Tailor-made tests for goodness of fit to semiparametric hypotheses

We introduce a new framework for constructing tests of general semiparametric hypotheses which have nontrivial power on the $n^{-1/2}$ scale in every direction, and can be tailored to put substantial power on alternatives of importance. The approach is based on combining test statistics based on stochastic processes of score statistics with bootstrap critical values.Comment: Published at http://dx.doi.org/10.1214/009053606000000137 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Constrained-Realization Monte-Carlo Method for Hypothesis Testing

We compare two theoretically distinct approaches to generating artificial (or ``surrogate'') data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single ``best'' model to the original data, and then to generate surrogate data sets that are ``typical realizations'' of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data. The comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal. A statistic is ``pivotal'' if its distribution is the same for all processes consistent with the null hypothesis. The typical-realization method requires that the discriminating statistic satisfy this property. The constrained-realization approach, on the other hand, does not share this requirement, and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic.Comment: 19 pages, single spaced, all in one postscript file, figs included. Uncompressed .ps file is 425kB (sorry, it's over the 300kB recommendation). Also available on the WWW at http://nis-www.lanl.gov/~jt/Papers/ To appear in Physica

On the power of conditional independence testing under model-X

For testing conditional independence (CI) of a response Y and a predictor X given covariates Z, the recently introduced model-X (MX) framework has been the subject of active methodological research, especially in the context of MX knockoffs and their successful application to genome-wide association studies. In this paper, we study the power of MX CI tests, yielding quantitative explanations for empirically observed phenomena and novel insights to guide the design of MX methodology. We show that any valid MX CI test must also be valid conditionally on Y and Z; this conditioning allows us to reformulate the problem as testing a point null hypothesis involving the conditional distribution of X. The Neyman-Pearson lemma then implies that the conditional randomization test (CRT) based on a likelihood statistic is the most powerful MX CI test against a point alternative. We also obtain a related optimality result for MX knockoffs. Switching to an asymptotic framework with arbitrarily growing covariate dimension, we derive an expression for the limiting power of the CRT against local semiparametric alternatives in terms of the prediction error of the machine learning algorithm on which its test statistic is based. Finally, we exhibit a resampling-free test with uniform asymptotic Type-I error control under the assumption that only the first two moments of X given Z are known, a significant relaxation of the MX assumption

Generalized Wald-type Tests based on Minimum Density Power Divergence Estimators

In testing of hypothesis the robustness of the tests is an important concern. Generally, the maximum likelihood based tests are most efficient under standard regularity conditions, but they are highly non-robust even under small deviations from the assumed conditions. In this paper we have proposed generalized Wald-type tests based on minimum density power divergence estimators for parametric hypotheses. This method avoids the use of nonparametric density estimation and the bandwidth selection. The trade-off between efficiency and robustness is controlled by a tuning parameter $\beta$. The asymptotic distributions of the test statistics are chi-square with appropriate degrees of freedom. The performance of the proposed tests are explored through simulations and real data analysis.Comment: 26 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:1403.033