Optimal foldover designs

An obvious strategy for obtaining a Doptimal foldover design for p factors at two levels each in 2N runs is to fold a Doptimal main effects plan. We show that this strategy works except when N = 4t + 2 and s is even In that case there are two different classes of D-optimal main effects plans with N runs that have the same determinant. However folding them gives two different values foi the D-optimality criteiion One set of designs is D-optimal The other is not.     

Two-queue polling systems with switching policy based on the queue that is not being served

We study a system of two non-identical and separate M/M/1/? queues with capacities (buffers) C₁ < ∞ and C₂ = ∞, respectively, served by a single server that alternates between the queues. The server’s switching policy is threshold-based, and, in contrast to other threshold models, is determined by the state of the queue that is not being served. That is, when neither queue is empty while the server attends Q_i (i = 1, 2), the server switches to the other queue as soon as the latter reaches its threshold. When a served queue becomes empty we consider two switching scenarios: (i) Work-Conserving, and (ii) Non-Work-Conserving. We analyze the two scenarios using Matrix Geometric methods and obtain explicitly the rate matrix R, where its entries are given in terms of the roots of the determinants of two underlying matrices. Numerical examples are presented and extreme cases are investigated.     

A distribution function estimator for the difference of order statistics from two independent samples

For two independent populations X and Y we develop the empirical distribution function estimator for the difference of order statistics of the form X _(i)−Y _(j). The key practical application for this estimator pertains to inference between quantiles from two independent populations.     

Reliability analysis of systems with components having two dependent subcomponents

In this article, a system that consists of n independent components each having two dependent subcomponents (A_i, B_i), i = 1, …, n is considered. The system is assumed to compose of components that have two correlated subcomponents (A_i, B_i), and functions iff both systems of subcomponents A₁, A₂, …, A_n and B₁, B₂, …, B_n work under certain structural rules. The expressions for reliability and mean time to failure of such systems are obtained. A sufficient condition to compare two systems of bivariate components in terms of stochastic ordering is also presented.     

Consistent directions for least-squares estimates

The least-squares estimate θ_n = X_n⁺y_n of the parameter θ in the linear model Y_n = X_nθ + ?_n may not be consistent, but there may be directions u such that u'θ_n tends to u'θ in some sense. This set of directions u has been characterized in two different papers: Drygas (1976) and Wu (1980). The conditions for consistency appear to be different in the two papers. The purpose of this note is to show that the two conditions are equivalent and that they both show that the consistent directions depend upon the geometry of the row vectors v_i, i = 1,…, n, of X_n with respect to the direction u.     

On kernel estimators of density ratio

Let f(x) and g(x) denote two probability density functions and g(x)≠0. There are two ways to estimate the density ratio f(x)/g(x). One is to estimate f(x) and g(x) first and then the ratio, the other is to estimate f(x)/g(x) directly. In this paper, we derive asymptotic mean square errors and central limit theorems for both estimators.     

Combination-Based Permutation Tests: Equipower Property and Power Behavior in Presence of Correlation

Multivariate combination-based permutation tests have been widely used in many complex problems. In this paper we focus on the equipower property, derived directly from the finite-sample consistency property, and we analyze the impact of the dependency structure on the combined tests. At first, we consider the finite-sample consistency property which assumes that sample sizes are fixed (and possibly small) and considers on each subject a large number of informative variables. Moreover, since permutation test statistics do not require to be standardized, we need not assume that data are homoscedastic in the alternative. The equipower property is then derived from these two notions: consider the unconditional permutation power of a test statistic T for fixed sample sizes, with V ? 2 independent and identically distributed variables and fixed effect δ, calculated in two ways: (i) by considering two V-dimensional samples sized m₁ and m₂, respectively; (ii) by considering two unidimensional samples sized n₁ = Vm₁ and n₂ = Vm₂, respectively. Since the unconditional power essentially depends on the non centrality induced by T, and two ways are provided with exactly the same likelihood and the same non centrality, we show that they are provided with the same power function, at least approximately. As regards both investigating the equipower property and the power behavior in presence of correlation we performed an extensive simulation study.     

On the admissibility of ordered poisson parameter under the entropy loss function

Let X ₁, X ₂be two independent Poisson random variables with means θ ₁and θ ₂respectively. Assume 0 ≤ θ ₁ ≤ θ ₂ ≤ ∞. The problem is estimation of the ordered parameters which has received considerable attention in statistical literature during the last two decades, In this paper the main portion of the study is devoted to the problem of estimating the smallest of the two ordered Poisson means, when it is known which population corresponds to each mean under the entropy loss function. An extension of the problem to the k-ordered Poisson means is also discussed.     

Computing factors for exact two-sided tolerance limits for a normal distribution

A self-contained FORTRAN subroutine is provided which computes factors for Wald-Wolfowitz type tolerance limits allowing arbitrary combinations of sample size n and degrees of freedom ν. The exact calculations from our program reveal inadequacies of two existing approximations, especially when ν ? n. Numerous applications where ν ≠ n ? 1 are cited; two of these are discussed and illustrated.     

Allocation strategies of standby redundancies in series/parallel system

This article deals with the topic of optimal allocation of two standby redundancies in a two-component series/parallel system. There are two original components C₁ and C₂ which can be used to construct a series/parallel system, and two spares R₁ (same as C₁) and R₂ (different from both C₁ and C₂) at hand with them being standby redundancies so as to enhance the reliability level of the system. The goal for an engineer is to seek after the optimal allocation policy in this framework. It is shown that, for the series structure, the engineer should allocate R₂ to C₁ and R₁ to C₂ provided that C₁ (or R₁) performs either the best or worst among all the units; otherwise, the allocation policy should be reversed. For the parallel structure, the optimal allocation strategy is just opposed to that of series case. We also provide some numerical examples for illustrating the theoretical results.     

Cross-correlation between simultaneously generated sequences of pseudo-random uniform deviates

Systematic patterns are revealed when sequences of pseudo-random uniform deviates are generated from multiplicative congruential generators. If the initial seeds, x ₀ and y ₀, for two such sequences are related by y ₀ = (n ₁/n ₂)x ₀, where n ₁ and n ₂ are relatively prime, positive integers, then an approximate argument suggests that the asymptotic correlation coefficient between corresponding members of the two sequences is (n ₁ n ₂)^–1. This unsettling phenomenon is discussed in the context of related, existing literature.     

Using a Truncated C p Statistic for Variable Selection in Multiple Linear Regression

In multiple linear regression analysis each lower-dimensional subspace L of a known linear subspace M of ? ⁿ corresponds to a non empty subset of the columns of the regressor matrix. For a fixed subspace L, the C _p statistic is an unbiased estimator of the mean square error if the projection of the response vector onto L is used to estimate the expected response. In this article, we consider two truncated versions of the C _p statistic that can also be used to estimate this mean square error. The C _p statistic and its truncated versions are compared in two example data sets, illustrating that use of the truncated versions may result in models different from those selected by standard C _p .     

Sums, products and ratios of generalized beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, products and ratios of independent beta random variables. In this paper, we extend professor Pham-Gia's results when X ₁ and X ₂ are independent random variables distributed according to two generalized beta distributions. For each of these distributions, we derive exact expressions for the densities of S=X ₁+X ₂, D=X ₁−X ₂, P=X ₁ X ₂, and R=X ₂/X ₁. The expressions turn out to involve the incomplete beta function as well as the hypergeometric functions of one and two variables.     

On the Sum of Symmetric Random Variables

Let U and V be two symmetric (about zero) random variables with U + V symmetric about C; here C is a constant. It is easy to see that if U and V are mutually independent, or if both U and V satisfy the weak law of large numbers, then C = 0. So, intuitively, we would suspect that C = 0 in general. However, we show that there exist two random variables U and V symmetric about 0 with U + V symmetric about C ≠ 0 The example given is closely related to one given by Alejandro D. De Acosta in another context.     

Testing for Generalized Lorenz Dominance

The Generalized Lorenz dominance can be used to take account of differences in mean income as well as income inequality in case of two income distributions possessing unequal means. Asymptotically distribution-free and consistent tests have been proposed for comparing two generalized Lorenz curves in the whole interval [p ₁, p ₂] where 0 < p ₁ < p ₂ < 1. Size and power of the test has been derived.     

Inferential analysis for the reliability parameter based on the three-parameter Lindley distribution

ABSTRACT

In this article, we consider the estimation of R = P(Y < X), when Y and X are two independent three-parameter Lindley (LI) random variables. On the basis of two independent samples, the modified maximum likelihood estimator along its asymptotic behavior and conditional likelihood-based estimator are used to estimate R. We also propose sample-based estimate of R and the associated credible interval based on importance sampling procedure. A real life data set involving the times to breakdown of an insulating fluid is presented and analyzed for illustrative purposes.     

Tests of regression coefficients under ridge regression models

This paper investigates two “non-exact” t-type tests, t( k₂) and t(k₂), of the individual coefficients of a linear regression model, based on two ordinary ridge estimators. The reported results are built on a simulation study covering 84 different models. For models with large standard errors, the ridge-based t-tests have correct levels with considerable gain in powers over those of the least squares t-test, t(0). For models with small standard errors, t(k₁) is found to be liberal and is not safe to use while, t(k₂) is found to slightly exceed the nominal level in few cases. When tie two ridge tests art: not winners, the results indicate that they don't loose much against t(0).     

A Revised Version of McNemar's Test for Paired Binary Data

In the formula of the McNemar test, a test on 2 × 2 classification tables with pairs of data, only the two categories A and D, which represent changes, are included; the retained parts B and C, which represent concordant responses, are not considered. Generally, it would be more reasonable for the significance of the changes to depend not only on A and D, but also on B and C, or on the sample size, n. To develop the test, two formulae, based on A, D, and n, and on A, D, B, C, and n, respectively, are proposed.     

Matching a correlation coefficient by a Gaussian copula

A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρ_x between two random variables with given marginal distributions, the correlation coefficient ρ_z between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρ_x to ρ_z. In a few cases, this equation has an explicit solution, but in other cases it must be solved numerically. This paper attempts to address this issue. If two continuous random variables are involved, the marginal transformation is approximated by a weighted sum of Hermite polynomials; via Mehler’s formula, a polynomial of ρ_z is derived to approximate the function relationship between ρ_x and ρ_z. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρ_x is expressed as a polynomial of ρ_z by Taylor expansion. For a given ρ_x, ρ_z can be efficiently determined by solving a polynomial equation.