Inequalities for generalized order statistics from some restricted family of distributions

The Steffensen inequality is applied to derive quantile bounds for the expectations of generalized order statistics from a distribution belonging to a particular subclass of distributions. The subclass consists of F having the property that F^?1(0+)=x₀>0 and that x →[1? F(x)]x^z is nonincreasing for all x > X₀ and some z > 0.     

A class of tests for testing ‘new is better than used’

A class of tests is proposed for testing H₀ F?(x) = e^?λx, λ > 0, x≥0 vs. H₁ F?(x + y) ≤ F?(x)F?(y), x, y≥0, with strict inequality for some x, y ≥ 0 (F = new is better than used). Efficiency comparisons of some tests within the class are made and a new test is proposed on the basis of these comparisons. Consistency and the asymptotic normality of the class of tests is proved under fairly broad conditions on the underlying entities.     

Singular gauss-markov models

For the general linear model Y = X$sZ + e in which e has a singular dispersion matrix $sG²A, $sG > 0, where A is n x n and singular, Mitra [2] considers the problem of testing F$sZ, where F is a known q x q matrix and claims that the sum of squares (SS) due to hypothesis is not distributed (as a x² variate with degrees of freedom (d. f.) equal to the rank of F) independent of the SS due to error, when a generalized inverse of A is chosen as (A + X'X)^–. This claim does not hold if a pseudo-inverse of A is taken to be (A + X'X)⁺ where A⁺ denotes the unique Moore-Penrose inverse (MPI) of A.     

Statistics of Shape,Direction and Cylindrical Variables

In statistical shape analysis, the shape of an object is understood to be what remains after the effects of location, scale and rotation are removed. We consider the distributional problem of triangular shape and an associated direction; motivated by a data set of microscopic fossils. We begin by constructing a parallel transport system such that the data transform onto the space 𝒮²×𝒮². A joint shape distribution on 𝒮²×𝒮¹ is proposed based on Jupp & Mardia's bivariate distribution on 𝒮²×𝒮¹. For concentrated data, an approximation to the distribution on 𝒮²×𝒮¹ is given by a distribution on ?¹×𝒮¹, and we explore a distribution on this space by extending Mardia & Sutton's distribution on ?²×𝒮¹. In this distribution, the expected edgel direction varies linearly in the shape coordinates. This is found to be a useful model for the microfossil data.     

Efficient computation of the noncentral x2 distribution

In this article we compare and contrast finite algorithms for computing the noncentral x² distribution function for odd or even degrees of freedom with the algorithms proposed recently by Ashour & Abdel-Samad (1990). We also obtain an alternative error bound for Ruben's (1974a) algorithm for even degrees of freedom and analyze the rate of convergence of two common infinite series representations for computing the cdf     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

A Class of Multivariate Distributions

In this paper we introduce a class of multivariate distributions, known as the generalized Liouville distribution and defined by the functional form (0 ≤x_i < ∞, αi > 0, β_i > 0, q_i > 0). It is shown that such distributions can be used to derive both the Dirichlet distribution and the beta distribution of the second kind.     

A Distribution-Free Median-Unbiased Quantile Estimator

Given λ∈(0-,l), let x_λ(F) denote the unique λ-quantile of the distribution F. A distribution-free median-unbiased estimator of x_λ(F) is explicitly constructed     

Comments on “a generalization of fisher's exact test in pxq contingency tables using more concordant relations”

The purpose of this note is to criticize Nguyen (1985) for his account of the literature on the generalization of Fisher's exact test and to point out parallels with existing algorithms of the algorithm proposed by Nguyen. Subsequently we will briefly raise some questions on the methodology proposed by Nguyen.

Nguyen (1985) suggests that all literature on exact testing prior to Nguyen & Sampson (1985) is based on the “more probable” relation or Exact Probability Test (EPT) as a test statistic. This is not correct. Yates (1934 - Pearson's X²), Lewontin & Felsenstein (1965 - X²), Agresti & Wackerly (1977 - X², Kendall's tau, Kruskal & Goodman's gamma), Klotz (1966 - Wilcoxon), Klotz & Teng (1977 - Kruskall & Wallis' H), Larntz (1978 - X², loglike-lihood-ratio statistic G², Freeman & Tukey statistic), and several others have investigated exact tests with other statistics than the EPT. In fact, Bennett & Nakamura (1963) are incorrectly cited as they investigated both X² and G², rather than EPT. Also, Freeman & Halton (1951) are incorrectly cited for they generalized Fisher's exact test to pxq tables and not 2xq tables as stated. And they are even predated by Yates (1934) who extended the test to 2×3 tables.     

Frequentist and Bayesian Interval Estimators for the Normal Mean

It is well known that a Bayesian credible interval for a parameter of interest is derived from a prior distribution that appropriately describes the prior information. However, it is less well known that there exists a frequentist approach developed by Pratt (1961 Pratt , J. W. ( 1961 ). Length of confidence intervals . J. Amer. Statist. Assoc. 56 : 549 – 657 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) that also utilizes prior information in the construction of frequentist confidence intervals. This frequentist approach produces confidence intervals that have minimum weighted average expected length, averaged according to some weight function that appropriately describes the prior information. We begin with a simple model as a starting point in comparing these two distinct procedures in interval estimation. Consider X ₁,…, X _n that are independent and identically N(μ, σ²) distributed random variables, where σ² is known, and the parameter of interest is μ. Suppose also that previous experience with similar data sets and/or specific background and expert opinion suggest that μ = 0. Our aim is to: (a) develop two types of Bayesian 1 ? α credible intervals for μ, derived from an appropriate prior cumulative distribution function F(μ) more importantly; (b) compare these Bayesian 1 ? α credible intervals for μ to the frequentist 1 ? α confidence interval for μ derived from Pratt's frequentist approach, in which the weight function corresponds to the prior cumulative distribution function F(μ). We show that the endpoints of the Bayesian 1 ? α credible intervals for μ are very different to the endpoints of the frequentist 1 ? α confidence interval for μ, when the prior information strongly suggests that μ = 0 and the data supports the uncertain prior information about μ. In addition, we assess the performance of these intervals by analyzing their coverage probability properties and expected lengths.     

Minimum Hellinger distance estimation in a nonparametric mixture model

In this paper, we investigate the estimation problem of the mixture proportion λ$\lambda$ in a nonparametric mixture model of the form λF(x)+(1-λ)G(x)$\lambda F(x)+(1-\lambda )G(x)$ using the minimum Hellinger distance approach, where F and G are two unknown distributions. We assume that data from the distributions F and G   as well as from the mixture distribution λF+(1-λ)G$\lambda F+(1-\lambda )G$ are available. We construct a minimum Hellinger distance estimator of λ$\lambda$ and study its asymptotic properties. The proposed estimator is chosen to minimize the Hellinger distance between a parametric mixture model and a nonparametric density estimator. We also develop a maximum likelihood estimator of λ$\lambda$. Theoretical properties such as the existence, strong consistency, asymptotic normality and asymptotic efficiency of the proposed estimators are investigated. Robustness properties of the proposed estimator are studied using a Monte Carlo study. Two real data examples are also analyzed.     

Rate of consistency of one sample tests of location

Let X₁,…,X_n be a sample from a population with continuous distribution function F(x?θ) such that F(x)+F(-x)=1 and 0<F(x)<1, x?R¹. It is shown that the power- function of a monotone test of H: θ=θ₀ against K: θ>θ₀ cannot tend to 1 as θ?θ₀ → ∞ more than n times faster than the tails of F tend to 0. Some standard as well as robust tests are considered with respect to this rate of convergence.