Asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable. More importantly, asymptotic analysis can also provide insight into the inference process itself, suggesting what information is available and how this information may be extracted. The development of likelihood inference over the past twenty-some years provides an illustration of the interplay between techniques of approximation and statistical theory.

Siegmund and Worsley considered the problem of testing for a signal with unknown location and scale in a Gaussian random field defined on~$\mathbb{R}^N$. The test statistic was the maximum of a Gaussian random field in an $(N+1)$-dimensional "scale space," N dimensions for location and one dimension for the scale of a smoothing kernel. Siegmund and Worsley used two methods, one involving the expected Euler characteristic of the excursion set and the other involving the volume of tubes, to derive an approximate null distribution. The purpose of this paper is to extend the scale space result to the rotation space random field when N=2, where the maximum is taken over all rotations of the filter as well as scales. We apply this result to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

This paper studies classes of nonstationary processes, such as warped processes and frequency-modulated processes, that result from the deformation of stationary processes. Estimating deformations can often provide important information about an underlying physical phenomenon. A computational harmonic analysis viewpoint shows that the deformed autocovariance satisfies a transport equation at small scales, with a velocity proportional to a deformation gradient. We derive an estimator of the deformation from a single realization of the deformed process, with a proof of consistency under appropriate assumptions.

We discuss the mathematical properties of a recently introduced method for computing geometric structures in a digital image without any a priori information. This method is based on a basic principle of perception which we call the Helmholtz principle. According to this principle, an observed geometric structure is perceptually "meaningful" if the expectation of its number of occurrences (in other words, its number of false alarms, NF) is very small in a random image. It is "maximal meaningful" if its NF is minimal among the meaningful structures of the same kind which it contains or is contained in. This definition meets the gestalt theory requirement that parts of a whole are not perceived. We explain by large-deviation estimates why this definition leads to an a priori knowledge-free method, compatible with phenomenology. We state a principle according to which maximal structures do not meet. We prove this principle in the large-deviations framework in the case of alignments in a digital image. We show why these results make maximal meaningful structures computable and display several applications.

Confidence bands for regression curves and their first p derivatives are obtained via local pth order polynomial estimation. The method allows for multiparameter local likelihood estimation as well as other unbiased estimating equations. As an alternative to the confidence bands obtained by asymptotic distribution theory, we also study smoothed bootstrap confidence bands. Simulations illustrate the finite sample properties of the methodology.

The asymptotic behavior of jackknife estimators and jackknife variance estimators is investigated for nonsmooth functions of the sample mean vector. An application of jackknifing a suitable estimator of the intrinsic diversity profile is also presented.

We propose two Bayesian bootstrap extensions, the binomial and Poisson forms, for proportional hazards models. The binomial form Bayesian bootstrap is the limit of the posterior distribution with a beta process prior as the amount of the prior information vanishes, and thus can be considered as a default nonparametric Bayesian analysis. It is also the same as Lo's Bayesian bootstrap for censored data when covariates are absent. The Poisson form Bayesian bootstrap is equivalent to the Bayesian analysis with Cox's profile likelihood. When the baseline distribution is discrete, thus when the data set has many ties, simulation study suggests that the binomial form Bayesian bootstrap performs better than standard frequentist procedures in the frequentist sense. An advantage of the proposed Bayesian bootstrap procedures over the standard Bayesian analysis is conceptual and computational simplicity. Finally, it is shown that both Bayesian bootstrap posteriors are asymptotically equivalent to the sampling distribution of the maximum likelihood estimator.

A bootstrap methodology for the periodogram of a stationary process is proposed which is based on a combination of a time domain parametric and a frequency domain nonparametric bootstrap. The parametric fit is used to generate periodogram ordinates that imitate the essential features of the data and the weak dependence structure of the periodogram while a nonparametric (kernel-based) correction is applied in order to catch features not represented by the parametric fit. The asymptotic theory developed shows validity of the proposed bootstrap procedure for a large class of periodogram statistics. For important classes of stochastic processes, validity of the new procedure is also established for periodogram statistics not captured by existing frequency domain bootstrap methods based on independent periodogram replicates.

A large part of the theory of extreme value index estimation is developed for positive extreme value indices. The best-known estimator of a positive extreme value index is probably the Hill estimator. This estimator belongs to the category of moment estimators, but can also be interpreted as a quasi-maximum likelihood estimator. It has been generalized to a kernel-type estimator, but this kernel-type estimator can, similarly to the Hill estimator, only be used for the estimation of positive extreme value indices. In the present paper, we introduce kernel-type estimators which can be used for estimating the extreme value index over the whole (positive and negative) range. We present a number of results on their distributional behavior and compare their performance with the performance of other estimators, such as moment-type estimators for the whole range and the quasi-maximum likelihood estimator, based on the generalized Pareto distribution. We also discuss an automatic bandwidth selection method and introduce a kernel estimator for a second-order parameter, controlling the speed of convergence.

We prove that when the distribution of a stochastic process in C[0,1]$ is in the domain of attraction of a max-stable process, then natural estimators for the extreme-value index (which is now a continuous function) and for the mean measure of the limiting Poisson process are consistent in the appropriate topologies. The ultimate goal, estimating probabilities of small (failure) sets, will be considered later.

Multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously. One multiple hypothesis testing error measure is the false discovery rate (FDR), which is loosely defined to be the expected proportion of false positives among all significant hypotheses. The FDR is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests. In this work, we introduce a modified version of the FDR called the "positive false discoveryrate" (pFDR). We discuss the advantages and disadvantages of the pFDR and investigate its statistical properties. When assuming the test statistics follow a mixture distribution, we show that the pFDR can be written as a Bayesian posterior probability and can be connected to classification theory. These properties remain asymptotically true under fairly general conditions, even under certain forms of dependence. Also, a new quantity called the "$q$-value" is introduced and investigated, which is a natural "Bayesian posterior p-value," or rather the pFDR analogue of the p-value.

The notion of efficient test for a Euclidean parameter in a semiparametric model was introduced by Stein [Proc. Third Berkeley Symp. Math. Statist. Probab. 1 (1956) 187-195]. Such tests are locally most powerful for a wide class of infinite-dimensional nuisance parameters. The first formal application of this notion to a suitably parametrized two-sample problem was provided by Hájek [Ann. Math. Statist. 33 (1962) 1124-1147]. However, this and subsequent solutions appear to be not well-suited for practical applications. This article aims to show that an adaptive two-sample test introduced recently by Janic-Wróblewska and Ledwina [Scand. J. Statist. 27 (2000) 281-297] is locally most powerful under a more realistic setting.

Let (x,z) be a pair of observable random vectors. We construct a new "smoothed" empirical likelihood-based test for the hypothesis $\E\{ g(z,\break \theta)|x \} = 0$ w.p.1, where g is a vector of known functions and $\theta$ an unknown finite-dimensional parameter. We show that the test statistic is asymptotically normal under the null hypothesis and derive its asymptotic distribution under a sequence of local alternatives. Furthermore, the test is shown to possess an optimality property in large samples. Simulation evidence suggests that it also behaves well in small samples.