A TEST FOR THE NONPARAMETRIC TWO SAMPLE SCALE PROBLEM

We propose a distribution-free test for the nonparametric two sample scale problem. Unlike the other tests for this problem, we do not assume that the two distribution functions have a common median. We assume that they have a common quantile of order a (not necessarily 1/2). The test statistic is a modification of the Sukhatme statistic for the scale problem and the Wilcoxon-Mann-Whitney statistic for stochastic dominance. It is shown that the new test is uniformly more efficient (in the Pitman sense) than the Sukhatme test and has very good efficiency when compared to the Mood test.     

THE POWER AND SIZE OF NONPARAMETRIC TESTS FOR COMMON DISTRIBUTIONAL CHARACTERISTICS

This paper considers the power and size properties of some well known nonparametric linear rank tests for location and scale as well as the Kolmogorov-Smirnov omnibus test and proposed alternatives to it. Independence between some classes of linear rank tests is established facilitating their joint application. Monte Carlo study confirms the asymptotic power properties of the linear rank tests but raises concerns about their application in more general and practically relevant circumstances. It also indicates that the new omnibus tests constitute viable alternatives with superior properties to the Kolmogorov-Smirnov test in certain circumstances.     

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.     

ASYMPTOTIC EFFICIENCY OF THE BLEST‐TYPE TESTS FOR INDEPENDENCE

Blest (2000, Aust. N. Z. J. Stat. 42 , 101–111) proposed a new measure of rank correlation that is sensitive to discrepancies in the small ranks. This paper investigates the efficiency properties of non‐parametric tests for independence based on Blest's correlation coefficient and its modifications. Pitman efficiency comparisons are made with analogous tests existing in the literature. Conditions for Pitman optimality of the Blest‐type tests are established.     

ESTIMATING THE ROOT OF A NONPARAMETRIC REGRESSION FUNCTION IN A ROBUST FASHION

For a nonparametric regression model y = m(x)+e with n independent observations, we analyze a robust method of finding the root of m(x) based on an M-estimation first discussed by Härdle & Gasser (1984). It is shown here that the robustness properties (minimaxity and breakdown function) of such an estimate are quite analogous to those of an M -estimator in the simple location model, but the rate of convergence is somewhat limited due to the nonparametric nature of the problem.     

Distribution- free test for slope in linear regression model with replications

Some distribution-free methods are suggested in the paper for testing the hypothesis about the slope parameter in a one-sample linear regression model with multiple observations at each level of independent variable. Asymptotic relative efficiencies of these tests are discussed, and the tests are compared with their nonparametric competitors.     

Linear rank tests for homogeneity against ordered alternatives in anova and anocova

A unified method of constructing rank tests for homogeneity against ordered alternatives in unbalanced analysis of variance and analysis of covariance is considered. The relationship between these tests with some of the existing methods are studied. The normal theory likelihood ratio tests are also derived and the asymptotic relative efficiency comparisons, in Pitman sense, of the rank tests with respect to the likelihood ratio tests are carried out.     

ASYMPTOTIC TESTS FOR THE CONTINUOUS TIME MARKOV BRANCHING PROCESS WITH IMMIGRATION

This paper presents limit distributions for the modified score and the likelihood-ratio (LR) statistic for testing a composite hypothesis involving the split intensity and mean of the offspring distribution of the supercritical continuous time Markov branching process allowing immigration (CBPI). The immigration intensity and mean are treated as nuisance parameters.     

A REVIEW OF SYSTEMS COINTEGRATION TESTS

The literature on systems cointegration tests is reviewed and the various sets of assumptions for the asymptotic validity of the tests are compared within a general unifying framework. The comparison includes likelihood ratio tests, Lagrange multiplier and Wald type tests, lag augmentation tests, tests based on canonical correlations, the Stock-Watson tests and Bierens' nonparametric tests. Asymptotic results regarding the power of these tests and previous small sample simulation studies are discussed. Further issues and proposals in the context of systems cointegration tests are also considered briefly. New simulations are presented to compare the tests under uniform conditions. Special emphasis is given to the sensitivity of the test performance with respect to the trending properties of the DGP.     

OPTIMALITY OF NONPARAMETRIC TESTS FOR ASSOCIATION BETWEEN SURVIVAL TIME AND CONTINUOUS COVARIATES

Abstract

Recently, Chen (Chen, Z. (2000 Chen, Z. 2000. A new two-parameter lifetime distribution with bathtub-shape or increasing failure rate function. Statistics &; Probability Letters, 49: 155–161. [Crossref], [Web of Science ®] , [Google Scholar]). A new two-parameter lifetime distribution with bathtub-shape or increasing failure rate function. Statistics &; Probability Letters 49:155–161.) proposed a two-parameter model that can be used to model bathtub-shaped failure rate. Although this model has several interesting properties, it does not contain a scale parameter and hence not flexible in modeling real data. A generalized model including the scale parameter has shown to be interesting and it has the traditional Weibull distribution as an asymptotic case. In this article, a detailed analysis of this model is presented. Shapes of the density and failure rate function are studied. The asymptotic confidence intervals for the parameters are also derived from the Fisher information matrix. The likelihood ratio test is applied to test the goodness of fit of Weibull extension model. Some examples are shown to illustrate the application of the model and analysis.     

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.     

A TEST OF HOMOGENEITY AGAINST SCALE ALTERNATIVES USING SUBSAMPLE EXTREMA1

In this paper a new class of non-parametric tests for testing homogeneity of several populations against scale alternatives is proposed. For this, independent samples of fixed sizes are drawn from each population and from these samples, all possible sub-samples of the same size are drawn and their maxima and minima are computed. Using these extreme the class of tests is obtained. Tests of this type have been offered for the two-sample slippage problem by Kochar (1978). Under certain conditions, this class of tests is shown to be consistent against ‘difference in scale’ alternatives. The test has been compared with Bhapkar's V-test (1961), Deshpande's D-test (1965), Sugiura's D_rs-test (1965) and with a classical test given by Lehmann (1959, pp. 273–275). It is shown that some members of this proposed class of tests are more efficient than the first three tests in the case of uniform, Laplace and normal distributions, when the number of populations compared is small.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

A NONPARAMETRIC TEST FOR THE EQUALITY OF DEPENDENT CORRELATION COEFFICIENTS UNDER NORMALITY

ABSTRACT

A nonparametric testing method for the equality of two correlation coefficients in trivariate normal distribution, namely, one of the variables are common, is discussed. Using a permutation test, we obtain asymptotically exact solutions. The performance of this test is compared with the likelihood ratio test and a method of using the limiting distribution of correlation coefficients.     

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     

GOODNESS-OF-FIT AND DISCORDANCY TESTS FOR SAMPLES FROM THE WATSON DISTRIBUTION ON THE SPHERE

The only parametric model in current use for axial data from a rotationally symmetric bipolar or girdle distribution on the sphere is the Watson distribution. This paper develops methods for evaluating the model as a fit to data using graphical and formal goodness-of-fit tests, and tests of discordancy.     

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.     

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.     

THE BEHAVIOR OF HEGY TESTS FOR QUARTERLY TIME SERIES WITH SEASONAL MEAN SHIFTS

ABSTRACT

This paper studies the behavior of the HEGY statistics for quarterly data, for seasonal autoregressive unit roots, when the analyzed time series is deterministic seasonal stationary but exhibits a change in the seasonal pattern. We analyze also the HEGY test for the nonseasonal unit root. the data generation process being trend stationary too. Our results show that when the break magnitudes are finite, the HEGY test statistics are not asymptotically biased toward the nonrejection of the seasonal and nonseasonal unit root hypotheses. However, the finite sample power properties may be substantially affected, the behavior of the tests depending on the type of the break.