EVALUATION OF POWER OF TESTS INVOLVING COVARIATES UNDER NONCONSTANT VARIANCE

In many experiments where pre-treatment and post-treatment measurements are taken, investigators wish to determine if there is a difference between two treatment groups. For this type of data, the post-treatment variable is used as the primary comparison variable and the pre-treatment variable is used as a covariate. Although most of the discussion in this paper is written with the pre-treatment variable as the covariate the results are applicable to other choices of a covariate. Tests based on residuals have been proposed as alternatives to the usual covariance methods. Our objective is to investigate how the powers of these tests are affected when the conditional variance of the post-treatment variable depends on the magnitude of the pre-treatment variable. In particular, we investigate two cases. [1] Crager, Michael R. 1987. Analysis of Covariance in Parallel-Group Clinical Trials With Pretreatment Baselines. Biometrics, 43: 895–901. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable gradually increases as the magnitude of the pre-treatment variable increases. (In many biological models this is the case.) [2] Knoke, James D. 1991. Nonparametric Analysis of Covariance for Comparing Change in Randomized Studies with Baseline Values Subject to Error. Biometrics, 47: 523–533. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable is dependent upon natural or imposed subgroups contained within the pre-treatment variable. Power comparisons are made using Monte Carlo techniques.     

TWO-STAGE PROCEDURES FOR TESTING HOMOGENEITY OF MEANS AGAINST ORDERED ALTERNATIVES IN ONE-WAY ANOVA WITH COMMON UNKNOWN VARIANCE

In this paper we present two-stage procedures for testing the equality of normal means against ordered alternatives in one-way ANOVA with common and unknown variance. The procedures proposed are analogous to Williams's and Bartholomew's single-stage methods. A table of percentiles needed for implementation is provided. Some Monte Carlo results for estimating the power of both procedures are given.     

ON TESTS OF SYMMETRY FOR DISCRETE POPULATIONS

In this paper problems of tests of symmetry about the origin with discrete samples are considered. Recently Vorli?ková established the asymptotic normality of linear rank statistics and signed rank statistics in [5] and [6]. Here we propose statistics which are conditionally the sum of independent variables, including the locally most powerful tests for a one sided one parameter family. Their asymptotic distributions are derived under the null hypothesis and the contiguous rounding off location alternatives. We propose four types of signed rank tests and investigate their properties.     

ON COMPONENTS OF VARIANCE and MODELLING EXCEEDANCES OVER A THRESHOLD

Variance component estimates from a simple hierarchical model are used to model the distribution of large data values exceeding a threshold. The advantages of interrelating the components of variance and the exceedances are stressed. The theory is outlined and an analysis of some data on blood pressure is discussed.     

ON THE PERFORMANCE OF KOLMOGOROV-SMIRNOV TESTS ON STABLE RANDOM VARIABLES1

This paper provides two procedures to perform the Kolmogorov–Smirnov (K-S) tests on stable random variables. One utilizes the integral representation of the cumulative distribution function of this random variable to perform the K-S test; the other utilizes the Gil-Pelaez-Roseń transformation of data (Csörg(1983)) to test uniformity on (0,1) for the transformed sample. Both procedures are examined and evaluated by a simulation study.     

ON THE ROBUST ANALYSIS OF VARIANCE COMPONENTS MODELS FOR PEDIGREE DATA

Quantitative traits measured over pedigrees of individuals may be analysed using maximum likelihood estimation, assuming that the trait has a multivariate normal distribution. This approach is often used in the analysis of mixed linear models. In this paper a robust version of the log likelihood for multivariate normal data is used to construct M-estimators which are resistant to contamination by outliers. The robust estimators are found using a minimisation routine which retains the flexible parameterisations of the multivariate normal approach. Asymptotic properties of the estimators are derived, computation of the estimates and their use in outlier detection tests are discussed, and a small simulation study is conducted.     

BAHADUR SLOPES FOR COMPARISONS OF TESTS BASED ON RESAMPLING

Let X₁,…, X_n be random variables symmetric about θ from a common unknown distribution F_θ(x) =F(x–θ). To test the null hypothesis H₀:θ= 0 against the alternative H₁:θ > 0, permutation tests can be used at the cost of computational difficulties. This paper investigates alternative tests that are computationally simpler, notably some bootstrap tests which are compared with permutation tests. Of these the symmetrical bootstrap-f test competes very favourably with the permutation test in terms of Bahadur asymptotic efficiency, so it is a very attractive alternative.     

ON ASYMPTOTIC OPTIMALITY OF SOME TESTS FOR EXPONENTIAL DISTRIBUTIONS

The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?^-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρ_i, s?_i), 0 < ρ_i < ∞, 0 < s?_i < s?_i < and f(x; ρ, s?) is as given above. Let X_i1 < X_i2 <… < X_iri denote the first r_i order statistics in a random sample of size n_i, drawn from the ith population with pdf f(x; ρ_i, s?_i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρ_i, s?_i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions.     

ON THE POWER BREAKDOWN OF THE NEGATIVE EXPONENTIAL DISPARITY TESTS

The generalized negative exponential disparity, discussed in Bhandari et al. (Robust inference in parametric models using the family of generalized negative exponential disparities, 2006, ANZJS, 48 , 95–114), represents an important class of disparity measures that generates efficient estimators and tests with strong robustness properties. In their paper, however, Bhandari et al. failed to provide a sharp lower bound for the power breakdown point of the corresponding tests. This was acknowledged by the authors, who indicated the possible existence of a sharper bound, but noted that they did not “have a proof at this point”. In this paper we provide an improved bound for this power breakdown point, and show with an example how this can enhance the existing results.     

MODEL-ASSISTED HIGHER-ORDER CALIBRATION OF ESTIMATORS OF VARIANCE

In survey sampling, interest often centres on inference for the population total using information about an auxiliary variable. The variance of the estimator used plays a key role in such inference. This study develops a new set of higher‐order constraints for the calibration of estimators of variance for various estimators of the population total. The proposed strategy requires an appropriate model for describing the relationship between the response and auxiliary variable, and the variance of the auxiliary variable. It is therefore referred to as a model‐assisted approach. Several new estimators of variance, including the higher‐order calibration estimators of the variance of the ratio and regression estimators suggested by Singh, Horn & Yu and Sitter & Wu are special cases of the proposed technique. The paper presents and discusses the results of an empirical study to compare the performance of the proposed estimators and existing counterparts.     

EXAMPLES OF MINIMUM VARIANCE ESTIMATION1

In this sequel to a previous discussion of minimum variance estimation (Bartlett, 1982), the gain with conditional estimation procedures is illustrated for the location parameter for (i) the rectangular distribution; (ii) a triangular distribution (typifying an asymmetric case). This note concludes with further remarks on the multi-sample and multi-parameter cases.     

COMPONENTS OF VARIANCE MODELS WITH TRANSFORMATIONS

Power transformations are a popular way to improve the agreement between the observations and the assumptions in a statistical model. In this paper it is assumed that the data, after appropriate power transformation Λ, satisfies a variance components model, with independent Gaussian components. The focus is on inference for quantities which have an interpretation regardless of the choice of Λ (Carroll & Ruppert, 1981) – in particular the intraclass correlation coefficient ρ, the predicted probability of a new observation being less than a specified value and the predicted quantile. It is shown that, in the case Λ= 0, the asymptotic variance of ρ is the same, whether or not one treats Δ as estimated or as known. This supports an empirical conjecture of Solomon (1985). For predicted probabilities and predicted quantiles the variance when A is estimated is shown to be only slightly greater than the variance assuming Δ is known, except in the tails of the distribution where there can be substantial difference between the two variances.     

ON VARIANCE ESTIMATION IN UNEQUAL PROBABILITY SAMPLING1

The purpose of this article is to propose a model-based estimator of the variance of the Horvitz-Thompson estimator. Empirical investigations reveal that the estimator is seldom greatly biased and is quite satisfactory from the stability point of view.     

OPTIMAL TESTS OF SIGNIFICANCE

To perform a test of significance of a null hypothesis, a test statistic is chosen which is expected to be small if the hypothesis is false. Then the significance level of the test for an observed sample is the probability that the test statistic, under the assumptions of the hypothesis, is as small, or smaller than, its observed value. A "good" test statistic is taken to be one which is stochastically small when the null hypothesis is false. Optimal test statistics are defined using this criterion and the relationship of these methods to the Neyman-Pearson theory of hypothesis testing is considered.     

ON ENTROPY BASED TESTS FOR EXPONENTIALITY

ABSTRACT

Recent literature has proposed a test for exponentiality based on sample entropy. We consider transformations of the observations which turn the test of exponentiality into one of uniformity and use a corresponding test based on entropy. The test based on the transformed variables performs better in many cases of interest.     

MINIMUM VARIANCE UNBIASED ESTIMATION OF PROBABILITY DENSITIES

The purpose of this note is to point out an elementary method for deriving the minimum variance unbiased (mvu) estimators of probability densities. The method is illustrated by estimating the densities of some well known important distributions. We believe that this note would be of interest especially to both teachers and students of a first course in mathematical statistics.     

ON THE USE OF THE STEIN VARIANCE ESTIMATOR IN THE DOUBLE k-CLASS ESTIMATOR IN REGRESSION

This paper investigates the predictive mean squared error performance of a modified double k-class estimator by incorporating the Stein variance estimator. Recent studies show that the performance of the Stein rule estimator can be improved by using the Stein variance estimator. However, as we demonstrate below, this conclusion does not hold in general for all members of the double k-class estimators. On the other hand, an estimator is found to have smaller predictive mean squared error than the Stein variance-Stein rule estimator, over quite large parts of the parameter space.