Recent developments towards optimality in multiple hypothesis testing

There are many different notions of optimality even in testing a single hypothesis. In the multiple testing area, the number of possibilities is very much greater. The paper first will describe multiplicity issues that arise in tests involving a single parameter, and will describe a new optimality result in that context. Although the example given is of minimal practical importance, it illustrates the crucial dependence of optimality on the precise specification of the testing problem. The paper then will discuss the types of expanded optimality criteria that are being considered when hypotheses involve multiple parameters, will note a few new optimality results, and will give selected theoretical references relevant to optimality considerations under these expanded criteria.Comment: Published at http://dx.doi.org/10.1214/074921706000000374 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Viscoplastic constitutive relationships with dependence on thermomechanical history

Experimental evidence of thermomechanical history dependence in the cyclic hardening behavior of some common high-temperature structural alloys is presented with special emphasis on dynamic metallurgical changes. The inadequacy of formulating nonisothermal constitutive equations solely on the basis of isothermal testing is discussed. A representation of thermoviscoplasticity is proposed that qualitatively accounts for the observed hereditary behavior. This is achieved by formulating the scalar evolutionary equation in an established viscoplasticity theory to reflect thermomechanical path dependence. To assess the importance of accounting for thermomechanical history dependence in practical structural analyses, two qualitative models are specified: (1) formulated as if based entirely on isothermal information; (2) to reflect thermomechanical path dependence using the proposed thermoviscoplastic representation. Predictions of the two models are compared and the impact the calculated differences in deformation behavior may have on subsequent lifetime predictions is discussed

An overview of the goodness-of-fit test problem for copulas

We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc, are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized

FDR Control in the Presence of an Unknown Correlation Structure

The false discovery rate (FDR, Benjamini and Hochberg 1995) is a powerful approach to multiple testing. However, the original approach developed by Benjamini and Hochberg (1995) applies only to independent tests. Yekutieli (2008) showed that a modification of the Benjamini-Hochberg (BH) approach can be used in the presence of dependent tests and labelled his procedure as separate subsets BH (ssBH). However, Yekutieli (2008) left the practical specification of the subsets of p values largely unresolved. In this paper we propose a modification of the ssBH procedure based on a selection of the subsets that guarantees that the dependence properties needed to control the FDR are satisfied. We label this new procedure as the separate pairs BH (spBH). An extensive Monte Carlo analysis is presented that compares the properties of the BH and spBH procedures.Multiple testing, False discovery rate, Copulas

A One-Sample Test for Normality with Kernel Methods

We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure

Entropy-Based Tests for Complex Dependence in Economic and Financial Time Series with the R Package tseriesEntropy

Testing for complex serial dependence in economic and financial time series is a crucial task that bears many practical implications. However, the linear paradigm remains pervasive among practitioners as the autocorrelation function, because, despite its known shortcomings, it is still one of the most used tools in time series analysis. We propose a solution to the problem, by introducing the R package tseriesEntropy, dedicated to testing for serial/cross dependence and nonlinear serial dependence in time series, based on the entropy metric S-rho. The package implements tests for both continuous and categorical data. The nonparametric tests, based on S-rho, rely on minimal assumptions and have also been shown to be powerful for small sample sizes. The measure can be used as a nonlinear auto/cross-dependence function, both as an exploratory tool, or as a diagnostic measure, if computed on the residuals from a fitted model. Different null hypotheses of either independence or linear dependence can be tested by means of resampling methods, backed up by a sound theoretical background. We showcase our methods on a panel of commodity price time series. The results hint at the presence of a complex dependence in the conditional mean, together with conditional heteroskedasticity, and indicate the need for an appropriate nonlinear specification

Large-Scale Multiple Testing of Composite Null Hypotheses Under Heteroskedasticity

Heteroskedasticity poses several methodological challenges in designing valid and powerful procedures for simultaneous testing of composite null hypotheses. In particular, the conventional practice of standardizing or re-scaling heteroskedastic test statistics in this setting may severely affect the power of the underlying multiple testing procedure. Additionally, when the inferential parameter of interest is correlated with the variance of the test statistic, methods that ignore this dependence may fail to control the type I error at the desired level. We propose a new Heteroskedasticity Adjusted Multiple Testing (HAMT) procedure that avoids data reduction by standardization, and directly incorporates the side information from the variances into the testing procedure. Our approach relies on an improved nonparametric empirical Bayes deconvolution estimator that offers a practical strategy for capturing the dependence between the inferential parameter of interest and the variance of the test statistic. We develop theory to show that HAMT is asymptotically valid and optimal for FDR control. Simulation results demonstrate that HAMT outperforms existing procedures with substantial power gain across many settings at the same FDR level. The method is illustrated on an application involving the detection of engaged users on a mobile game app