Hypothesis testing results often rely on simple, yet important assumptions about the behaviour of the distribution of p-values under the null and the alternative. We examine tests for one dimensional parameters of interest that converge to a normal distribution, possibly in the presence of nuisance parameters, and characterize the distribution of the p-values using techniques from the higher order asymptotics literature. We show that commonly held beliefs regarding the distribution of p-values are misleading when the variance and location of the test statistic are not well-calibrated or when the higher order cumulants of the test statistic are not negligible. Corrected tests are proposed and are shown to perform better than their first order counterparts in certain settings.
In this paper, we compare the powers of several discrete goodness-of-fit test statistics considered by Steele and Chaseling [10] under the null hypothesis of a 'zig-zag' distribution. The results suggest that the Discrete Kolmogorov-Smirnov test statistic is generally more powerful for the decreasing trend alternative. The Pearson Chi-Square statistic is generally more powerful for the increasing, unimodal, leptokurtic, platykurtic and bath-tub shaped alternatives. Finally, both the Nominal Kolmogorov- Smirnov and the Pearson Chi-Square test statistic are generally more powerful for the bimoda…
The logical and practical difficulties associated with research interpretation using P values and null hypothesis significance testing have been extensively documented. This paper describes an alternative, likelihood-based approach to P-value interpretation. The P-value and sample size of a research study are used to derive a likelihood function with a single parameter, the estimated population effect size, and the method of maximum likelihood estimation is used to calculate the most likely effect size. Comparison of the likelihood of the most likely effect size and the likelihood of the minim…
In this paper, we consider the hypothesis testing of correlation between two $m$-uniform hypergraphs on $n$ unlabelled nodes. Under the null hypothesis, the hypergraphs are independent, while under the alternative hypothesis, the hyperdges have the same marginal distributions as in the null hypothesis but are correlated after some unknown node permutation. We focus on two scenarios: the hypergraphs are generated from the Gaussian-Wigner model and the dense Erdös-Rényi model. We derive the sharp information-theoretic testing threshold. Above the threshold, there exists a powerful test to distin…
We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^2$-distances - the new…
A non parametric method based on the empirical likelihood is proposed for detecting the change in the coefficients of high-dimensional linear model where the number of model variables may increase as the sample size increases. This amounts to testing the null hypothesis of no change against the alternative of one change in the regression coefficients. Based on the theoretical asymptotic behaviour of the empirical likelihood ratio statistic, we propose, for a fixed design, a simpler test statistic, easier to use in practice. The asymptotic normality of the proposed test statistic under the null…
Markov regime switching models have been used in numerous empirical studies in economics and finance. However, the asymptotic distribution of the likelihood ratio test statistic for testing the number of regimes in Markov regime switching models has been an unresolved problem. This paper derives the asymptotic distribution of the likelihood ratio test statistic for testing the null hypothesis of $M_0$ regimes against the alternative hypothesis of $M_0 + 1$ regimes for any $M_0 \geq 1$ both under the null hypothesis and under local alternatives. We show that the contiguous alternatives converge…
Null Hypothesis Significance Testing (NHST) has long been of central importance to psychology as a science, guiding theory development and underlying the application of evidence-based intervention and decision-making. Recent years, however, have seen growing awareness of serious problems with NHST as it is typically used; this awareness has led to proposals to limit the use of NHST techniques, to abandon these techniques and move to alternative statistical approaches, or even to ban the use of NHST entirely. These proposals are premature, because the observed problems with NHST arise as a cons…
Local smoothing testing that is based on multivariate nonparametric regression estimation is one of the main model checking methodologies in the literature. However, relevant tests suffer from the typical curse of dimensionality resulting in slow convergence rates to their limits under the null hypotheses and less deviation from the null under alternatives. This problem leads tests to not well maintain the significance level and to be less sensitive to alternatives. In this paper, a dimension-reduction model-adaptive test is proposed for generalized linear models. The test behaves like a local…
In this paper, in order to test whether changes have occurred in a nonlinear parametric regression, we propose a nonparametric method based on the empirical likelihood. Firstly, we test the null hypothesis of no-change against the alternative of one change in the regression parameters. Under null hypothesis, the consistency and the convergence rate of the regression parameter estimators are proved. The asymptotic distribution of the test statistic under the null hypothesis is obtained, which allows to find the asymptotic critical value. On the other hand, we prove that the proposed test statis…
The bivariate Poisson distribution is commonly used to model bivariate count data. In this paper we study a goodness-of-fit test for this distribution. We also provide a review of the existing tests for the bivariate Poisson distribution, and its multivariate extension. The proposed test is consistent against any fixed alternative. It is also able to detect local alternatives converging to the null at the rate $n^{-\frac{1}{2}}$. The bootstrap can be employed to consistently estimate the null distribution of the test statistic. Through a simulation study we investigated the goodness of the boo…
We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location $θ$ and allows to derive locally asymptotically most powerful tests under specified $θ$. The second one, that addresses the Fisher-von Mises-Langevin (FvML) case, relates to the unspecified-$θ$ problem and shows that the high-dimensional Rayleigh test is locally asymptotically most powerful invariant. Under mild assumption…
A Transferability Criterion for Null-Optimized Variance Reduction in Cumulant-Based Error-Independence Testing
Cherkasy State Business College（切尔卡瑟州立商业学院）；State Scientific Research Institute of Armament and Military Equipment Testing and Certification（国家武器装备测试与认证科学研究院）；Uzhhorod National University（乌日霍罗德国立大学）
The normalized maximum likelihood (NML) is a recent penalized likelihood that has properties that justify defining the amount of discrimination information (DI) in the data supporting an alternative hypothesis over a null hypothesis as the logarithm of an NML ratio, namely, the alternative hypothesis NML divided by the null hypothesis NML. The resulting DI, like the Bayes factor but unlike the p-value, measures the strength of evidence for an alternative hypothesis over a null hypothesis such that the probability of misleading evidence vanishes asymptotically under weak regularity conditions a…
Classical multiple testing theory prescribes the null distribution, which is often a too stringent assumption for nowadays large scale experiments. This paper presents theoretical foundations to understand the limitations caused by ignoring the null distribution, and how it can be properly learned from the (same) data-set, when possible. We explore this issue in the case where the null distributions are Gaussian with an unknown rescaling parameters (mean and variance) and the alternative distribution is let arbitrary. While an oracle procedure in that case is the Benjamini Hochberg procedure a…
In model checking for regressions, nonparametric estimation-based tests usually have tractable limiting null distributions and are sensitive to oscillating alternative models, but suffer from the curse of dimensionality. In contrast, empirical process-based tests can, at the fastest possible rate, detect local alternatives distinct from the null model, but is less sensitive to oscillating alternative models and with intractable limiting null distributions. It has long been an issue on how to construct a test that can fully inherit the merits of these two types of tests and avoid the shortcomin…
In this paper a new class of uniformity tests is proposed. It is shown that those tests are applicable to the cases of any simple null hypothesis as well as for the composite null hypothesis of rectangular distributions on arbitrary support. The asymptotic properties of test statistics are examined. The tests are compared with some standard and some recent uniformity tests. For each test the Bahadur efficiencies against some common local alternatives are calculated. A class of locally optimal alternatives is found for each proposed test. The power study is also provided. Some applications in t…
In this paper we propose a new test of heteroscedasticity for parametric regression models and partial linear regression models in high dimensional settings. When the dimension of covariates is large, existing tests of heteroscedasticity perform badly due to the \curse of dimensionality". To attack this problem, we construct a test of heteroscedasticity by using a projection-based empirical process. We study the asymptotic properties of the test statistic under the null hypothesis and alternative hypotheses. It is shown that the test can detect local alternatives departure from the null hypoth…
Null Hypothesis Significance Testing (NHST) has long been central to the scientific project, guiding theory development and supporting evidence-based intervention and decision-making. Recent years, however, have seen growing awareness of serious problems with NHST as it is typically used, and hence to proposals to limit the use of NHST techniques, to abandon these techniques and move to alternative statistical approaches, or even to ban the use of NHST entirely. These proposals are premature, because the observed problems with NHST all arise as a consequence of a contingent and in many cases i…
$P$-values that are derived from continuously distributed test statistics are typically uniformly distributed on $(0,1)$ under least favorable parameter configurations (LFCs) in the null hypothesis. Conservativeness of a $p$-value $P$ (meaning that $P$ is under the null hypothesis stochastically larger than a random variable which is uniformly distributed on $(0,1)$) can occur if the test statistic from which $P$ is derived is discrete, or if the true parameter value under the null is not an LFC. To deal with both of these sources of conservativeness, we present two approaches utilizing random…
In this note we derive the exact null distribution for the test statistic proposed by Anis and Mitra (2011) for testing exponentiality against NBUE alternatives. As a special case, we obtain the exact null distribution for the test statistic proposed by Hollander and Proschan (1975). Selected critical values for different size are tabulated for these two statistics. Some remarks concerning the benefits of using the exact distribution are made.
We study statistical models for one-dimensional diffusions which are recurrent null. A first parameter in the drift is the principal one, and determines regular varying rates of convergence for the score and the information process. A finite number of other parameters, of secondary importance, introduces additional flexibility for the modelization of the drift, and does not perturb the null recurrent behaviour. Under time-continuous observation we obtain local asymptotic mixed normality (LAMN), state a local asymptotic minimax bound, and specify asymptotically optimal estimators.
This paper is concerned with testing normality in a Hilbert space based on the maximum mean discrepancy. Specifically, we discuss the behavior of the test from two standpoints: asymptotics and practical aspects. Asymptotic normality of the test under a fixed alternative hypothesis is developed, which implies that the test has consistency. Asymptotic distribution of the test under a sequence of local alternatives is also derived, from which asymptotic null distribution of the test is obtained. A concrete expression for the integral kernel associated with the null distribution is derived under t…
We propose a procedure to decide between the null hypothesis of (strict) stationarity and the alternative of non-stationarity, in the context of a Random Coefficient AutoRegression (RCAR). The procedure is based on randomising a diagnostic which diverges to positive infinity under the null, and drifts to zero under the alternative. Thence, we propose a randomised test which can be used directly and - building on it - a decision rule to discern between the null and the alternative. The procedure can be applied under very general circumstances: albeit developed for an RCAR model, it can be used …
We propose and analyze nonparametric tests of the null hypothesis that a function belongs to a specified parametric family. The tests are based on BIC approximations, π_{BIC}, to the posterior probability of the null model, and may be carried out in either Bayesian or frequentist fashion. We obtain results on the asymptotic distribution of π_{BIC} under both the null hypothesis and local alternatives. One version of π_{BIC}, call it π_{BIC}^*, uses a class of models that are orthogonal to each other and growing in number without bound as sample size, n, tends to infinity. We show that \sqrtn(1…
A new test is proposed for the weak white noise null hypothesis. The test is based on a new automatic choice of the order for a Box-Pierce or Hong test statistic. The test uses Lobato (2001) or Kuan and Lee (2006) HAC critical values. The data-driven order choice is tailored to detect a new class of alternatives with autocorrelation coefficients which can be $o(n^{-1/2})$ provided there are enough of them. A simulation experiment illustrates the good behavior of the test both under the weak white noise null and the alternative.
