Original Test Statistics Research Articles

The Cauchy Combination Test under Arbitrary Dependence Structures

Combining individual p-values to perform an overall test is often encountered in statistical applications. The Cauchy combination test (CCT) (Journal of the American Statistical Association, 2020, 115, 393–402) is a powerful and computationally efficient approach to integrate individual p-values under arbitrary dependence structures for sparse signals. We revisit this test to additionally show that (i) the tail probability of the CCT can be approximated just as well when more relaxed assumptions are imposed on individual p-values compared to those of the original test statistics; (ii) such assumptions are satisfied by six popular copula distributions; and (iii) the power of the CCT is no less than that of the minimum p-value test when the number of p-values goes to infinity under some regularity conditions. These findings are confirmed by both simulations and applications in two real datasets, thus, further broadening the theory and applications of the CCT.

Smoothed Nested Testing on Directed Acyclic Graphs.

We consider the problem of multiple hypothesis testing when there is a logical nested structure to the hypotheses. When one hypothesis is nested inside another, the outer hypothesis must be false if the inner hypothesis is false. We model the nested structure as a directed acyclic graph, including chain and tree graphs as special cases. Each node in the graph is a hypothesis and rejecting a node requires also rejecting all of its ancestors. We propose a general framework for adjusting node-level test statistics using the known logical constraints. Within this framework, we study a smoothing procedure that combines each node with all of its descendants to form a more powerful statistic. We prove a broad class of smoothing strategies can be used with existing selection procedures to control the familywise error rate, false discovery exceedance rate, or false discovery rate, so long as the original test statistics are independent under the null. When the null statistics are not independent but are derived from positively-correlated normal observations, we prove control for all three error rates when the smoothing method is arithmetic averaging of the observations. Simulations and an application to a real biology dataset demonstrate that smoothing leads to substantial power gains.

Hypothesis testing for high-dimensional time series via self-normalization

Self-normalization has attracted considerable attention in the recent literature of time series analysis, but its scope of applicability has been limited to low-/fixed-dimensional parameters for low-dimensional time series. In this article, we propose a new formulation of self-normalization for inference about the mean of high-dimensional stationary processes. Our original test statistic is a U-statistic with a trimming parameter to remove the bias caused by weak dependence. Under the framework of nonlinear causal processes, we show the asymptotic normality of our U-statistic with the convergence rate dependent upon the order of the Frobenius norm of the long-run covariance matrix. The self-normalized test statistic is then constructed on the basis of recursive subsampled U-statistics and its limiting null distribution is shown to be a functional of time-changed Brownian motion, which differs from the pivotal limit used in the low-dimensional setting. An interesting phenomenon associated with self-normalization is that it works in the high-dimensional context even if the convergence rate of original test statistic is unknown. We also present applications to testing for bandedness of the covariance matrix and testing for white noise for high-dimensional stationary time series and compare the finite sample performance with existing methods in simulation studies. At the root of our theoretical arguments, we extend the martingale approximation to the high-dimensional setting, which could be of independent theoretical interest.

Resampling in the frequency domain of time series to determine critical values for change-point tests

We study a model with an abrupt change in the mean and dependent errors that form a linear process. Dierent kinds of statistics are considered, such as maximum-type statistics (particularly dierent CUSUM procedures) or sum-type statistics. Approximations of the critical values for change-point tests are obtained through permutation methods in the frequency domain. The theoretical results show that the original test statistics and their corresponding frequency permutation counterparts follow the same distributional asymptotics. The main step in the proof is to obtain limit theorems for the corresponding rank statistics and then deduce the permutation asymptotics conditionally on the given data. Some simulation studies illustrate that the permutation tests usually behave better than the original tests if performance is measured by the - and -errors respectively.

Block permutation principles for the change analysis of dependent data

We study an AMOC model with an abrupt change in the mean and dependent errors that form a linear process. Different kinds of statistics are considered, such as maximum-type statistics (particularly different CUSUM procedures) or sum-type statistics. Approximations of the critical values for change-point tests are obtained through permutation methods. The theoretical results show that the original test statistics and their corresponding block permutation counterparts follow the same distributional asymptotics. The main step in the proof is to obtain limit theorems for the corresponding rank statistics and then use laws of large numbers to obtain the permutation asymptotics conditionally on the given data. Some simulation studies illustrate that the permutation tests usually behave better than the original tests if performance is measured by the α - and β -errors, respectively.

Permutation principles for the change analysis of stochastic processes under strong invariance

Approximations of the critical values for change-point tests are obtained through permutation methods. Both, abrupt and gradual changes are studied in models of possibly dependent observations satisfying a strong invariance principle, as well as gradual changes in an i.i.d. model. The theoretical results show that the original test statistics and their corresponding permutation counterparts follow the same distributional asymptotics. Some simulation studies illustrate that the permutation tests behave better than the original tests if performance is measured by the α - and β -error, respectively.

Testing poolability in a system of dynamic regressions with nonspherical disturbances

This study analyzes the testing of cross-equation restrictions within a set of regression equations. Through Monte Carlo experiments we examine the actual size of various asymptotic procedures for testing the poolability hypothesis, i.e., equal slope vectors across individual equations. Regression models with both lagged dependent variable regressors and nonspherical disturbances are considered. In these models we find that the performance in finite samples of classical asymptotic test procedures using critical values from either ℱ or χ2 approximations is often rather poor. However, employing the original test statistics with bootstrapped critical values leads to much more accurate inference in finite samples. In an empirical analysis of panel data on GDP growth and unemployment rates in OECD countries it is shown that classical asymptotic tests and bootstrap procedures may lead to conflicting test outcomes.

How do bootstrap and permutation tests work?

Resampling methods are frequently used in practice to adjust critical values of nonparametric tests. In the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented. Under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points. From this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality. The results leads to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses. They are conditionally correct iff a central limit theorem holds for the original test statistic. We prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs. Special cases are the m (n) out of k (n) bootstrap, the weighted bootstrap, the wild bootstrap and all kinds of permutation statistics. The program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail. These results are used to compare power functions of conditional resampling tests and their unconditional counterparts. The proof uses the method of random scores for permutation type statistics.

A formula to improve score test statistics

Bartlett corrections are widely used to improve the chi–squared approximation to the log–likelihood ratio statistic. Bartlett-type corrections are a recent extension of Bartlett corrections to statistics other than the log–likelihood ratio. For any test statistic having a chi-squared distribution asymptotically, Cordeiro and Ferrari (1991) derived a polynomial transformation in the original test statistic whose distribution function to O(n -1)is chi–squared, but it does not preserve monotonicity, where n is the sample size. We present a formula for the Bartlett-type correction which is equivalent to order n -1 to this polynomial and is a monotone transformation of the original statistic. Our formula has some advantages for obtaining improved score test statistics. The formula is investigated by means of simulation and is compared with both the polynomial transfondation and the original score statistic.

Rank Covariance Methods for the Analysis of Survival Data

A procedure for comparing survival times between several groups of patients through rank analysis of covariance was introduced by WOOLSON and LACHENBRUCH (1983). It is a modification of Quade' rank analysis of covariance procedure (1967) and can be used for the analysis of right-censored data. In this paper, two additional modifications of Quade' original test statistic are proposed and compared to the original modification introduced by Woolson and Lachenbruch. These statistics are compared to one another and to the score test from Cox' proportional hazards model by way of a limited Monte Carlo study. One of the statistics, QR2, is recommended for general use for the rank analysis of covariance of right-censored survivorship data.

Tests for Autocorrelation and Randomness in Multiple Time Series

Abstract There exist several tests for autocorrelation and randomness in multiple time series. Unfortunately, the exact distributions of these statistics are unknown and the asymptotic distributions that are known do not provide adequate approximation to the exact ones in small samples. In this article, the test statistics are modified. Asymptotically, these modified statistics are equivalent to their original counterparts; however, it is found that the asymptotic distributions of these statistics provide adequate approximation to the exact ones in relatively small samples and possibly when the time series are nonnormal. The adequacy of the approximations is examined by simulation experiments. The original test statistics are based on sample lag cross-covariances, autocovariances, cross-correlations, and autocorrelations standardized by their asymptotic means and covariances. The modified statistics are obtained when the asymptotic means and covariances in the standardization are replaced by the exact means and covariances. The expressions for these moments are derived on the assumption that the time series is Gaussian. These moments (both asymptotic and exact) involve nuisance parameters. In constructing the test statistics, these nuisance parameters are replaced by their sample counterparts, which are consistent estimates of the parameters.

Tests for Autocorrelation and Randomness in Multiple Time Series

Abstract There exist several tests for autocorrelation and randomness in multiple time series. Unfortunately, the exact distributions of these statistics are unknown and the asymptotic distributions that are known do not provide adequate approximation to the exact ones in small samples. In this article, the test statistics are modified. Asymptotically, these modified statistics are equivalent to their original counterparts; however, it is found that the asymptotic distributions of these statistics provide adequate approximation to the exact ones in relatively small samples and possibly when the time series are nonnormal. The adequacy of the approximations is examined by simulation experiments. The original test statistics are based on sample lag cross-covariances, autocovariances, cross-correlations, and autocorrelations standardized by their asymptotic means and covariances. The modified statistics are obtained when the asymptotic means and covariances in the standardization are replaced by the exact mea...

An efficiency property of linear nonparametric tests on trend

The efficiency of nonparametric trend tests is considered in the randomized block design: For a large number of rank test statistics the projection on the subset of linear rank statistics defines an even uniformly more efficient rank test than the original test statistics.