution is IG, and 2) using sampling plans from the existing standards instead of the optimal plans derived by the methodology developed in this paper. Sensitivities of some of the model input parameters are also studied using the analysis of variance technique. The information obtained on the parameter sensitivities can be used by the model users on prudently allocating resources for estimation of input parameters.     

The effects of normal mixtures and autocorrelation on the fraction non-conforming

In this article, the effects of mixtures of two normal distributions on the fraction non-conforming are studied in the context of capability analysis. When the output from several processes is mixed, the quality characteristic variables of the resulting mix may result in a normal mixture distribution. This can happen in cases such as monitoring an output from several suppliers, several machines, or several workers. This study considered the independence case and autocorrelated processes for a mixture of two normal distributions, using an autoregressive model of order one, AR(1). It is shown that the true attained process fraction non-conforming (corresponding to specific values for some capability index) can be very different from what is expected when the data are independent normal random variables.     

On the ratio of generalized variances

This paper examines the ratio of the determinants of two independent central Wishart matrices. The ratio is represented as a product of independent beta type-2 random variables and then the exact density is evaluated for the general case. Different representations and particular cases are also discussed.     

Saddlepoint Approximations for a Product and Its Application

A technique of the saddlepoint approximation with double exponential base, SPA_D is developed to evaluate the probability of a product of two random variables, which may be independent or dependent, normal or contaminated normal random variables. The SPA_D shows a slightly better approximation as compared to the saddlepoint approximation with Lagannani–Rice formula. However, both methods get remarkable results when applied to evaluate the tail probabilities of the Reynolds stress for soil erosion prediction.     

Likelihood-based Estimation of the Regression Model with Scrambled Responses

A significant problem in the collection of responses to potentially sensitive questions, such as relating to illegal, immoral or embarrassing activities, is non-sampling error due to refusal to respond or false responses. Eichhorn & Hayre (1983) suggested the use of scrambled responses to reduce this form of bias. This paper considers a linear regression model in which the dependent variable is unobserved but for which the sum or product with a scrambling random variable of known distribution, is known. The performance of two likelihood-based estimators is investigated, namely of a Bayesian estimator achieved through a Markov chain Monte Carlo (MCMC) sampling scheme, and a classical maximum-likelihood estimator. These two estimators and an estimator suggested by Singh, Joarder & King (1996) are compared. Monte Carlo results show that the Bayesian estimator out-performs the classical estimators in all cases, and the relative performance of the Bayesian estimator improves as the responses become more scrambled.     

Screening procedures using quadratic forms

In quality control, a performance variable having a two-sided specification limit is common for assessing lot quality. Sometimes it is difficult or impossible to measure the performance variable directly; for example, when testing is destructive, expensive, or when the performance variable is related to the lifetime of the product. However, it may happen that there are several concomitant variables which are easily measured and which correlate highly with the variable of interest. Thus, one may use measurements on these variables to select or screen product which will have a high conditional probability of meeting product specification. We consider this situation when all variables have a joint multivariate normal distribution and the specification limits on the performance variable are two-sided.     

Approximating the equilibrium quantity traded and welfare in large markets

We consider the efficient outcome of a canonical economic market model involving buyers and sellers with independent and identically distributed random valuations and costs, respectively. When the number of buyers and sellers is large, we show that the joint distribution of the equilibrium quantity traded and welfare is asymptotically normal. Moreover, we bound the approximation rate. The proof proceeds by constructing, on a common probability space, a representation consisting of two independent empirical quantile processes, which in large markets can be approximated by independent Brownian bridges. The distribution of interest can then be approximated by that of a functional of a Gaussian process. This methodology applies to a variety of mechanism design problems.     

Score tests in the two - way layout of counts

For the balanced two-way layout of a count response variable Y classified by fixed or random factors A and B, we address the problems of (i) testing for individual and interactive effects on Y of two fixed factors, and (ii) testing for the effect of a fixed factor in the presence of a random factor and conversely. In case (i), we assume independent Poisson responses with µ_ij= E(Y| A=i,B=j) = α_iβ_jγ_ij corresponding respectively to the multiplicative

interactive and non-interactive cases. For case (ii) with factor A random, we derive a multivariate gamma-Poisson model by mixing on the random variable associated with each level of A. In each case Neyman C(α) score tests are derived. We present simulation results,and apply the interaction test to a data set, to evaluate and compare the size and power of the score test for interaction between two fixed factors, the competing Poisson-based likelihood ratio test, and the F-tests based on the assumptions that √Y+1 or log(Y+1) are approximately normal. Our results provide strong evidence that the normal-theory based F-tests typically are very far from nominal size, and that the likelihood ratio test is somewhat more liberal than the score test.     

The Representation of Hypergeometric Random Variables using Independent Bernoulli Random Variables

In this paper, we show that a hypergeometric random variable can be represented as a sum of independent Bernoulli random variables that are, except in degenerate cases, not identically distributed. In the proof, we use the factorial moment generating function. An asymptotic result on the probabilities of the Bernoulli random variables in the sum is also presented. Numerical examples are used to illustrate the results.     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

A robust multivariate measurement error model with skew-normal/independent distributions and Bayesian MCMC implementation

Skew-normal/independent distributions are a class of asymmetric thick-tailed distributions that include the skew-normal distribution as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in multivariate measurement errors models. We propose the use of skew-normal/independent distributions to model the unobserved value of the covariates (latent variable) and symmetric normal/independent distributions for the random errors term, providing an appealing robust alternative to the usual symmetric process in multivariate measurement errors models. Among the distributions that belong to this class of distributions, we examine univariate and multivariate versions of the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.     

Approximations of distributions for some standardized partial sums in sequential analysis

In sequential analysis it is often necessary to determine the distributions of √t Y _t and/or √a Y _t , where t is a stopping time of the form t = inf{ n ≥ 1 : n+S_n+ξ_n> a }, Y _n is the sample mean of n independent and identically distributed random variables (iidrvs) Y_i with mean zero and variance one, S_n is the partial sum of iidrvs X_i with mean zero and a positive finite variance, and { ξ_n } is a sequence of random variables that converges in distribution to a random variable ξ as n →∞ and ξ_n is independent of ( X_n+1, Y_n+1), (X_n+2, Y_n+2), . . . for all n ≥ 1. Anscombe's (1952) central limit theorem asserts that both √t Y _t and √a Y _t are asymptotically normal for large a , but a normal approximation is not accurate enough for many applications. Refined approximations are available only for a few special cases of the general setting above and are often very complex. This paper provides some simple Edgeworth approximations that are numerically satisfactory for the problems it considers.     

The Beta Product Distribution with Complex Parameters

The family consisting of the distributions of products of two independent beta variables is extended to include cases where some of the parameters are not positive but negative or complex. This “beta product” distribution is expressible as a Meijer G function. An example (from risk theory) where such a distribution arises is given: an infinite sum of products of independent random variables is shown to have a distribution that is the product convolution of a complex-parameter beta product and an independent exponential. The distribution of the infinite sum is a new explicit solution of the stochastic equation X = (in law) B(X + C). Characterizations of some G distributions are also proved.     

The distribution of the product of independent beta variables

We investigate the classic distribution and approximate distribution of the product of Beta variables which are independent. We show that the product of independent Beta variables is a Beta variable under the some assumptions. We also obtain the approximate distribution of the product of independent Beta variables.     

Phase I monitoring of generalized linear model-based regression profiles

In some industrial applications, the quality of a process or product is characterized by a relationship between the response variable and one or more independent variables which is called as profile. There are many approaches for monitoring different types of profiles in the literature. Most researchers assume that the response variable follows a normal distribution. However, this assumption may be violated in many cases. The most likely situation is when the response variable follows a distribution from generalized linear models (GLMs). For example, when the response variable is the number of defects in a certain area of a product, the observations follow Poisson distribution and ignoring this fact will cause misleading results. In this paper, three methods including a T²-based method, likelihood ratio test (LRT) method and F method are developed and modified in order to be applied in monitoring GLM regression profiles in Phase I. The performance of the proposed methods is analysed and compared for the special case that the response variable follows Poisson distribution. A simulation study is done regarding the probability of the signal criterion. Results show that the LRT method performs better than two other methods and the F method performs better than the T²-based method in detecting either small or large step shifts as well as drifts. Moreover, the F method performs better than the other two methods, and the LRT method performs poor in comparison with the F and T²-based methods in detecting outliers. A real case, in which the size and number of agglomerates ejected from a volcano in successive days form the GLM profile, is illustrated and the proposed methods are applied to determine whether the number of agglomerates of each size is under statistical control or not. Results showed that the proposed methods could handle the mentioned situation and distinguish the out-of-control conditions.     

Order statistics from non-identical power function random variables

In this paper, we derive some recurrence relations satisfied by the single and the product moments of order statistics arising from n independent and non-identically distributed power function random variables. These recurrence relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the multiple-outlier model are deduced as special cases. The results are further generalized to the case of truncated power function random variables.