The behavior of locally most powerful tests

summary:The locally most powerful (LMP) tests of the hypothesis $H: \theta =\theta _0$ against one-sided as well as two-sided alternatives are compared with several competitive tests, as the likelihood ratio tests, the Wald-type tests and the Rao score tests, for several distribution shapes and for location, shape and vector parameters. A simulation study confirms the importance of the condition of local unbiasedness of the test, and shows that the LMP test can sometimes dominate the other tests only in a very restricted neighborhood of $H.$ Hence, we cannot recommend a universal application of the LMP tests in practice. The tests with a high Bahadur efficiency, though not exactly LMP, also seem to be good in the local sense

A class of optimal tests for symmetry based on local Edgeworth approximations

The objective of this paper is to provide, for the problem of univariate symmetry (with respect to specified or unspecified location), a concept of optimality, and to construct tests achieving such optimality. This requires embedding symmetry into adequate families of asymmetric (local) alternatives. We construct such families by considering non-Gaussian generalizations of classical first-order Edgeworth expansions indexed by a measure of skewness such that (i) location, scale and skewness play well-separated roles (diagonality of the corresponding information matrices) and (ii) the classical tests based on the Pearson--Fisher coefficient of skewness are optimal in the vicinity of Gaussian densities.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ298 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Locally most powerful sequential tests of a simple hypothesis vs one-sided alternatives

Let $X_1,X_2,...$ be a discrete-time stochastic process with a distribution $P_\theta$, $\theta\in\Theta$, where $\Theta$ is an open subset of the real line. We consider the problem of testing a simple hypothesis $H_0:$ $\theta=\theta_0$ versus a composite alternative $H_1:$ $\theta>\theta_0$, where $\theta_0\in\Theta$ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test $(\psi,\phi)$ with a (randomized) stopping rule $\psi$ and a (randomized) decision rule $\phi$ let $\alpha(\psi,\phi)$ be the type I error probability, $\dot \beta_0(\psi,\phi)$ the derivative, at $\theta=\theta_0$, of the power function, and $\mathscr N(\psi)$ an average sample number of the test $(\psi,\phi)$. Then we are concerned with the problem of maximizing $\dot \beta_0(\psi,\phi)$ in the class of all sequential tests such that $$\alpha(\psi,\phi)\leq \alpha\quad{and}\quad \mathscr N(\psi)\leq \mathscr N,$$ where $\alpha\in[0,1]$ and $\mathscr N\geq 1$ are some restrictions. It is supposed that $\mathscr N(\psi)$ is calculated under some fixed (not necessarily coinciding with one of $P_\theta$) distribution of the process $X_1,X_2...$. The structure of optimal sequential tests is characterized.Comment: 30 page

Asymptotic power of sphericity tests for high-dimensional data

This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and contiguous alternatives, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643-1697] the limiting process is degenerate, and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the limiting process is nondegenerate, and the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy-Widom-type tests is trivial (i.e., equals the asymptotic size), whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1100 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sign Tests for Long-memory Time Series

This paper proposes sign-based tests for simple and composite hypotheses on the long-memory parameter of a time series process. The tests allow for nonstationary hypothesis, such as unit root, as well as for stationary hypotheses, such as weak dependence or no integration. The proposed generalized Lagrange multiplier sign tests for simple hypotheses on the long-memory parameter are exact and locally optimal among those in their class. We also propose tests for composite hypotheses on the parameters of ARFIMA processes. The resulting tests statistics have a standard normal limiting distribution under the null hypothesis.Publicad