A CLASS OF NONPARAMETRIC TESTS FOR THE ONE-SAMPLE LOCATION PROBLEM

A class of “optimal”U-statistics type nonparametric test statistics is proposed for the one-sample location problem by considering a kernel depending on a constant a and all possible (distinct) subsamples of size two from a sample of n independent and identically distributed observations. The “optimal” choice of a is determined by the underlying distribution. The proposed class includes the Sign and the modified Wilcoxon signed-rank statistics as special cases. It is shown that any “optimal” member of the class performs better in terms of Pitman efficiency relative to the Sign and Wilcoxon-signed rank statistics. The effect of deviation of chosen a from the “optimal” a on Pitman efficiency is also examined. A Hodges-Lehmann type point estimator of the location parameter corresponding to the proposed “optimal” test-statistics is also defined and studied in this paper.     

A COMPARISON OF TESTS OF UNIFORMITY FOR SPHERICAL DATA

A comparison is made af three tests of uniformity for spherical vector data, and of the corresponding tests of uniformity for spherical axial data.     

ROBUSTNESS OF PROCEDURES FOR THE BEHRENS-FISHER PROBLEMS: EXTENSION TO BIVARIATE NORMAL MIXTURES

In the applied sciences, it is often important to be able to compare the mean values of two populations. However, testing a hypothesis can be complex, if the two populations are heteroscedastic and exhibit non-normality in the data. This paper reviews currently available strategies for the multivariate Behrens-Fisher problem. It then carries out Monte Carlo comparisons of selected procedures to assess their robustness when applied to data from normal mixture distributions. The overall conclusion is that Johansen's procedure appears to work best for small sample data both in terms of empirical power and significance level. Johansen's procedure works reasonably well even with mixture data. The simulation also provides researchers with specific guidelines to follow at the early designing and planning stages of the investigation.     

A NEW FAMILY OF DIVERGENCE MEASURES FOR TESTS OF FIT

The aim of this work is to investigate a new family of divergence measures based on the recently introduced Basu, Harris, Hjort and Jones (BHHJ) measure of divergence (Biometrika 85 , 549–559). The new family is investigated in connection with hypothesis testing problems, and new test statistics are proposed. Simulations are performed to check the appropriateness of the proposed test statistics.     

ASYMPTOTIC INFERENCE FOR A CLASS OF CHAIN BINOMIAL MODELS

Chain binomial models axe commonly used to model the spread of an epidemic through a population. This paper shows that for a flexible class of chain binomial models an approximate maximum likelihood estimator of the infection rate, derived from a Poisson approximation to the binomial distribution, has an asymptotically normal distribution, as do some other related'estimators.     

GOODNESS OF FIT TESTS FOR HETEROGENEITY IN A CAPTURE-RECAPTURE EXPERIMENT

A class of goodness of fit tests is proposed to test for heterogeneity in a population in a capture-recapture experiment when individuals' capture intensities may depend on time in an unspecified manner.     

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.     

APPROXIMATIONS TO SOME TEST STATISTICS FOR PERMUTATION TESTS IN A COMPLETELY RANDOMIZED DESIGN1

Under a randomization model for a completely randomized design permutation tests are considered based on the usual F statistic and on a multi-response permutation procedure statistic. For the first statistic the first two moments are obtained so a comparision with the distribution under the normal theory model can be made. The second statistic is shown to converge in distribution to an infinite weighted sum of chi-squared variates, the weights being the limits of the eigenvalues of a matrix depending on the distance measure used and the order statistics of the observations.     

ON A GENERAL DISTRIBUTION THEORY FOR A CLASS OF LIKELIHOOD RATIO CRITERIA

In this paper the exact distribution of a statistic with general moment function has been derived in a gamma series and also in a beta series. It is shown that the Box expansion can be obtained from the gamma series by collecting terms of the same order.     

USE OF A LOG-LINEAR MODEL TO COMPUTE THE EMPIRICAL SURVIVAL CURVE FROM INTERVAL-CENSORED DATA, WITH APPLICATION TO DATA ON TESTS FOR HIV POSITIVITY

We describe how a log-linear model can be used to compute the nonparametric maximum likelihood estimate of the survival curve from interval-censored data. This permits such computation to be performed with the aid of readily available statistical software such as GLIM or SAS. The method is illustrated with reference to data from a cohort of Danish homosexual men, each of whom was tested for HIV positivity on one or more of six possible follow-up times.     

A MONTE CARLO COMPARISON OF ELLIPTICAL-THEORY AND OTHER TESTS FOR EQUALITY OF COVARIANCE MATRICES

A Monte Carlo study of the size and power of tests of equality of two covariance matrices is carried out. Tests based upon normality assumptions, elliptical distribution assumptions as well as distribution-free tests are compared. Samples are generated from normal, elliptical and non-elliptical populations. The elliptical-theory tests, in particular, have poor size properties for both elliptical distributions with moderate sample sizes and for non-elliptical distributions.     

MODIFICATION OF KELLY AND RICE METHOD FOR SYNERGISM TESTS

ABSTRACT

In this paper, we present a modified Kelly and Rice method for testing synergism. This approach is consistent with Berenbaum's1-3 Berenbaum, M.C. 1977. Synergism, Additivism, and Antagonism in Immuno-Suppression: A Critical Review. Clinical and Experimental Immunology, 28: 1–18. Berenbaum, M.C. 1985. The Expected Effect of a Combination of Agents: The General Solution. Journal of Theoretical Biology, 114: 413–431. Berenbaum, M.C. 1989. What is Synergism? Pharmacological Reviews. 41: 93–141.   framework for additivity. The delta method[4] Bishop, Y.M., Fienberg, S.E. and Holland, P.W. 1975. Discrete Multivariate Analysis: Theory and Practice Massachusetts: MIT Press.  [Google Scholar] is applied to obtain the estimated variance for the predicted additivity proportion. A Monte Carlo simulation study for the evaluation of the method's performance, i.e., global overall tests for synergism, is also discussed. Kelly and Rice[5] Kelly, C. and Rice, J. 1990. Monotone Smoothing with Application to Dose-Response Curves and the Assessment of Synergism. Biometrics, 46: 1071–1085. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] do not provide a correct test statistic because the variance is underestimated. Hence, the performance of the Kelly–Rice[5] Kelly, C. and Rice, J. 1990. Monotone Smoothing with Application to Dose-Response Curves and the Assessment of Synergism. Biometrics, 46: 1071–1085. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] method is generally anti-conservative, based on the simulation findings. In addition, the overall test of synergism with χ²(r) from the modified Kelly and Rice method for larger sample sizes is better than that with χ²(1) from the modified Kelly and Rice method.     

CRITICAL VALUE APPROXIMATIONS FOR TESTS OF LINEAR REGRESSION DISTURBANCES

Two important classes of tests for non-spherical disturbances in the linear regression model involve test statistics whose null distributions and hence critical values depend on the regressors. This paper investigates the accuracy of the normal, two moment beta and four moment beta approximations to the critical values of such tests. An empirical experiment aimed at evaluating the accuracy of the approximations for a variety of tests against autocorrelation and heteroscedasticity is conducted. Overall the approximations are found to provide reasonably accurate critical values with skewness being a factor determining the degree of accuracy.     

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.     

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.     

TESTS FOR INDEPENDENCE IN THE PRESENCE OF MISSING VALUES

The locally most powerful rank test is derived for testing independence of two random variables with possible missing observations on both responses. The test statistic has a simple form and can be easily obtained. For the purpose of practical use, the asymptotic null distribution of the test statistic is also provided.     

COMPARISON OF THREE NON-PARAMETRIC TESTS FOR BIVARIATE INTERCHANGEABILITY1

Given a random sample(X₁, Y₁), …,(X_n, Y_n) from a bivariate (BV) absolutely continuous c.d.f. H (x, y), we consider rank tests for the null hypothesis of interchangeability H₀: H(x, y). Three linear rank test statistics, Wilcoxon (W_N), sum of squared ranks (SSR_N) and Savage (S_N), are described in Section 1. In Section 2, asymptotic relative efficiency (ARE) comparisons of the three types of tests are made for Morgenstern (Plackett, 1965) and Moran (1969)BV alternatives with marginal distributions satisfying G(x) = F(x/θ) for some θ≠ 1. Both gamma and lognormal marginal distributions are used.     

A METHOD FOR DETERMINING THE ASYMPTOTIC EFFICIENCY OF SOME SEQUENTIAL PROBABILITY RATIO TESTS

This paper gives a method for decomposing many sequential probability ratio tests into smaller independent components called “modules”. A function of some characteristics of modules can be used to determine the asymptotically most efficient of a set of statistical tests in which a, the probability of type I error equals β, the probability of type II error. The same test is seen also to give the asymptotically most efficient of the corresponding set of tests in which a is not equal to β. The “module” method is used to give an explanation for the super-efficiency of the play-the-winner and play-the-loser rules in two-sample binomial sampling. An example showing how complex cases can be analysed numerically using this method is also given.     

BURR CRITICAL VALUE APPROXIMATIONS FOR TESTS OF AUTOCORRELATION AND HETEROSCEDASTICITY

The paper evaluates the accuracy of Burr approximations of critical values and p-values for test a of autocorrelation and heteroscedasticity in the linear regression model.