Generalized Wald-type tests based on minimum density power divergence estimators

In testing of hypothesis, the robustness of the tests is an important concern. Generally, the maximum likelihood-based tests are most efficient under standard regularity conditions, but they are highly non-robust even under small deviations from the assumed conditions. In this paper, we have proposed generalized Wald-type tests based on minimum density power divergence estimators for parametric hypotheses. This method avoids the use of nonparametric density estimation and the bandwidth selection. The trade-off between efficiency and robustness is controlled by a tuning parameter β. The asymptotic distributions of the test statistics are chi-square with appropriate degrees of freedom. The performance of the proposed tests is explored through simulations and real data analysis.     

For what use are tests of hypotheses and tests of significance

Tests of sharp null hypotheses, although frequently computed, are.rarely appropriate as the major end product of statistical analyses, except possibly in some, areas of the natural sciences. But procedures closely akin to tests, although often less formal, are needed in almost every investigation in which exploratory data analysis is used to help to decide upon the statistical model appropriate for the final analysis. The term “diagnostic check”has been suggested by Box and Jenkins for these procedures. Traditional statistical tests often suggest useful diagnostic checks -and this, in my view, is what tests are mainly good for-but visual examination and interpretation of data plots are often equally important. Biere is also much to be gained by the development of new diagnostic checks, and testing theory may be useful as one guide to this development.     

Robust I-sample analysis of means type randomization tests for variances

Four Analysis of Means (ANOM) type randomization tests for testing the equality of I variances are presented. Randomization techniques for testing statistical hypotheses can be used when parametric tests are inappropriate. Suppose that I independent samples have been collected. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples "in a bowl" forms the combined sample. Drawing samples "from the bowl" forms a shuffle. Shuffles can be made with replacement (bootstrap shuffling) or without replacement (permutation shuffling). The tests that are presented offer two advantages. They are robust to non-normality and they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. The decision chart facilitates easy assessment of both statistical and practical significance. A Monte Carlo study is used to identify robust randomization tests that exhibit excellent power when compared to other robust tests.     

A comparison of alternative unit root tests

In this paper we evaluate the performance of three methods for testing the existence of a unit root in a time series, when the models under consideration in the null hypothesis do not display autocorrelation in the error term. In such cases, simple versions of the Dickey-Fuller test should be used as the most appropriate ones instead of the known augmented Dickey-Fuller or Phillips-Perron tests. Through Monte Carlo simulations we show that, apart from a few cases, testing the existence of a unit root we obtain actual type I error and power very close to their nominal levels. Additionally, when the random walk null hypothesis is true, by gradually increasing the sample size, we observe that p-values for the drift in the unrestricted model fluctuate at low levels with small variance and the Durbin-Watson (DW) statistic is approaching 2 in both the unrestricted and restricted models. If, however, the null hypothesis of a random walk is false, taking a larger sample, the DW statistic in the restricted model starts to deviate from 2 while in the unrestricted model it continues to approach 2. It is also shown that the probability not to reject that the errors are uncorrelated, when they are indeed not correlated, is higher when the DW test is applied at 1% nominal level of significance.     

The Use of Noncentral F Approximations for Calculation of Power and Sample Size

Approximations to the noncentral F distribution yield surprisingly accurate results for power and sample size problems arising from linear hypotheses about normal random variables. The approximations are easy to use with a desk (or hand-held) calculator that computes cumulative F probabilities. These approximations are particularly advantageous for testing the hypothesis that differences among the means are small against the alternative that the differences are large.     

Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.     

On the procedure of hodges and lehmann for testing composite hypotheses of the mean of a normal distribution with unknown variance

In 1954 Hodges and Lehmann gave a test procedure for testing the hypothesis that the mean of an identically independently normally distributed random sample with unknown variance is contained within a certain interval [μ1, μ2]. The test is similar on the boundary of the zero-hypothesis and superior in power to the composite t-test usually applied to this problem. However Hodges and Lehmann could prove the unbiasedness of their test only for the special case that the sample consists of two elements. From numerical computations they guessed that unbiasedness would be valid for arbitrary sample sizes. This question is discussed here and partially answered.     

Nonparametric repeated significance tests for some analysis of covariance models

For a general class of nonparametric analysis of covariance problems (with stochastic covariates), some repeated significance testing procedures are developed. These procedures rest on the construction of suitable rank order statistics based on the partial sequence of sample sizes and allow for a monitoring of experimentation with the objective of a possible early termination of experimentation. The basic theory is based on the weak convergence of certain stochastic processes relating to the rank order statistics. Various properties of the proposed tests are discussed.     

Rank tests for independence based on partially right-censored pairs

Linear rank procedures are developed for testing independence with right-censored matched pairs. It is assumed that censoring Is Independent of the random variables under study. The test statistics are derived as score statistics (Hajek and Sidak, 1967) based on the probability of the generalised rank vectors (Prentice, 1978). Applications to survival data analysis are also discussed.     

The problem of outlier tests in sample surveys

Outliers can occur as readily in samples from the finite populations (e.g. in sample surveys) as in samples from infinite populations. However, in the vast literature on outliers there is almost no mention of outlier tests for data from sample surveys. We examine the behaviour of some standard outlier test statistics for infinite populations when these are applied to finite populations, examining their properties by extensive simulation studies. Some anomalous results are obtained Nsuggesting a fundamental difficulty in testing outliers for the finite population case.     

STEPWISE TESTS WHEN THE TEST STATISTICS ARE INDEPENDENT

Consider the problem of simultaneously testing a nonhierarchical finite family of hypotheses based on independent test statistics. A general stepwise test is defined, of which the well known step-down and step-up tests are special cases. The step-up test is shown to dominate the other stepwise tests, including the step-down test, for situations of practical importance. When testing against two-sided alternatives, it is pointed out that if the step-up test is augmented to include directional decisions then the augmented step-up test controls the type I and III familywise error jointly at the original level q. The definition of the adjusted p values for the step-up test is justified. The results are illustrated by a numerical example.     

Robust empirical Bayes tests for continuous distributions

In this paper, we study the empirical Bayes two-action problem under linear loss function. Upper bounds on the regret of empirical Bayes testing rules are investigated. Previous results on this problem construct empirical Bayes tests using kernel type estimators of nonparametric functionals. Further, they have assumed specific forms, such as the continuous one-parameter exponential family for {F_θ:θΩ}, for the family of distributions of the observations. In this paper, we present a new general approach of establishing upper bounds (in terms of rate of convergence) of empirical Bayes tests for this problem. Our results are given for any family of continuous distributions and apply to empirical Bayes tests based on any type of nonparametric method of functional estimation. We show that our bounds are very sharp in the sense that they reduce to existing optimal or nearly optimal rates of convergence when applied to specific families of distributions.     

Empirical behavior of some tests for normality

The behavior of a range of tests assessing the normality of a sequence of independent and identically distributed random variables is investigated.An examination of the empirical significance level of the tests is undertaken for different sample sizes. The empirical power associated with these tests is also calculated under some alternative distributions.     

Predictive discordancy tests for exponential observations

Using a Bayesian approach, unconditional and conditional predictive testing procedures are proposed for the detection of discordant observations for the translated exponential distribution in the presence of censored observations.     

Small sample equivalence tests for exponentiality

We consider small sample equivalence tests for exponentialy. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require much larger sample sizes, in comparison with classical “difference tests,” to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of small sample equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. We perform small sample simulation studies to assess the bias of our various tests and show that all of the Bayes posteriors we consider are integrable. Our simulation studies show that the saddlepoint-approximated UMPU method performs remarkably well for small sample sizes and is the only method that consistently exhibits an empirical significance level close to the nominal 5% level.     

Deriving a lower bound for the proportion of perceivers in replicated difference tests by means of multiple test theory

Analyzing repeated difference tests aims in significance testing for differences as well as in estimating the mean discrimination ability of the consumers. In addition to the average success probability, the proportion of consumers that may detect the difference between two products and therefore account for any increase of this probability is of interest. While some authors address the first two goals, for the latter one only an estimator directly linked to the average probability seems to be used. However, this may lead to unreasonable results. Therefore we propose a new approach based on multiple test theory. We define a suitable set of hypotheses that is closed under intersection. From this, we derive a series of hypotheses that may be sequentially tested while the overall significance level will not be violated. By means of this procedure we may determine a minimal number of assessors that must have perceived the difference between the products at least once in a while. From this, we can find a conservative lower bound for the proportion of perceivers within the consumers. In several examples, we give some insight into the properties of this new method and show that the knowledge about this lower bound might indeed be valuable for the investigator. Finally, an adaption of this approach for similarity tests will be proposed.     

A comparison of uniformity tests

Problems of goodness-of-fit to a given distribution can usually be reduced to test uniformity. The uniform distribution appears due to natural random events or due to the application of methods for transforming samples from any other distribution to the samples with values uniformly distributed in the interval (0, 1). Thus, one can solve the problem of testing if a sample comes from a given distribution by testing whether its transformed sample is distributed according to the uniform distribution. For this reason, the methods of testing for goodness-of-fit to a uniform distribution have been widely investigated. In this paper, a comparative power analysis of a selected set of statistics is performed in order to give suggestions on which one to use for testing uniformity against the families of alternatives proposed by Stephens [Stephens, M.A., 1974, EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730–737.]. Definition and some relevant features of the considered test statistics are given in section 1. Implemented numerical processes to calculate percentage points of every considered statistic are described in section 2. Finally, a Monte Carlo simulation experiment has been carried out to fulfill the mentioned target of this paper.     

A re-appraisal of tests for normality

Tests for normality can be divided into two groups - those based upon a function of the empirical distribution function and those based upon a function of the original observations. The latter group of statistics test spherical symmetry and not necessarily normality. If the distribution is completely specified then the first group can be used to test for ‘spherical’ normality. However, if the distribution is incompletely specified and F‘‘x_i - x’/s’ is used these test statistics also test sphericity rather than normality. A Monte Carlo study was conducted for the completely specified case, to investigate the sensitivity of the distance tests to departures from normality when the alternative distributions are non-normal spherically symmetric laws. A “new” test statistic is proposed for testing a completely specified normal distribution     

Likelihood-based tests in zero-inflated power series models

We address the issue of performing testing inference in the class of zero-inflated power series models. These models provide a straightforward way of modelling count data and have been widely used in practical situations. The likelihood ratio, Wald and score statistics provide the basis for testing the parameter of inflation of zeros in this class of models. In this paper, in addition to the well-known test statistics, we also consider the recently proposed gradient statistic. We conduct Monte Carlo simulation experiments to evaluate the finite-sample performance of these tests for testing the parameter of inflation of zeros. The numerical results show that the new gradient test we propose is more reliable in finite samples than the usual likelihood ratio, Wald and score tests. An empirical application to real data is considered for illustrative purposes.     

Smooth goodness of fit tests for the weibull distribution with singly censored data

The smooth goodness of fit tests are generalized to singly censored data and applied to the problem of testing Weibull (or extreme value) fit. Smooth tests, Pearson-type tests, and the spacings tests proposed by Mann, Schemer, and Fertig (1973) are compared on the basis of local asymptotic relative efficiency with respect to the asymptotic best test against generalized gamma alternatives, The smooth test of order one Is found to be most efficient for the generalized gamma alternatives.