Higher Criticism, or second-level significance testing, is a multiple comparisons concept mentioned in passing by Tukey (1976). It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested to compare the fraction of observed significances at a given α-level to the expected fraction under the joint null, in fact he suggested to standardize the difference of the two quantities and form a z-score; the resulting z-score tests the significance of the body of significance tests. We consider a generalization, where we maximize this z-score over a range of significance levels 0 < α ≤ α0. We are able to show that the resulting Higher Criticism statistic is effective at resolving a very subtle testing problem: testing whether n normal means are all zero versus the alternative that a small fraction is nonzero. The subtlety of this ‘sparse normal means ’ testing problem can be seen from work of Ingster (1999) and Jin (2002), who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so

With modern technology, massive data can easily be collected in a form of multiple sets of curves. New statistical challenge includes testing whether there is any statistically significant difference among these sets of curves. In this paper, we propose some new tests for comparing two groups of curves based on the adaptive Neyman test and the wavelet thresholding techniques introduced in Fan (1996). We demonstrate that these tests inherit the properties outlined in Fan (1996) and they are simple and powerful for detecting di erences between two sets of curves. We then further generalize the idea to compare multiple sets of curves, resulting in an adaptive high-dimensional analysis of variance, called HANOVA. These newly developed techniques are illustrated by using a dataset on pizza commercial where observations are curves and an analysis of cornea topography in ophthalmology where images of individuals are observed. A simulation example is also presented to illustrate the power of the adaptive Neyman test.

Several new tests are proposed for examining the adequacy of a family of parametric models against large nonparametric alternatives. These tests formally check if the bias vector of residuals from parametric ts is negligible by using the adaptive Neyman test and other methods. The testing procedures formalize the traditional model diagnostic tools based on residual plots. We examine the rates of contiguous alternatives that can be detected consistently by the adaptive Neyman test. Applications of the procedures to the partially linear models are thoroughly discussed. Our simulation studies show that the new testing procedures are indeed powerful and omnibus. The power of the proposed tests is comparable to the F-test statistic even in the situations where F -test is known to be suitable and can be far more powerful than the F-test statistic in other situations. An application to testing linear models versus additive models are discussed.

Abstract: We study so–called augmented GARCH sequences, which include many sub-models of considerable interest, such as polynomial and exponential GARCH. To model the returns of speculative assets, it is particularly important to understand the behaviour of the squares of the observations. The main aim of this paper is to present a strong approximation for the sum of the squares. This will be achieved by an approximation of the volatility sequence with a sequence of blockwise independent random variables. Furthermore, we derive a necessary and sufficient condition for the existence of a unique (strictly) stationary solution of the general augmented GARCH equations. Also, necessary and sufficient conditions for the finiteness of moments are provided.

Proofs are given of the limiting null distributions of the statistics of Berk and Jones (1979) and of Einmahl and McKeague (2002). 1.

Abstract. We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray-Knight theorems), and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed. Research partially supported by NSF grant DMS-9700721. 1 1. Introduction. Let Bt be a continuous function of t, let t0, x0, β1, β2 ∈ R, and consider the ordinary differential equation dXt

A unified family of goodness-of-fit tests based on φ-divergences is introduced and studied. The new family of test statistics Sn(s) includes both the supremum version of the Anderson–Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47–59] as special cases (s = 2 and s = 1, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk

Abstract: Berk and Jones (1979) introduced a goodness of fit test statistic Rn which is the supremum of pointwise likelihood ratio tests for testing H0: F (x) =F0(x) versusH1: F (x) � = F0(x). They showed that their statistic does not always converge almost surely to a constant under alternatives F, and, in fact that there exists an alternative distribution function F such Rn →d sup t>0 N(t)/t where N is a standard Poisson process on [0, ∞). We call the particular distribution function F which leads to this limiting Poisson behavior the Poisson boundary distribution function for Rn. Weinvestigate Poisson boundaries for weighted Kolmogorov statistics Dn(ψ) for various weight functions ψ and comment briefly on the history of results concerning Bahadur efficiency of these statistics. One result of note is that the logarithmically weighted Kolmogorov statistic of Groeneboom and Shorack (1981) has the same Poisson boundary as the statistic of Berk and Jones (1979). Keywords and phrases: Bahadur efficiency, Berk–Jones statistic, consistency, fixed alternatives, goodness of fit, Kolmogorov statistic, Poisson process,

Let {Xi,i = 1,2,...} be i.i.d. standard gaussian variables. Let Sn = X1 +... + Xn be the sequence of partial sums and

Abstract: We study a CUSUM–type monitoring scheme designed to sequentially detect changes in the regression parameter of an underlying linear model. The test statistic used is based on recursive residuals. Main aim of this paper is to derive the limiting extreme value distribution under the null hypothesis of structural stability. The model assumptions are flexible enough to include very general classes of error sequences such as augmented GARCH(1,1) processes. The result is underlined by an illustrative simulation study.