Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT

Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a “projection design”. In this article, we have identified all the geometric non isomorphic projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k = 3,4, and 5 factors when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.     

A Mixed Bivariate Distribution Connected with Geometric Maxima of Exponential Variables

We study the joint distribution of X and N, where N has a geometric distribution and X is the maximum of N i.i.d. exponential variables, independent of N. We present basic properties of these mixed bivariate distributions and discuss parameter estimation for this model. An example from finance, where N represents the number of consecutive positive daily log-returns of currency exchange rates, illustrates stochastic modeling potential of these laws.     

Asymptotic Distributions of the Generalized Range,Midrange, Extremal Quotient,and Extremal Product,with a Comparison Study

Necessary and sufficient conditions for the weak convergence of the generalized range, midrange, extremal quotient, and extremal product are obtained. The classes of possible non degenerate limit distribution functions of these simple statistics are characterized. Comparison study between these statistics with some examples for the most important distribution functions are given.     

On Discrete Gamma Distribution

A new discrete counterpart of gamma distribution for modelling discrete life data is defined based on similar mathematical form and properties of the continuous version. The main statistical and reliability properties of this distribution are derived and it is shown that this model can deal with both over and under-dispersed data. Geometric variables and finite sum of geometric variables, i.e., negative binomial are shown to be special cases of the proposed discrete gamma. Also, the size-biased discrete gamma distribution is derived and discussed. Moreover, different estimation methods of the underlying parameters of this distribution are utilized and comparisons of their performance have been made. Finally, an application in real-life data is used to elucidate the earlier results of this article.     

Admissibility intervals for linear correlation coefficients

The correlation coefficient is widely used to quantify the degree of association between two quantitative variables. By resorting to the geometric representation of the linear correlation coefficient, it is possible to calculate the upper and lower bounds of the correlation coefficient between two variables x ₁,x ₂ when the correlation coefficients with a third variable x ₃ are available. Implications in observational studies, where x ₃ could be a proxy of a target variable x ₂, whose direct measurement is too expensive or impractical, are discussed.     

Efficiency and Convergence Properties of Slice Samplers

The slice sampler (SS) is a method of constructing a reversible Markov chain with a specified invariant distribution. Given an independence MetropolisHastings algorithm (IMHA) it is always possible to construct a SS that dominates it in the Peskun sense. This means that the resulting SS produces estimates with a smaller asymptotic variance than the IMHA. Furthermore the SS has a smaller second-largest eigenvalue. This ensures faster convergence to the target distribution. A sufficient condition for uniform ergodicity of the SS is given and an upper bound for the rate of convergence to stationarity is provided.     

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.     

Optimal confidence intervals for the geometric parameter

Abstract

This article discusses optimal confidence estimation for the geometric parameter and shows how different criteria can be used for evaluating confidence sets within the framework of tail functions theory. The confidence interval obtained using a particular tail function is studied and shown to outperform others, in the sense of having smaller width or expected width under a specified weight function. It is also shown that it may not be possible to find the most powerful test regarding the parameter using the Neyman-Pearson lemma. The theory is illustrated by application to a fecundability study.     

Control charts for high-quality processes: MAX or CUMAX?

For attribute data with (very) small failure rates control charts were introduced which are based on subsequent groups of r   failure times, for some r≥1$r\ge 1$. Within this family, it was shown to be attractive to stop once the maximum of such a group is sufficiently small, because this choice allows a very satisfactory nonparametric adaptation. The question we address here is whether a cumulative approach offers even further improvement. Thus instead of fixed groups, we shall use the first sequence of r consecutive sufficiently small failure times to produce a signal. A further reason for considering this type of chart is the fact that it forms the nonparametric counterpart of the well-known sets method.     

Weighted geometric distribution with new characterizations of geometric distribution

In this article, we derive a new generalized geometric distribution through a weight function, which can also be viewed as a discrete analog of weighted exponential distribution introduced by Gupta and Kundu (2009 Gupta, R. D., and D. Kundu. 2009. A new class of weighted exponential distributions. Statistics 43:621–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We derive some distributional properties like moments, generating functions, hazard function, and infinite divisibility followed by different estimation methods to estimate the parameters. New characterizations of the geometric distribution are presented using the proposed generalized geometric distribution. The superiority of the proposed distribution to other competing models is demonstrated with the help of two real count datasets.