Testing for spherical symmetry via the empirical characteristic function

Kolmogorov–Smirnov-type and Cramér–von Mises-type goodness-of-fit tests are proposed for the null hypothesis that the distribution of a random vector X is spherically symmetric. The test statistics utilize the fact that X has a spherical symmetric distribution if, and only if, the characteristic function of X is constant over surfaces of spheres centred at the origin. Both tests come in convenient forms that are straightforwardly applicable with the computer. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. Since both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of X, a conditional Monte Carlo procedure is used to actually carry out the tests. Results on the behaviour of the test in finite-samples are included along with a real-data example.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Maximum Average-Power (MAP) Tests

The objective of this article is to propose and study frequentist tests that have maximum average power, averaging with respect to some specified weight function. First, some relationships between these tests, called maximum average-power (MAP) tests, and most powerful or uniformly most powerful tests are presented. Second, the existence of a maximum average-power test for any hypothesis testing problem is shown. Third, an MAP test for any hypothesis testing problem with a simple null hypothesis is constructed, including some interesting classical examples. Fourth, an MAP test for a hypothesis testing problem with a composite null hypothesis is discussed. From any one-parameter exponential family, a commonly used UMPU test is shown to be also an MAP test with respect to a rich class of weight functions. Finally, some remarks are given to conclude the article.     

Goodness-of-Fit Tests for the Gamma Distribution Based on the Empirical Laplace Transform

We propose a class of goodness-of-fit tests for the gamma distribution that utilizes the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity, the test statistics approach limit values related to the first non zero component of Neyman's smooth test for the gamma law. The new tests are compared with other omnibus tests for the gamma distribution.     

A modified Mack–Wolfe test for the umbrella alternative problem

ABSTRACT

The Mack–Wolfe test is the most frequently used non parametric procedure for the umbrella alternative problem. In this paper, modifications of the Mack–Wolfe test are proposed for both known peak and unknown peak umbrellas. The exact mean and variance of the proposed tests in the null hypothesis are also derived. We compare these tests with some of the existing tests in terms of the type I error rate and power. In addition, a real data example is presented.     

On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.     

A weighted cramer-von mises statistic derived from a decomposition of the anderson-darling statistic

Two goodness of fit statistics with asymmetric weight function are derived from a decomposition of the Anderson-Darling statistic, For each one, the asymptotic null distribution is found for a simple null hypothesis and some upper percentties are calculated. The asymptotic power of the tests are obtained for some contiguous alternatives around a normal null hypothesis. The tests allow the user to choose to which tail to give more weight and it is intended to be used for that purpose. Therefore it should be not considered as a competitor of the classical goodness of fit tests.     

Modified Wilcoxon–Mann–Whitney Test and Power Against Strong Null

The Wilcoxon–Mann–Whitney (WMW) test is a popular rank-based two-sample testing procedure for the strong null hypothesis that the two samples come from the same distribution. A modified WMW test, the Fligner–Policello (FP) test, has been proposed for comparing the medians of two populations. A fact that may be under-appreciated among some practitioners is that the FP test can also be used to test the strong null like the WMW. In this article, we compare the power of the WMW and FP tests for testing the strong null. Our results show that neither test is uniformly better than the other and that there can be substantial differences in power between the two choices. We propose a new, modified WMW test that combines the WMW and FP tests. Monte Carlo studies show that the combined test has good power compared to either the WMW and FP test. We provide a fast implementation of the proposed test in an open-source software. Supplementary materials for this article are available online.     

Nonparametric Two‐Sample Tests of the Marginal Mark Distribution with Censored Marks

Occasionally, investigators collect auxiliary marks at the time of failure in a clinical study. Because the failure event may be censored at the end of the follow‐up period, these marked endpoints are subject to induced censoring. We propose two new families of two‐sample tests for the null hypothesis of no difference in mark‐scale distribution that allows for arbitrary associations between mark and time. One family of proposed tests is a nonparametric extension of an existing semi‐parametric linear test of the same null hypothesis while a second family of tests is based on novel marked rank processes. Simulation studies indicate that the proposed tests have the desired size and possess adequate statistical power to reject the null hypothesis under a simple change of location in the marginal mark distribution. When the marginal mark distribution has heavy tails, the proposed rank‐based tests can be nearly twice as powerful as linear tests.     

Inferences on the lognormal mean for complete samples

In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be confusing if it is remembered that this implies that their equivalent confidence intervals are conservative. No implication of optimality is intended for these conservative procedures. It is emphasized that these are direct statistical inference methods for the lognormal mean, as opposed to the already well-known methods for the parameters of the underlying normal distribution. The method currently employed in MIL-STD-471A for statistical demonstration of MTTR is analyzed and compared to the new method in terms of asymptotic relative efficiency. The new methods are also compared to the optimum methods derived by Land (1971, 1973).     

Robust tests for the common principal components model

When dealing with several populations, the common principal components (CPC) model assumes equal principal axes but different variances along them. In this paper, a robust log-likelihood ratio statistic allowing to test the null hypothesis of a CPC model versus no restrictions on the scatter matrices is introduced. The proposal plugs into the classical log-likelihood ratio statistic robust scatter estimators. Using the same idea, a robust log-likelihood ratio and a robust Wald-type statistic for testing proportionality against a CPC model are considered. Their asymptotic distributions under the null hypothesis and their partial influence functions are derived. A small simulation study allows to compare the behavior of the classical and robust tests, under normal and contaminated data.     

Testing for changes using permutations of U-statistics

The critical values for various tests based on U-statistics to detect a possible change are obtained through permutations of the observations. We obtain the same approximations for the permutated U-statistics under the no change null hypothesis as well as under the exactly one change alternative. The results are used to show that the simulated critical values are asymptotically valid under the null hypothesis and the tests reject with the probability tending to one under the alternative.     

Fourier–type tests involving martingale difference processes

We develop testing procedures which detect if the observed time series is a martingale difference sequence. Furthermore, tests are developed that detect change–points in the conditional expectation of the series given its past. The test statistics are formulated following the approach of Fourier–type conditional expectations first proposed by Bierens (1982 Bierens, H. J. (1982). Consistent model speci?cation tests. J. Econometr. 20:105–134.[Crossref], [Web of Science ®] , [Google Scholar]) and have the advantage of computational simplicity. The limit behavior of the test statistics is investigated under the null hypothesis as well as under alternatives. Since the asymptotic null distribution contains unknown parameters, a bootstrap procedure is proposed in order to actually perform the test. The performance of the bootstrap version of the test is compared in finite samples with other methods for the same problem. A real–data application is also included.     

The Use of Permutation Tests for Variance Components in Linear Mixed Models

Standard asymptotic chi-square distribution of the likelihood ratio and score statistics under the null hypothesis does not hold when the parameter value is on the boundary of the parameter space. In mixed models it is of interest to test for a zero random effect variance component. Some available tests for the variance component are reviewed and a new test within the permutation framework is presented. The power and significance level of the different tests are investigated by means of a Monte Carlo simulation study. The proposed test has a significance level closer to the nominal one and it is more powerful.     

Extensions of empirical likelihood and chi-squared-based tests for ordered alternatives

Several methods for comparing k populations have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. We focus on two important alternative hypotheses: monotone and umbrella ordering. Two new families of test statistics are proposed, including two known tests, as well as two new powerful tests under monotone ordering. Furthermore, these families are adapted for testing umbrella ordering. We compare some members of the families with respect to power and Type I errors under different simulation scenarios. Finally, the methods are illustrated in several applications to real data.     

The Use of Posterior Predictive P-Values in Testing Goodness-of-Fit

In this article, we introduce two goodness-of-fit tests for testing normality through the concept of the posterior predictive p-value. The discrepancy variables selected are the Kolmogorov-Smirnov (KS) and Berk-Jones (BJ) statistics and the prior chosen is Jeffreys’ prior. The constructed posterior predictive p-values are shown to be distributed independently of the unknown parameters under the null hypothesis, thus they can be taken as the test statistics. It emerges from the simulation that the new tests are more powerful than the corresponding classical tests against most of the alternatives concerned.     

On Goodness-of-Fit Tests for Aalen's Additive Risk Model

Abstract.  In this paper we propose goodness-of-fit tests for Aalen's additive risk model. They are based on test statistics the asymptotic distributions of which are determined under both the null and alternative hypotheses. The results are derived using martingale techniques for counting processes. An important feature of these tests is that they can be adjusted to particular alternatives. One of the alternatives we consider is Cox's multiplicative risk model. It is perhaps remarkable that such a test needs no estimate of the baseline hazard in the Cox model. We present simulation studies which give an impression of the performance of the proposed tests. In addition, the tests are applied to real data sets.     

Convergence of posterior odds

The problem of approximating an interval null or imprecise hypothesis test by a point null or precise hypothesis test under a Bayesian framework is considered. In the literature, some of the methods for solving this problem have used the Bayes factor for testing a point null and justified it as an approximation to the interval null. However, many authors recommend evaluating tests through the posterior odds, a Bayesian measure of evidence against the null hypothesis. It is of interest then to determine whether similar results hold when using the posterior odds as the primary measure of evidence. For the prior distributions under which the approximation holds with respect to the Bayes factor, it is shown that the posterior odds for testing the point null hypothesis does not approximate the posterior odds for testing the interval null hypothesis. In fact, in order to obtain convergence of the posterior odds, a number of restrictive conditions need to be placed on the prior structure. Furthermore, under a non-symmetrical prior setup, neither the Bayes factor nor the posterior odds for testing the imprecise hypothesis converges to the Bayes factor or posterior odds respectively for testing the precise hypothesis. To rectify this dilemma, it is shown that constraints need to be placed on the priors. In both situations, the class of priors constructed to ensure convergence of the posterior odds are not practically useful, thus questioning, from a Bayesian perspective, the appropriateness of point null testing in a problem better represented by an interval null. The theories developed are also applied to an epidemiological data set from White et al. (Can. Veterinary J. 30 (1989) 147–149.) in order to illustrate and study priors for which the point null hypothesis test approximates the interval null hypothesis test. AMS Classification: Primary 62F15; Secondary 62A15     

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behavior under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behavior of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. We also provide a numerical comparison with other competing tests.     

An extension of Monte Carlo hypothesis tests

There are many hypothesis testing settings in which one can calculate a “reasonable” test statistic, but in which the null distribution of the statistic is unknown or completely intractable. Fortunately, in many such situations, it is possible to simulate values of the test statistic under the null hypothesis, in which case one can conduct a Monte Carlo test. A difficulty however arises in that Monte Carlo tests, as they are currently structured, are applicable only if ties cannot occur among the values of the test statistics. There is a frequently occurring scenario in which there are lots of ties, namely that in which the null distribution of the test statistic has a (single) point mass. It turns out that one can modify the current form of Monte Carlo tests so as to accommodate such settings. Developing this modification leads to an intriguing identity involving the binomial probability function and its derivatives. In this article, we will briefly explain the modified procedure, discuss simulation studies which demonstrate its efficacy, and provide a proof of the identity referred to above.