Comparisons of the asymptotic efficiencies of two sample tests for discrete distributions

The asymptotic efficiencies are computed for several popular two sample rank tests when the underlying distributions are Poisson, binomial, discrete uniform, and negative binomial The rank tests examined include the Mann-Whitney test, the van der Waerden test, and the median test. Three methods for handling ties are discussed and compared. The computed asymptotic efficiencies apply also to the k-sample extensions of the above tests, such as the Kruskal-Wallis test, etc.     

Estimates of regression coefficients based on the sign covariance matrix

Summary. A new estimator of the regression parameters is introduced in a multivariate multiple-regression model in which both the vector of explanatory variables and the vector of response variables are assumed to be random. The affine equivariant estimate matrix is constructed using the sign covariance matrix (SCM) where the sign concept is based on Oja's criterion function. The influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the SCM regression estimate. The estimate is shown to be consistent with a limiting multinormal distribution. The influence function, as a function of the length of the contamination vector, is shown to be linear in elliptic cases; for the least squares (LS) estimate it is quadratic. The asymptotic relative efficiencies with respect to the LS estimate are given in the multivariate normal as well as the t -distribution cases. The SCM regression estimate is highly efficient in the multivariate normal case and, for heavy-tailed distributions, it performs better than the LS estimate. Simulations are used to consider finite sample efficiencies with similar results. The theory is illustrated with an example.     

Least Squares estimation for the multivariate weibull model of bougaard based on accelerated life test of system and component

Accelerated life testing of products quickly yields information on life. In this article, we present a simple method to incorporate the information collected from accelerated life tests of both components and (series) systems. The multivariate Weibull distribution of Hougaard is applied to model lifetimes of components. Least-squares (LS) estimators of the model parameters and their joint asymptotic distribution are derived. The effects of the dependence parameter and the proportion of the system-data to the asymptotic relative efficiencies of the LS estimators are investigated.     

On Estimating Finite Population Mean in Simple and Stratified Random Sampling

The likelihood equations are given for the Weibull distribution with qtouped data* A numerical technique is described for calculating the asymptotic variance-covatiance matrix with grouped data. Results from a Monte Carlo study are then used to show that in son© cases the measured efficiencies obtained with grouped data are substantially better than those rrteasiired Cor partially grouped (censored! dataf contradicting the asymptotic efficiency predictions.     

Tests of multinormality based on location vectors and scatter matrices

Classical univariate measures of asymmetry such as Pearson’s (mean-median)/σ or (mean-mode)/σ often measure the standardized distance between two separate location parameters and have been widely used in assessing univariate normality. Similarly, measures of univariate kurtosis are often just ratios of two scale measures. The classical standardized fourth moment and the ratio of the mean deviation to the standard deviation serve as examples. In this paper we consider tests of multinormality which are based on the Mahalanobis distance between two multivariate location vector estimates or on the (matrix) distance between two scatter matrix estimates, respectively. Asymptotic theory is developed to provide approximate null distributions as well as to consider asymptotic efficiencies. Limiting Pitman efficiencies for contiguous sequences of contaminated normal distributions are calculated and the efficiencies are compared to those of the classical tests by Mardia. Simulations are used to compare finite sample efficiencies. The theory is also illustrated by an example.     

A consistent correlation-type goodness-of-fit test;with application to the two-parameter weibull distribution

In this article the properties of a general univariate JiT-sample rank tests for complete block designs are investigated. Especially, the asymptotic distribution of the test .statistic under H0 and under contiguous alternatives is derived. Some asymptotic relative'PITMAN efficiencies are computed.

AMSX 1980 subject classifications: Primary 62G10; secondary 62K10     

On an asymptotic relative efficiency concept based on expected volumes of confidence regions

ABSTRACT

The paper deals with an asymptotic relative efficiency concept for confidence regions of multidimensional parameters that is based on the expected volumes of the confidence regions. Under standard conditions the asymptotic relative efficiencies of confidence regions are seen to be certain powers of the ratio of the limits of the expected volumes. These limits are explicitly derived for confidence regions associated with certain plugin estimators, likelihood ratio tests and Wald tests. Under regularity conditions, the asymptotic relative efficiency of each of these procedures with respect to each one of its competitors is equal to 1. The results are applied to multivariate normal distributions and multinomial distributions in a fairly general setting.     

Ultrastructural elliptical models

Dolby's (1976) ultrastructural model with no replications is investigated within the class of the elliptical distributions. General asymptotic results are given for the sample covariance matrix S in the presence of incidental parameters. These results are used to study the asymptotic behaviour of some estimators of the slope parameter, unifying and extending existing results in the literature. In particular, under some regularity conditions they are shown to be consistent and asymptotically normal. For the special case of the structural model, some asymptotic relative efficiencies are also reported which show that generalized least squares and the method of moment estimators can be highly inefficient under nonnormality.     

k-sample rank tests using pair-specific scoring functions

In this paper we consider generalizations of the distribution-free one-way ANOVA test procedures studied by Puri (1965), such generalizations allowing for possibly different scoring functions for different pairs of samples. The asymptotic distributions of the test statistics are derived and formulas for the asymptotic relative efficiencies of these tests with respect to a parametric competitor are obtained. Applications of these procedures to settings of ordered and umbrella alternatives are presented and potential advantages of this pair-specific scoring scheme are discussed.     

Simple nonparametric tests for a known standard survival based on censored data

It is often of interest in survival analysis to test whether the distribution of lifetimes from which the sample under study was derived is the same as a reference distribution. The latter can be specified on the basis of previous studies or on subject matter considerations. In this paper several tests are developed for the above hypothesis, suitable for right-censored observations. The tests are based on modifications of Moses' one-sample limits of some classical two-sample rank tests. The asymptotic distributions of the test statistics are derived, consistency is established for alternatives which are stochastically ordered with respect to the null, and Pitman asymptotic efficiencies are calculated relative to competing tests. Simulated power comparisons are reported. An example is given with data on the survival times of lung cancer patients.     

An efficiency result for the empirical characteristic function in stationary time-series models

It is shown under general conditions that arbitrarily high asymptotic efficiencies can be obtained when the parameters of a stationary time series are estimated by fitting the characteristic functions of the process to their empirical versions. A consistency and a central limit result are also given.     

Estimating the asymptotic relative efficiency of weighted rankings

Five tests for the hypothesis of no treatment effects in complete blocks designs are compared with respect to asymptotic relative efficiency: the ordinary two-way analysis of variance, the analysis of variance of ranks (Friedman, 1937), the method of ranking after alignment (Hodges and Lehmann, 1962), and a method of weighted rankings proposed by Quade (1972, 1979). This comparison required devising a procedure for estimating the asymptotic relative efficiencies of test statistics with asymp-totic chi-squared distributions.     

Estimating pareto quantiles using two order statistics

Asymptotically best linear unbiased estimators of the population quantiles for the location-scale Pareto distribution with fixed shape parameter are obtained using two suitably chosen order statistics. Formulae for the appropriate order statistics, coefficients, variances, and asymptotic relative efficiencies (relative to the usual non-parametric estimator for quantiles) are given     

Estimation of location and scale parameters of a distribution by U-statistics based on best linear functions of order statistics

In this work we propose a technique of estimating the location parameter μ$\mu$ and scale parameter σ$\sigma$ of a distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The method has been illustrated for estimating the location and scale parameters of type-I extreme value distribution. We have computed the asymptotic relative efficiencies of the proposed U-statistics with the appropriate maximum likelihood estimators based on samples drawn from each of type-I extreme value, logistic and normal distributions. In all cases very high asymptotic relative efficiencies are obtained.     

Linear signed rank tests for hultivariate location

Previously proposed linear signed rank tests for multivariate location are not invariant under linear transformations of the observations, The asymptotic relative efficiencies of the tests 2 with respect to Hotelling's T²test depend on the direction of shift and the covariance matrix of the alternative distributions. For distributions with highly correlated components, the efficiencies of some of these tests can be arbitrarily low; they approach zero for certain multivariate normal alternatives, This article proposes a transformation of the data to be performed prior to standard linear signed rank tests, The resulting procedures have attractive power and efficiency properties compared to the original tests, In particular, for elliptically symmetric contiguous alternafives, the efficiencies of the new tests equal those of corresponding univariate linear signed rank tests with respect to the t test.     

Estimation efficiency of grouped and censored observation schemes for a generalised regression model

A multiple regression model is considered in which the density of the response variable is a member of a very wide family which includes many well-known distributions. Schemes of observation in which the response observations are grouped or type 1 right censored are examined. Results on the asymptotic variance efficiencies of the maximum likelihood estimators of the regression coefficients and standard deviation of the error distribution are presented for the two schemes.     

Conditional even point estimation for bivariate discrete distributions

This paper presents a simple procedure for estimating the parameters of bivariate discrete distributions. The procedure uses the marginal means and certain observed frequencies in one or more conditional distributions. The bivariate Poisson and Negative Binomial distributions are used as illustrative examples, Parameter estimators are derived and asymptotic efficiencies are examined for various parameter values.     

Parametric and semiparametric models for recapture and removal studies: a likelihood approach

Capture–recapture processes are biased samplings of recurrent event processes, which can be modelled by the Andersen–Gill intensity model. The intensity function is assumed to be a function of time, covariates and a parameter. We derive the maximum likelihood estimators of both the parameter and the population size and show the consistency and asymptotic normality of the estimators for both recapture and removal studies. The estimators are asymptotically efficient and their theoretical asymptotic relative efficiencies with respect to the existing estimators of Yip and co-workers can be as large as ∞. The variance estimation and a numerical example are also presented.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

An evaluation of the huber estimator

The asymptotic efficiency for the Huber estimator is evalu¬ated for distributions in the class of normal scale mixtures, gamma power, and Student's t-distributions. Rather high efficien¬cies are found for small deviations from normality, while intract¬able efficiencies occur for larger deviations.