Markov chain sampling methods that adapt to characteristics of the distribution being sampled can be constructed using the principle that one can ample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal "slice" defined by the current vertical position, or more generally, with some update that leaves the uniform distribution over this slice invariant. Such "slice sampling" methods are easily implemented for univariate distributions, and can be used to sample from a multivariate distribution by updating each variable in turn. This approach is often easier to implement than Gibbs sampling and more efficient than simple Metropolis updates, due to the ability of slice sampling to adaptively choose the magnitude of changes made. It is therefore attractive for routine and automated use. Slice sampling methods that update all variables simultaneously are also possible. These methods can adaptively choose the magnitudes of changes made to each variable, based on the local properties of the density function. More ambitiously, such methods could potentially adapt to the dependencies between variables by constructing local quadratic approximations. Another approach is to improve sampling efficiency by suppressing random walks. This can be done for univariate slice sampling by "overrelaxation," and for multivariate slice sampling by "reflection" from the edges of the slice.

Resampling methods are frequently used in practice to adjust critical values of nonparametric tests. In the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented. Under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points. From this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality. The results leads to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses. They are conditionally correct iff a central limit theorem holds for the original test statistic. We prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs. Special cases are the m (n) out of k (n) bootstrap, the weighted bootstrap, the wild bootstrap and all kinds of permutation statistics. The program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail. These results are used to compare power functions of conditional resampling tests and their unconditional counterparts. The proof uses the method of random scores for permutation type statistics.

This paper describes the large sample properties of the likelihood ratio test statistic (LRTS) when the parameters characterizing the true null distribution are not unique. It is well known that the classical asymptotic theory for the likelihood ratio test does not apply to such problems and the LRTS may not have the typical chi-squared type limiting distribution. This paper establishes a general quadratic approximation of the log-likelihood ratio function in a Hellinger neighborhood of the true density which is valid with or without loss of identifiability of the true distribution. Under suitable conditions, the asymptotic null distribution of the LRTS under loss of identifiability can be obtained by maximizing the quadratic form. These results extend the work of Chernoff and Le Cam. In particular, applications to testing the number of mixture components in finite mixture models are discussed.

This paper discusses the behavior of the maximum likelihood estimator (MLE), in the case that the true parameter cannot be identified uniquely. Among many statistical models with unidentifiability, neural network models are the main concern of this paper. It is known in some models with unidentifiability that the asymptotics of the likelihood ratio of the MLE has an unusually larger order. Using the framework of locally conic models put forth by Dacunha-Castelle and Gassiat as a generalization of Hartigan's idea, a useful sufficient condition of such larger orders is derived. This result is applied to neural network models, and a larger order is proved if the true function is given by a smaller model. Also, under the condition that the model has at least two redundant hidden units, a log n lower bound for the likelihood ratio is derived.

Since statistical models are simplifications of reality, it is important in estimation theory to study the behavior of estimators also under distributions (slightly) different from the proposed model. In testing theory, when dealing with test statistics where nuisance parameters are estimated, knowledge of the behavior of the estimators of the nuisance parameters is needed under alternatives to evaluate the power. In this paper the moderate deviation behavior of minimum contrast estimators is investigated not only under the supposed model, but also under distributions close to the model. A particular example is the (multivariate) maximum likelihood estimator determined within the proposed model. The set-up is quite general, including also, for instance, discrete distributions.

The rate of convergence under alternatives is determined both when comparing the minimum contrast estimator with a "natural" parameter in the parameter space and when comparing it with the proposed "true" value in the parameter space. It turns out that under the model the asymptotic optimality of the maximum likelihood estimator in the local sense continues to hold in the moderate deviation area.

We propose a new test for the comparison of two regression curves that is based on a difference of two marked empirical processes based on residuals. The large sample behavior of the corresponding statistic is studied to provide a full nonparametric comparison of regression curves. In contrast to most procedures suggested in the literature, the new procedure is applicable in the case of different design points and heteroscedasticity. Moreover, it is demonstrated that the proposed test detects continuous alternatives converging to the null at a rate $N^{-1/2}$ and that, in contrast to all other available procedures based on marked empirical processes, the new test allows the optimal choice of bandwidths for curve estimation (e.g., $N^{-1/5}$ in the case of twice differentiable regression functions). As a by-product we explain the problems of a related test proposed by Kulasekera [J. Amer. Statist. Assoc. 90 (1995) 1085-1093] and Kulasekera and Wang [J. Amer. Statist. Assoc. 92 (1997) 500-511] with respect to accuracy in the approximation of the level. These difficulties mainly originate from the comparison with the quantiles of an inappropriate limit distribution.

A simulation study is conducted to investigate the finite sample properties of a wild bootstrap version of the new test and to compare it with the so far available procedures. Finally, heteroscedastic data is analyzed in order to demonstrate the benefits of the new test compared to the so far available procedures which require homoscedasticity.

In fault-line estimation in spatial problems it is sometimes possible to choose design points sequentially, by working one's way gradually through the "response plane," rather than distributing design points across the plane prior to conducting statistical analysis. For example, when estimating a change line in the concentration of resources on or under the sea bed, individual measurements can be particularly expensive to make. In such cases, sequential, design-adaptive methods are attractive. Appropriate methodology is largely lacking, however, and the potential advantages of taking a sequential approach are unclear. In the present paper we address both these problems. We suggest a methodology based on "sequential refinement with reassessment" that relies upon assessing the correctness of each sequential result, and reappraising previous results if significance tests show that there is reason for concern. We focus part of our attention on univariate problems, and we show how methods for the spatial case can be constructed from univariate ones.

In this paper, we study the problems of sequential probability ratio tests for parameterized hidden Markov models. We investigate in some detail the performance of the tests and derive corrected Brownian approximations for error probabilities and expected sample sizes. Asymptotic optimality of the sequential probability ratio test for testing simple hypotheses based on hidden Markov chain data is established. Next, we consider the cumulative sum (CUSUM) procedure for change point detection in this model. Based on the renewal property of the stopping rule, CUSUM can be regarded as a repeated one-sided sequential probability ratio test. Asymptotic optimality of the CUSUM procedure is proved in the sense of Lorden (1971). Motivated by the sequential analysis in hidden Markov models, Wald's likelihood ratio identity and Wald's equation for products of Markov random matrices are also given. We apply these results to several types of hidden Markov models: i.i.d. hidden Markov models, switch Gaussian regression and switch Gaussian autoregression, which are commonly used in digital communications, speech recognition, bioinformatics and economics.

Kunert [Ann. Statist. 12 (1984) 1006-1017] proved that, in the class of repeated measurement designs based on t treatments, p=t periods and $n=\lambda t$ experimental units, a balanced uniform design is universally optimal for direct treatment effects if $t \geq 3$ and $\lambda=1$, or if $t \geq 6$ and $\lambda=2$. This result is generalized to $t \geq 3$ as long as $\lambda \leq (t-1)/2$.

A two-level factorial design can be uniquely represented by a polynomial indicator function. Therefore, properties of factorial designs can be studied through their indicator functions. This paper shows that the indicator function is an effective tool in studying two-level factorial designs. The indicator function is used to generalize the aberration criterion of a regular two-level fractional factorial design to all two-level factorial designs. An important identity of generalized aberration is proved. The connection between a uniformity measure and aberration is also extended to all two-level factorial designs.

A general approach to studying fractional factorial designs with multiple groups of factors is proposed. A structure function is generated by the defining contrasts among different groups of factors and the remaining columns. The structure function satisfies a first-order partial differential equation. By solving this equation, general results about the structures and properties of the designs are obtained. As an important application, practical rules for the selection of "optimal" single arrays for robust parameter design experiments are derived.

In factor screening, often only a few factors among a large pool of potential factors are active. Under such assumption of effect sparsity, in choosing a design for factor screening, it is important to consider projections of the design onto small subsets of factors. Cheng showed that as long as the run size of a two-level orthogonal array of strength two is not a multiple of 8, its projection onto any four factors allows the estimation of all the main effects and two-factor interactions when the higher-order interactions are negligible. This result applies, for example, to all Plackett-Burman designs whose run sizes are not multiples of 8. It is shown here that the same hidden projection property also holds for Paley designs of sizes greater than 8, even when their run sizes are multiples of 8. A key result is that such designs do not have defining words of length three or four. Applications of this result to the construction of $E(s^2)$-optimal supersaturated designs are also discussed. In particular, certain designs constructed by using Wu's method are shown to be $E(s^2)$-optimal. The article concludes with some three-level designs with good projection properties.