Change in the coefficients or in the mean of the innovation distribution of an INAR(p) process is a sign of disturbance that is important to detect. The methods of this paper can test for change in any one of these quantities separately, or in any collection of them. They are available in forms that make one-sided tests possible, furthermore, they can be used to test for a temporary change. The tests are based on a CUSUM process using conditional least squares estimators of the parameters. Under alternative hypotheses consistency of the tests is proved and the large sample properties of the change-point estimator are also explored.
Hypothesis testing is a key part of empirical science and multiple testing as well as the combination of evidence from several tests are continued areas of research. In this article we consider the problem of combining the results of multiple hypothesis tests to i) test global hypotheses and ii) make marginal inference while controlling the k-FWER. We propose a new family of combination tests for joint hypotheses, called the 'Too Many, Too Improbable' (TMTI) statistics, which we show through simulation to have higher power than other combination tests against many alternatives. Furthermore, we…
We consider the problem of sequentially testing a simple null hypothesis versus a composite alternative hypothesis that consists of a finite set of densities. We study sequential tests that are based on thresholding of mixture-based likelihood ratio statistics and weighted generalized likelihood ratio statistics. It is shown that both sequential tests have several asymptotic optimality properties as error probabilities go to zero. First, for any weights, they minimize the expected sample size within a constant term under every scenario in the alternative hypothesis and at least to first order …
This paper proposes a class of origin-smooth approximators of indicators underlying the sum-of-negative-part statistic for testing multiple inequalities. The need for simulation or bootstrap to obtain test critical values is thereby obviated. A simple procedure is enabled using fixed critical values. The test is shown to have correct asymptotic size in the uniform sense that supremum finite-sample rejection probability over null-restricted data distributions tends asymptotically to nominal significance level. This applies under weak assumptions allowing for estimator covariance singularity. Th…
This article gives a survey of the e-value, a statistical significance measure a.k.a. the evidence rendered by observational data, X, in support of a statistical hypothesis, H, or, the other way around, the epistemic value of H given X. The $e$-value and the accompanying FBST, the Full Bayesian Significance Test, constitute the core of a research program that was started at IME-USP, is being developed by over 20 researchers worldwide, and has, so far, been referenced by over 200 publications. The e-value and the FBST comply with the best principles of Bayesian inference, including the likeliho…
Multiple testing literature contains ample research on controlling false discoveries for hypotheses classified according to one criterion, which we refer to as one-way classified hypotheses. Although simultaneous classification of hypotheses according to two different criteria, resulting in two-way classified hypotheses, do often occur in scientific studies, no such research has taken place yet, as far as we know, under this structure. This article produces procedures, both in their oracle and data-adaptive forms, for controlling the overall false discovery rate (FDR) across all hypotheses eff…
We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size $n$ goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially…
The most popular hypothesis testing procedure, the likelihood ratio test, is known to be highly non-robust in many real situations. Basu et al. (2013a) provided an alternative robust procedure of hypothesis testing based on the density power divergence; however, although the robustness properties of the latter test were intuitively argued for by the authors together with extensive empirical substantiation of the same, no theoretical robustness properties were presented in this work. In the present paper we will consider a more general class of tests which form a superfamily of the procedures d…
The behavior of the power function of autocorrelation tests such as the Durbin-Watson test in time series regressions or the Cliff-Ord test in spatial regression models has been intensively studied in the literature. When the correlation becomes strong, Krämer (1985) (for the Durbin-Watson test) and Krämer (2005) (for the Cliff-Ord test) have shown that the power can be very low, in fact can converge to zero, under certain circumstances. Motivated by these results, Martellosio (2010) set out to build a general theory that would explain these findings. Unfortunately, Martellosio (2010) does not…
Bayes factors represent the ratio of probabilities assigned to data by competing scientific hypotheses. Drawbacks of Bayes factors are their dependence on prior specifications that define null and alternative hypotheses and difficulties encountered in their computation. To address these problems, we define Bayes factor functions (BFF) directly from common test statistics. BFFs depend on a single non-centrality parameter that can be expressed as a function of standardized effect sizes, and plots of BFFs versus effect size provide informative summaries of hypothesis tests that can be easily aggr…
We formulate nonparametric and semiparametric hypothesis testing of multivariate stationary linear time series in a unified fashion and propose new test statistics based on estimators of the spectral density matrix. The limiting distributions of these test statistics under null hypotheses are always normal distributions, and they can be implemented easily for practical use. If null hypotheses are false, as the sample size goes to infinity, they diverge to infinity and consequently are consistent tests for any alternative. The approach can be applied to various null hypotheses such as the indep…
We propose some parametric tests for ergodic diffusion-plus-noise model, which is a version of state-space modelling in statistics for stochastic diffusion equations. The test statistics are classified into three types: likelihood-ratio-type test statistic; Wald-type one; and Rao-type one. All the test statistics are constructed with quasi-likelihood-functions for local mean sequence of noised observation. We also simulate the behaviour of them for several practical hypothesis tests and check the convergence in law of test statistics under null hypotheses and consistency of the test under alte…
The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative hypotheses which are a low rank departure from the null hypothesis. For rank one alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and linear statistics used in testing.
Semiparametric models are often considered for analyzing longitudinal data for a good balance between flexibility and parsimony. In this paper, we study a class of marginal partially linear quantile models with possibly varying coefficients. The functional coefficients are estimated by basis function approximations. The estimation procedure is easy to implement, and it requires no specification of the error distributions. The asymptotic properties of the proposed estimators are established for the varying coefficients as well as for the constant coefficients. We develop rank score tests for hy…
In this paper, we consider the problem of simultaneously testing many two-sided hypotheses when rejections of null hypotheses are accompanied by claims of the direction of the alternative. The fundamental goal is to construct methods that control the mixed directional familywise error rate, which is the probability of making any type 1 or type 3 (directional) error. In particular, attention is focused on cases where the hypotheses are ordered as $H_1 , \ldots, H_n$, so that $H_{i+1}$ is tested only if $H_1 , \ldots, H_i$ have all been previously rejected. In this situation, one can control the…
For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative requires complex analytic approximations and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say $p\le 20$. On the other hand, assuming that the data dimension $p$ as well as the number $q$ of regression variables are fixed while the sample size $n$ grows, several asymptotic approximations are proposed in the literature for Wilk's $\bLa$ …
In this paper we present a test for the maximal rank of the matrix-valued volatility process in the continuous Ito semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Ito semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. Finally, we demonstrate a homoscedasticity test for the rank process.
The False Discovery Rate (FDR) is a commonly used type I error rate in multiple testing problems. It is defined as the expected False Discovery Proportion (FDP), that is, the expected fraction of false positives among rejected hypotheses. When the hypotheses are independent, the Benjamini-Hochberg procedure achieves FDR control at any pre-specified level. By construction, FDR control offers no guarantee in terms of power, or type II error. A number of alternative procedures have been developed, including plug-in procedures that aim at gaining power by incorporating an estimate of the proportio…
The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest are defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-diverge…
A simple test is proposed for examining the correctness of a given completely specified response function against unspecified general alternatives in the context of univariate regression. The usual diagnostic tools based on residuals plots are useful but heuristic. We introduce a formal statistical test supplementing the graphical analysis. Technically, the test statistic is the maximum length of the sequences of ordered (with respect to the covariate) observations that are consecutively overestimated or underestimated by the candidate regression function. Note that the testing procedure can c…
This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite-dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi-likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to ex…
