Given a parametric model and an improper prior distribution, the formal posterior distribution induces decision rules in any decision problem. The results here provide conditions under which this formal Bayes method produces admissible decision rules for all quadratically regular decision problems. The conditions derived are shown to be equivalent to the recurrence of a natural symmetric Markov chain (on the parameter space) generated by the model and the improper prior. The results are also used to give conditions under which formal predictive distributions are admissible decision rules in certain prediction problems.

Biological cell membranes contain molecular structures known as ion channels. The electrical activity of these channels regulates a number of cellular functions, such as heartbeat and neural transmission. Often the experimenter can only observe the combined current conducted through a cell patch containing an unknown number of ion channels, each channel of equal conductance. The observed current record is a step function with unit up or down steps occurring at random times. Based on such data, we obtain a goodness-of-fit test for the hypothesis that the channels in a portion of cellular membrane are operating independently, and that $K$ types ($K$ small) of channels exist in the membrane. If the hypothesis is accepted, estimates for the mean behaviour of each type of channel in the experiment are obtained. These techniques are applied to simulated and biological data.

Trees of Polya urns are used to generate sequences of exchangeable random variables. By a theorem of de Finetti each such sequence is a mixture of independent, identically distributed variables and the mixing measure can be viewed as a prior on distribution functions. The collection of these Polya tree priors forms a convenient conjugate family which was mentioned by Ferguson and includes the Dirichlet processes of Ferguson. Unlike Dirichlet processes, Polya tree priors can assign probability 1 to the class of continuous distributions. This property and a few others are investigated.

Polya tree distributions are defined. They are generalizations of Dirichlet processes that allow for the possibility of putting positive mass on the set of continuous distributions. Predictive and posterior distributions are explained. A canonical construction of a Polya tree is given so that the Polya tree has any desired predictive distribution. Choices of the Polya tree parameters are discussed. Mixtures of Polya trees are defined and examples are given.

We investigate some of the possibilities for improvement of univariate and multivariate kernel density estimates by varying the window over the domain of estimation, pointwise and globally. Two general approaches are to vary the window width by the point of estimation and by point of the sample observation. The first possibility is shown to be of little efficacy in one variable. In particular, nearest-neighbor estimators in all versions perform poorly in one and two dimensions, but begin to be useful in three or more variables. The second possibility is more promising. We give some general properties and then focus on the popular Abramson estimator. We show that in many practical situations, such as normal data, a nonlocality phenomenon limits the commonly applied version of the Abramson estimator to bias of $O(\lbrack h / \log h\rbrack^2)$ instead of the hoped for $O(h^4)$.

The paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological (chemical, $\ldots$) process. Such samples of curves often possess the following features: There is a typical structural pattern common to all curves of the sample. On the other hand, individual realizations of the typical shape show different dynamics and intensity. In particular, typical peaks are shifted from individual to individual. Differences in dynamics complicate the analysis of samples of curves. For example, the cross-sectional average usually does not reflect an average pattern. Due to shifts, structure is smeared or might even disappear. Our approach consists in synchronizing the individual curves before determining the average or any further statistics. Pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity. The method requires the introduction of new statistical objects. They are defined mathematically, their properties are discussed, and possible estimators are proposed. The asymptotic bias and variance of the estimators are derived. An application to visually evoked brain potentials illustrates the approach.

Consider the problem of estimating the value of a functional $\Lambda(f)$ for $f$ an unknown density or regression function. The straightforward plug-in estimator $\Lambda(\hat f)$ with $\hat f$ a particular estimate of $f$ achieves the optimal rate of convergence in the sense of Stone over bounded subsets of a Sobolev space for a broad class of linear and nonlinear functionals. For many functionals the rate calculation depends on a Frechet-like derivative of the functional, which may be obtained using elementary calculus. For some classes of functionals, $\hat f$ is undersmoothed relative to what would be used to estimate $f$ optimally. Examples for which a plug-in estimator is optimal include $L^q$ norms of regression or density functions and their derivatives and the expected integrated squared bias. When interested in computing estimates over classes of functions which satisfy certain restrictions, such as strict positivity or boundary conditions, the plug-in estimator may or may not be optimal, depending on the functional and the function class. The functional calculus establishes conditions under which the plug-in estimator remains optimal, and sometimes suggests an appropriate modification when it does not.

Adaptive nonparametric kernel estimators for the location of a peak of the spectral density of a stationary time series are proposed and investigated. They are based on direct smoothing of the periodogram where the amount of smoothing is determined automatically in an asymptotically optimal fashion. These adaptive estimators minimize the asymptotic mean squared error. Adaptivity is derived from the weak convergence of a two-parameter stochastic process in a deviation and a bandwidth coordinate to a Gaussian limit process. Efficient global and local bandwidth choices which lead to adaptive peak estimators and practical aspects are discussed.

A maximum-penalized-likelihood method is proposed for estimating a mixing distribution and it is shown that this method produces a consistent estimator, in the sense of weak convergence. In particular, a new proof of the consistency of maximum-likelihood estimators is given. The estimated number of components is shown to be at least as large as the true number, for large samples. Also, the large-sample limits of estimators which are constrained to have a fixed finite number of components are identified as distributions minimizing Kullback-Leibler divergence from the true mixing distribution. Estimation of a Poisson mixture distribution is illustrated using the distribution of traffic accidents presented by Simar.

A Hellinger-type distance for hazard rate functions is defined. It is used to obtain a class of minimum distance estimators for data that are subject to a possible right censorship. The corresponding score process is shown to be approximated by a martingale, which is exploited to obtain the asymptotic normality under considerably weaker conditions than those normally assumed for minimum Hellinger distance estimators. It is also shown that under the parametric assumption the estimators are asymptotically as efficient as the maximum likelihood estimators.

Many methods have been proposed for modelling nonhomogeneous Poisson processes, including change point models and log-linear models. In this paper, we use likelihood ratio tests to choose which of these models are necessary. Of particular interest is the test for the presence of a change point, for which standard asymptotic theory is not valid. Large deviation methods are applied to approximate the significance level, and power approximations are given. Confidence regions for the change point and other parameters in the model are also derived. A British coal mining accident data set is used to illustrate the methodology.

A new test is derived for the hypothesis that a regression function has a prescribed parametric form. Unlike many recent proposals, this test does not depend on arbitrarily chosen smoothing parameters. In fact, the test statistic is itself a smoothing parameter which is selected to minimize an estimated risk function. The exact distribution of the test statistic is obtained when the error terms in the regression model are Gaussian, while the large sample distribution is derived for more general settings. It is shown that the proposed test is consistent against fixed alternatives and can detect local alternatives that converge to the null hypothesis at the rate $1/\sqrt n$, where $n$ is the sample size. More importantly, the test is shown by example to have an ability to adapt to the alternative at hand.

In this paper, we suggest tests of stationarity in the mean of autoregressive time series versus arbitrary trend alternatives. As an intermediate, though essential, step local asymptotic normality of autoregressive models with a nonparametric regression trend is established. Moreover, a functional central limit theorem for the underlying likelihood ratio processes is derived. These results then offer a general construction principle by which every goodness of fit test (case 0), which is based on comparing the empirical distribution function and the hypothetical distribution function, corresponds to a test of stationarity in the mean of AR processes. The asymptotic power of these tests is derived. A small simulation study illustrates the performance of Kolmogorov-Smirnov and Cramer-von Mises type tests of stationarity in the mean at hand of a particular AR(2) process.

A sequential procedure for estimating the regression parameter $\beta \in R^k$ in a regression model with symmetric errors is proposed. This procedure is shown to have asymptotically smaller regret than the procedure analyzed by Martinsek when $\mathbf{\beta} = \mathbf{0}$, and the same asymptotic regret as that procedure when $\mathbf{\beta} \neq \mathbf{0}$. Consequently, even when the errors are normally distributed, it follows that the asymptotic regret can be negative when $\mathbf{\beta} = \mathbf{0}$. These results extend a recent work of Takada dealing with the estimation of the normal mean, to both regression and nonnormal cases.

Simultaneous estimation of the vector of variance components under unbalanced mixed models w.r.t. the ordinary quadratic loss function is considered. A new method of constructing invariant quadratic admissible estimators, both with and without the condition of unbiasedness, is presented. Using this method, admissible estimators for the factorial models with imbalance at the last stage and the unbalanced $(p - 1)$-way nested factors design are constructed.

Three notions of depth for directional data, angular simplicial depth (ASD), angular Tukey's depth (ATD) and arc distance depth (ADD), are developed and studied. The empirical versions of these depths give rise to center-outward rankings of angular data which may be regarded as extensions of the usual center-outward ranking on the line. Three medians derived from these depths are examined and compared. Applications in nonparametric classification and in implementing the bootstrap to construct confidence regions for directional parameters are briefly discussed.

Bounds on the asymptotic relative efficiency (ARE) of the Box-Cox transformed two-sample $t$-test to the ordinary $t$-test are obtained under local alternatives. It is shown that the ARE is at least 1 for location-shift models. In the case of scale-shift models, a similar bound applies provided the limiting value of the estimated power transformation is greater than 1. If instead the Box-Cox transformed $t$-test is compared against the ordinary $t$-test applied to the log-transformed data, then the ARE is bounded below by 1 for all scale-shift models, regardless of the limiting value of the power transformation. The results extend naturally to the multisample $F$-test. A small simulation study to evaluate the validity of the asymptotic results for finite-sample sizes is also reported.

The usual confidence interval, based on Student's $t$ distribution, has conditional confidence that is larger than the nominal confidence level. Although this fact is known, along with the fact that increased conditional confidence can be used to improve a confidence assertion, the confidence assertion of Student's $t$ interval has never been critically examined. We do so here, and construct a confidence estimator that allows uniformly higher confidence in the interval and is closer (than $1 - \alpha$) to the indicator of coverage.

We consider $M$-estimators defined by minimization of a convex criterion function, not necessarily smooth. Our asymptotic results generalize some of those concerning the LAD estimators. We establish a Bahadur-type strong approximation and bounds on the rate of convergence.

The metrically trimmed mean is defined as the average of observations remaining after a fixed number of outlying observations have been removed. A metric, the distance from the median, is used to determine which points are outlying. The influence curve and the asymptotic normality of the metrically trimmed mean are derived using von Mises expansions. The relative merits of the median, the trimmed mean and the metrically trimmed mean are discussed in neighborhoods of nonparametric models with natural parameters. It is observed that the metrically trimmed mean works well for the center of symmetry of a symmetric distribution function with asymmetric contamination. A multivariate extension of the metrically trimmed mean is discussed.

Consider a multiple linear regression model $Y_i = x'_i\beta + \varepsilon_i$, where the $\varepsilon_i$'s are independent random variables with common distribution $F$ and the $x_i$'s are known design vectors. Let $\bar\beta_n$ be the $M$-estimator of $\beta$ corresponding to a score function $\psi$. Under some conditions on $F, \psi$ and the $x_i$'s, two-term Edgeworth expansions for the distributions of standardized and studentized $\bar\beta_n$ are obtained. Furthermore, it is shown that the bootstrap method is second order correct in the studentized case when the bootstrap samples are drawn from some suitable weighted empirical distribution or from the ordinary empirical distribution of the residuals.

Inference, including variance estimation, can be made from stratified samples by selecting a balanced set of subsamples. This balanced subsampling method is generically called the balanced repeated replication method in survey data analysis, which includes McCarthy's balanced half-samples method and its extensions for more general stratified designs. We establish the asymptotic consistency of the balanced repeated replication variance estimators when the parameter of interest is the population quantile. The consistency results also hold when balanced subsampling is replaced by random subsampling. As a key technical prerequisite, we prove a Bahadur-type representation for sample quantiles in stratified random sampling.

Techniques are developed for bootstrap estimation of conditional distributions, with application to confidence intervals and hypothesis tests for one parameter, conditional on the value of an estimator of another. Both Monte Carlo and saddlepoint methods for approximating bootstrap distributions are considered, and empirical methods are suggested for implementing these techniques. For example, in the case of Monte Carlo methods, we suggest empirical techniques for selecting both the smoothing parameter, necessary to define the estimator, and the importance resampling probabilities, required for efficient bootstrap simulation. The smoothing parameter depends critically on the number of Monte Carlo simulations, as well as on the data. Both our theoretical and numerical results indicate that pivoting can substantially improve performance.

A general notion of a bootstrapped mean constructed by exchangeably weighting sample points is introduced. Consistency of this generalized bootstrapped mean, which includes proposals of Efron and Rubin among others, is proved by classical linear rank statistics theory. The consistency of generalized bootstrapped empirical and quantile processes is also established.

For a multiarmed bandit problem with exponential discounting the optimal allocation rule is defined by a dynamic allocation index defined for each arm on its space. The index for an arm is equal to the expected immediate reward from the arm, with an upward adjustment reflecting any uncertainty about the prospects of obtaining rewards from the arm, and the possibilities of resolving those uncertainties by selecting that arm. Thus the learning component of the index is defined to be the difference between the index and the expected immediate reward. For two arms with the same expected immediate reward the learning component should be larger for the arm for which the reward rate is more uncertain. This is shown to be true for arms based on independent samples from a fixed distribution with an unknown parameter in the cases of Bernoulli and normal distributions, and similar results are obtained in other cases.

Let $(X, U)$ be jointly distributed on $\mathscr{X} \times \mathscr{R}^n$. Let $Y = E(U\mid X)$ and let $\mathscr{U}$ be the convex hull of the range of $U$. Let $C: \mathscr{X} \rightarrow \mathscr{C} = \{1,2,\ldots,k\}, k \geq 1$, induce a measurable $k$ way partition $\{\mathscr{X}_1,\ldots,\mathscr{X}_k\}$ of $\mathscr{X}$. Define the impurity of $\mathscr{X}_c = C^{-1}(c)$ to be $\phi(c, E(U\mid C(X) = c))$, where $\phi: \mathscr{C} \times \mathscr{U} \rightarrow \mathscr{R}^1$ is a concave function in its second argument. Define the impurity $\Psi$ of the partition as the average impurity of its members: $\Psi(C) = E\phi(C(X), E(U\mid C(X))$. We show that for any $C: \mathscr{X} \rightarrow \mathscr{C}$ there exists a mapping $\tilde{C}: \mathscr{U} \rightarrow \mathscr{C}$, such that $\Psi(\tilde{C}(Y)) \leq \Psi(C)$ and such that $\tilde{C}^{-1}(c)$ is convex, that is, for each $i, j \in C, i \neq j$, there exists a separating hyperplane between $\tilde{C}^{-1}(i)$ and $\tilde{C}^{-1}(j)$. This generalizes some results in statistics and information theory. Suitable choices of $U$ and $\phi$ lead to optimal partitions of simple form useful in the construction of classification trees and multidimensional regression trees.

We consider the problem of bandwidth selection for kernel density estimators. Let $H_n$ denote the bandwidth computed by the least squares cross-validation method. Furthermore, let $H^\ast_n$ and $h^\ast_n$ denote the minimizers of the integrated squared error and the mean integrated squared error, respectively. The main theorem establishes asymptotic normality of $H_n - H^\ast_n$ and $H_n - h^\ast_n$, for three classes of densities with comparable smoothness properties. Apart from densities satisfying the standard smoothness conditions, we also consider densities with a finite number of jumps or kinks. We confirm the $n^{-1/10}$ rate of convergence to 0 of the relative distances $(H_n - H^\ast_n)/H^\ast_n$ and $(H_n - h^\ast_n)/h^\ast_n$ derived by Hall and Marron in the smooth case. Unexpectedly, in turns out that these relative rates of convergence are faster in the nonsmooth cases.

In this paper we obtain uniform upper bounds for the $L_1$ error of kernel estimators in estimating monotone densities and densities of bounded variation. The bounds are nonasymptotic and optimal in $n$, the sample size. For the bounded variation class, it is also optimal wrt an upper bound of the total variation. The proofs employ a one-sided kernel technique and are extremely simple.