Multiple outlier test for upper outliers in an exponential sample

In this paper, a test statistic for testing upper outliers with a slippage alternative, in an exponential sample is proposed. No tables for critical values are required as they can be calculated easily for any sample size. A simulation study is also carried out to compare the performance of the test with the maximum likelihood ratio test and other existing tests.     

Some nonparametric methods for changepoint problems

A general model for changepoint problems is discussed from a nonparametric viewpoint. The test statistics introduced are based on Cramér-von Mises functionals of certain processes and are shown to converge in distribution to corresponding Gaussian functionals (under the assumption of no change in distribution, H₀). We also demonstrate how the distribution of the limiting Gaussian functionals may be tabulated. Finally, properties of the tests under the alternative hypothesis of exactly one changepoint occurring are studied, and some examples are given.     

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

Union-Intersection testing for outliers in multivariate normal data

The union-intersection approach to multivariate test construction is used to develop an alternative to Wilks' likelihood ratio test statistic for testing for two or more outliers in multivariate normal data. It is shown that critical values of both statistics are poorly approximated by Bonferroni bounds. Simulated critical values are presented for both statistics for significance levels 1% and 5%, for sample sizes 10(5)30, 40, 50, 75 and 100 for 2, 3, 4 and 5 dimensions. A power comparison of the two tests in the slippage of the mean model for generating outliers indicates that the union-intersection test is the more powerful when the slippages are close to collinear. Although Wilks' test remains the preference for general use, the union-intersection test could be valuable when such special structure in the data is suspected.     

On lachin'S formulae for sample sizes of survival tests

Lachin [1981] and Lachin and Foulkes [1986] consider two groups of identically independently exponentially distributed random variables and four models of data sampling. The test problem they treat is to decide whether the two distributions are identical (null-hypothesis H⁰) or not (alternative hypothesis H¹). Basing the test on maximum-likelihood estimators and their asymptotic normal densities they obtain formulae for the group sizes necessary to yield asymptotic tests with guaranteed power under a prescribed level for specified hypotheses. It is intuitively reasonable to expect the sizes decrease the more the hypotheses differ. It the distance betwen H⁰ and H¹ is measured by the difference of the exponential parameters this assumption time or the deviation of the exponential parameter ratio from unity is the measure larger distances between the hypotheses do not necessarily lead to smaller sample sizes.     

A class of k-sample distribution-free tests for location against ordered alternatives

This paper introduces a new class of distribution-free tests for testing the homogeneity of several location parameters against ordered alternatives. The proposed class of test statistics is based on a linear combination of two-sample U-statistics based on subsample extremes. The mean and variance of the test statistic are obtained under the null hypothesis as well as under the sequence of local alternatives. The optimal weights are also determined. It is shown via Pitman ARE comparisons that the proposed class of test statistics performs better than its competitor tests in case of heavy-tailed and long-tailed distributions     

A tool for systematically comparing the power of tests for normality

In this article, we describe a new approach to compare the power of different tests for normality. This approach provides the researcher with a practical tool for evaluating which test at their disposal is the most appropriate for their sampling problem. Using the Johnson systems of distribution, we estimate the power of a test for normality for any mean, variance, skewness, and kurtosis. Using this characterization and an innovative graphical representation, we validate our method by comparing three well-known tests for normality: the Pearson χ² test, the Kolmogorov–Smirnov test, and the D'Agostino–Pearson K ² test. We obtain such comparison for a broad range of skewness, kurtosis, and sample sizes. We demonstrate that the D'Agostino–Pearson test gives greater power than the others against most of the alternative distributions and at most sample sizes. We also find that the Pearson χ² test gives greater power than Kolmogorov–Smirnov against most of the alternative distributions for sample sizes between 18 and 330.     

The Slippage Problem for the Circular Normal Distribution

The slippage problem occurs when an unspecified observation in a given random sample is from a distribution other than that for all the remaining observations. This paper studies the problem in terms of the 'slip' in the mean direction of a circular normal distribution. The slippage problem is first treated as a multiple decision problem with a prior which is invariant under the permutations of the hypotheses. The probabilities of accepting the various hypotheses for the Bayes rule with respect to this prior are explicitly obtained. The likelihood ratio tests for this slippage problem, for the cases when the mean directions are both known and unknown, are shown to be easily computable. The tests are illustrated through two well-known datasets. The performances of a range of tests are compared using extensive simulation.     

On the use of priors in goodness‐of‐fit tests

Priors are introduced into goodness‐of‐fit tests, both for unknown parameters in the tested distribution and on the alternative density. Neyman–Pearson theory leads to the test with the highest expected power. To make the test practical, we seek priors that make it likely a priori that the power will be larger than the level of the test but not too close to one. As a result, priors are sample size dependent. We explore this procedure in particular for priors that are defined via a Gaussian process approximation for the logarithm of the alternative density. In the case of testing for the uniform distribution, we show that the optimal test is of the U‐statistic type and establish limiting distributions for the optimal test statistic, both under the null hypothesis and averaged over the alternative hypotheses. The optimal test statistic is shown to be of the Cramér–von Mises type for specific choices of the Gaussian process involved. The methodology when parameters in the tested distribution are unknown is discussed and illustrated in the case of testing for the von Mises distribution. The Canadian Journal of Statistics 47: 560–579; 2019 © 2019 Statistical Society of Canada     

Discordancy Tests Based on the Likelihood Ratio for the Bipolar Watson Distribution on the Hypersphere

As the Watson distribution is frequently used for modeling axial data, it is important to investigate the existence of possible outliers in samples from this distribution. Then, we develop for the bipolar Watson distribution defined on the hypersphere, some tests of discordancy of an outlier or several outliers en bloc based on the likelihood ratio, supposing an alternative model of contamination of slippage type. We evaluate the performance of these tests of discordancy of an outlier and we also compare some tests of discordancy of an outlier available for this distribution.     

Testing order restricted hypotheses with proportional hazards

Inferences for survival curves based on right censored continuous or grouped data are studied. Testing homogeneity with an ordered restricted alternative and testing the order restriction as the null hypothesis are considered. Under a proportional hazards model, the ordering on the survival curves corresponds to an ordering on the regression coefficients. Approximate likelihood methods are obtained by applying order restricted procedures to the estimates of the regression coefficients. Ordered analogues to the log rank test which are based on the score statistics are considered also. Chi-bar-squared distributions, which have been studied extensively, are shown to provide reasonable approximations to the null distributions of these tests statistics. Using Monte Carlo techniques, the powers of these two types of tests are compared with those that are available in the literature.     

The power of the circular cone test: A noncentral chi-bar-squared distribution

Pincus (1975) derived the null distribution of the likelihood-ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood-ratio test statistic has a chi-bar-squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood-ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi-bar-squared” distribution. To our knowledge, this is the first order-restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear-model setting. That is, the distribution of the likelihood-ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.     

On optimal tests for separate hypotheses and conditional probability integral transformations

Consider the problem of testing the composite null hypothesis that a random sample X₁,…,X_n is from a parent which is a member of a particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions. It is shown here that in many cases a uniformly most powerful similar (UMPS) test exists for this problem, and, moreover, that this test is equivalent to a uniformly most powerful invariant (UMPI) test. It is also seen in the method of proof used that the UMPS test statistic Is a function of the statistics U₁,…,U_n?k obtained by the conditional probability integral transformations (CPIT), and thus that no Information Is lost by these transformations, It is also shown that these optimal tests have power that is a nonotone function of the null hypothesis class of distributions, so that, for example, if one additional parameter for the distribution is assumed known, then the power of the test can not lecrease. It Is shown that the statistics U₁, …, U_n?k are independent of the complete sufficient statistic, and that these statistics have important invariance properties. Two examples at given. The UMPS tests for testing the two-parameter uniform family against the two-parameter exponential family, and for testing one truncation parameter distribution against another one are derived.     

Comparing Survival Times for Treatments with Those for a Control Under Proportional Hazards

Inferences for survival curves based on right censored data are studied for situations in which it is believed that the treatments have survival times at least as large as the control or at least as small as the control. Testing homogeneity with the appropriate order restricted alternative and testing the order restriction as the null hypothesis are considered. Under a proportional hazards model, the ordering on the survival curves corresponds to an ordering on the regression coefficients. Approximate likelihood methods, which are obtained by applying order restricted procedures to the estimates of the regression coefficients, and ordered analogues to the log rank test, which are based on the score statistics, are considered. Mau's (1988) test, which does not require proportional hazards, is extended to this ordering on the survival curves. Using Monte Carlo techniques, the type I error rates are found to be close to the nominal level and the powers of these tests are compared. Other order restrictions on the survival curves are discussed briefly.     

Skewness and mixtures of normal distributions

The effect of skewness on hypothesis tests for the existence of a mixture of univariate and bivariate normal distributions is examined through a Monte Carlo study. A likelihood ratio test based on results of the simultaneous estimation of skewness parameters, derived from power transformations, with mixture parameters is proposed. This procedure detects the difference between inherent distributional skewness and the apparent skewness which is a manifestation of the mixture of several distributions. The properties of this test are explored through a simulation study.     

Goodness of fit tests for discrete distributions

Some alternative procedures for testing goodness of fit in discrete distributions are discussed here.. These procedures are based on the probability generating functions.. The methods considered are quite general, being applicable in multidimensional situations., The strength of the tests lies in that no ambiguity as to classification of the data arises.. Hov-ever, some difficulties in the proposed procedures are also pointed out.     

On the robustness of the power function of the one-sample test for the negative exponential distribution

The robustness of the power function of the standard one-sample parametric test for the mean of the negative exponential distribution is examined. The main form of departure from the exponential assumption is a mixture of negative exponential components although an alternative Gamma distribution is also examined. It is found that the test is sensitive to these departures although the effect of mixtures with short tails is less dramatic than those with long tails.     

An alternative to the kolmogorov-smirnov test for goodness of fit

A test is proposed which requires a better fit in the extremes of a distribution than the Kolmogorov-Smirnov test for H₀. not to be rejected. Critical values are calculated for sample sizes up to 100, and approximate critical values are found for larger samples. The power of the test is obtained for a number of distributions, and it is shown that the test is more powerful than some existing tests for a wide range of cases     

A k-sample slippage test for location parameter

Lewis (1972) has proposed a test for the slippage of the location parameter of k- populations when the direction of the slippage is unknown. In this paper, an extension of this test is proposed. The distribution of the test statistic is obtained under the null hypothesis of no slippage and the power of the test is compared with that of another competitive test proposed by Conover (1968).     

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.