Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

Demand Systems Estimation With Microdata: A Censored Regression Approach

Demand systems estimation increasingly makes use of household-level microdata, mainly to measure the effects of demographic variables. Data based on these household-expenditure surveys present a major estimation problem. For any given household, many of the goods have zero consumption, implying a censored dependent variable. Techniques which do not take this censored dependent variable into account will yield biased results. We utilize a censored regression approach that is computationally simple, consistent, and asymptotically efficient. The results are then presented and compared with those obtained using an uncensored technique.     

An Empirical Likelihood Ratio Test for Normality

The empirical likelihood ratio (ELR) test for the problem of testing for normality is derived in this article. The sampling properties of the ELR test and four other commonly used tests are provided and analyzed using the Monte Carlo simulation technique. The power comparisons against a wide range of alternative distributions show that the ELR test is the most powerful of these tests in certain situations.     

Non-Normal Bivariate Distributions with Normal Marginals

The purpose of this note is to indicate that Fieller's Theorem can be expressed in the matrix formulation of the general linear model. The practical consequence is that one general computer program which can estimate the parameters and test the validity of a pertinent model, can also compute confidence limits for the ratios of any linear combinations of the parameters.     

Fisher's Hypergeometric Test for a Comparison in a Finite Population

Graphical methods have played a central role in the development of statistical theory and practice. This presentation briefly reviews some of the highlights in the historical development of statistical graphics and gives a simple taxonomy that can be used to characterize the current use of graphical methods. This taxonomy is used to describe the evolution of the use of graphics in some major statistical and related scientific journals.

Some recent advances in the use of graphical methods for statistical analysis are reviewed, and several graphical methods for the statistical presentation of data are illustrated, including the use of multicolor maps.     

Extensions of Samuelson's Inequality

In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.     

On Sharp Jensen's Inequality and Some Unusual Applications

The purpose of this article is two-fold. First, we find it very interesting to explore a kind of notion of optimality of the customary Jensen-bound among all Jensen-type bounds. Without this result, the customary Jensen-bound stood alone simply as just another bound. The proposed notion and the associated optimality are important given that in some situations the Jensen's inequality does leave us empty handed.

When it comes to highlighting Jensen's inequality, unfortunately only a handful of nearly routine applications continues to recycle time after time. Such encounters rarely produce any excitement. This article may change that outlook given its second underlying purpose, which is to introduce a variety of unusual applications of Jensen's inequality. The collection of our important and useful applications and their derivations are new.     

Recurrence formula on the joint distribution of sample skewness and kurtosis under normality

Abstract

Two recurrence relations with respect to sample size are given concerning the joint distribution of skewness and kurtosis of random observations from a normal population: one between the probability density functions and the other between the product moments. As a consequence, the latter yields a recurrence formula for the moments of sample kurtosis. The exact moments of Jarque-Bera statistic is also given.     

A robust procedure for testing an assumed value of the population correlation coefficient

For testing that the population correlations coefficientp Q, Tiku and Balakrishnan1986) developed a robust test. This test is extended here to situcitions where one wants to test that p p , p being a specified non-zero value of p. o o     

Test for Parameter Change in Linear Processes Based on Whittle's Estimator

In this article, we develop a cusum test for testing for parameter changes in linear processes based on Whittle's estimator. It is shown that under regularity conditions, the test statistic converges to the sup of a Brownian bridge. The result is particularly useful in handling the change point test in stationary ARMA processes. A simulation result is provided for illustration.