On the rate of convergence of the Robbins–Monro's algorithm in a linear stochastic ill-posed problem with α-mixing data

In this paper we consider a recursive method of Robbins–Monro type to estimate the solution of the linear problem Ax = u, in which the second member is measured with α-mixing errors. We also show the almost complete convergence (a.co) of this algorithm specifying its convergence rate.     

Densities and distribution functions of the ratio of two linear functions and the product of Gamma variates: A saddlepoint approach

Saddlepoint approximations for the densities and the distribution functions of the ratio of two linear functions of gamma random variables and the product of gamma random variables are derived. Ratios of linear functions with positive and negative weights and non identical gamma variables are considered. The saddlepoint approximations are very accurate in the tails as in the center of the distribution. Extensive simulation studies are used to evaluate the accuracy of the proposed methods.     

Optimal inclusion probabilities for balanced sampling

When auxiliary information is available at the design stage, samples may be selected by means of balanced sampling. The variance of the Horvitz-Thompson estimator is then reduced, since it is approximately given by that of the residuals of the variable of interest on the balancing variables. In this paper, a method for computing optimal inclusion probabilities for balanced sampling on given auxiliary variables is studied. We show that the method formerly suggested by Tillé and Favre (2005) enables the computation of inclusion probabilities that lead to a decrease in variance under some conditions on the set of balancing variables. A disadvantage is that the target optimal inclusion probabilities depend on the variable of interest. If the needed quantities are unknown at the design stage, we propose to use estimates instead (e.g., arising from a previous wave of the survey). A limited simulation study suggests that, under some conditions, our method performs better than the method of Tillé and Favre (2005).     

Tables of two-sided tolerance factors for a normal distribution

Given a random sample of size N from a normal distribution, we consider tolerance intervals of the form X ? ks to X + ks, where X is the sample mean and s is the sample standard deviation. The value of k is chosen so that the interval covers a given proportion P of the population with confidence γ. Exact values of k, computed from numerical integration, are given for N = 2(1)100; P = 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, 0.999; and γ = 0.5, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995. The exact values are compared with the values obtained from an approximation developed by Wald and Wolfowitz (1946).     

Techniques for nonlinear least squares and robust regression

Recently, the authors and others have made considerable progress in developing algorithms for solving certain large-residual nonlinear least-squares problems where Gauss-Newton (GN) methods can be expected to perform poorly. These methods take account of the term in the Hessian ignored by the GN methods and use quasi-Newton procedures to update this term explicitly. This paper reviews these new approaches and discusses how they can be modified to give good performance on nonlinear models with robust loss functions where lack of scale invariance causes several new problems to arise.     

Algorithms for restricted least absolute value estimation

We present a concise summary of recent progress in developing algorithms for restricted least absolute value (LAV) estimation (i. e. ?₁ approximation subject to linear constraints). The emphasis is on our own new algorithm, and we provide some numerical results obtained with it.     

On Symmetrizing Transformation of the Sample Coefficient of Variation from a Normal Population

Variance-stabilizing transformation (VST) for the sample coefficient of variation is often used as a normalizing transformation and may be used for inference on the population coefficient of variation. However, for small samples, the VST may not be symmetric and hence there is a scope of improvement in its performance by seeking a symmetrizing transformation. This article investigates such a transformation that has been obtained by solving a differential equation. The solution may be complex; hence, a numerical strategy is employed in order to make the approximation practically useful. This transformation has been compared with explicitly available VST. The approach has been illustrated on real data from an agricultural experiment concentrating on inference on single samples; however, the method may be generally applicable to multiple samples when testing the homogeneity of coefficients of variation for many populations by following usual normal-theory-based methods applied on transformed statistics.     

A simulation study of a test for the equality of the coefficients of variation

Simulation study results are given for the size and power of a test for the equality of the coefficients of variation from r normal populations. Independent samples of equal and unequal size from the normal and three other distributions were used. The size and power of the test compare favorably to two tests developed by Doornbos and Dijkstra and the test statistic is simpler to compute.     

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.     

Extended Bayesian model averaging for heritability in twin studies

Family studies are often conducted to examine the existence of familial aggregation. Particularly, twin studies can model separately the genetic and environmental contribution. Here we estimate the heritability of quantitative traits via variance components of random-effects in linear mixed models (LMMs). The motivating example was a myopia twin study containing complex nesting data structures: twins and siblings in the same family and observations on both eyes for each individual. Three models are considered for this nesting structure. Our proposal takes into account the model uncertainty in both covariates and model structures via an extended Bayesian model averaging (EBMA) procedure. We estimate the heritability using EBMA under three suggested model structures. When compared with the results under the model with the highest posterior model probability, the EBMA estimate has smaller variation and is slightly conservative. Simulation studies are conducted to evaluate the performance of variance-components estimates, as well as the selections of risk factors, under the correct or incorrect structure. The results indicate that EBMA, with consideration of uncertainties in both covariates and model structures, is robust in model misspecification than the usual Bayesian model averaging (BMA) that considers only uncertainty in covariates selection.