An analog of the spectral analysis of time series is developed for data in general spaces. This is applied to data from an election in which 5738 people rank ordered five candidates. Group theoretic considerations offer an analysis of variance like decomposition which seems natural and fruitful. A variety of inferential tools are suggested. The spectral ideas are then extended to general homogeneous spaces such as the sphere.

Consider a continuous Gaussian random field $z(x)$ defined on a compact set $R \subset \mathbb{R}^d$ with covariance function of the form $\operatorname{cov}(z(x), z(x')) = \sum^k_{i = 1}\theta_iK_i(x,x')$, where the $K_i$'s are specified and $\theta = (\theta_1, \ldots, \theta_k)'$ is to be estimated. Let $\{x_l\}^\infty_{l = 1}$ be a sequence of distinct points in $R$. Based on $z(x_1), \ldots, z(x_N)$, minimum norm quadratic estimation can be used to estimate $\theta$. Suppose $K_1, \ldots, K_k$ are compatible covariance functions on $R$, which means that the Gaussian measures with means zero and covariance functions $K_1, \ldots, K_k$ are mutually absolutely continuous. Then, as the number of observations $N$ increases, the minimum norm quadratic estimator of $\sum^k_{i = 1}\theta_i$ is asymptotically normal with variance of order $N^{-1}$. The minimum norm quadratic estimator of any other linear combination of the $\theta_i$'s converges (in $L^2$) to some nondegenerate random variable. This limit is the same for any two dense sequence of points in $R$. Thus, a definition of a minimum norm quadratic estimator of $\theta$ when $z(\cdot)$ is observed everywhere in $R$ is obtained.

The problem of constructing honest confidence regions for nonparametric regression is considered. A lower rate of convergence, $n^{-1/4}$, for the size of the confidence region is established. The achievability of this rate is demonstrated using Stein's estimates and the associated unbiased risk estimates. Practical implications are discussed.

We study the behavior of regression analysis when there might be some violation of the assumed link function, the functional form of the model which relates the outcome variable $y$ to the regressor variable $\mathbf{x}$ and the random error. We allow the true link function to be completely arbitrary, except that $y$ depends on $\mathbf{x}$ only through a linear combination $\beta\mathbf{x}$. The slope vector $\beta$ is identified only up to a multiplicative scalar. Under appropriate conditions, any maximum likelihood-type regression estimate is shown to be consistent for $\beta$ up to a multiplicative scalar, even though the estimate might be based on a misspecified link function. The crucial conditions are (1) the estimate is based on minimizing a criterion function $L(\theta, y)$ which is convex in $\theta$, where $\theta = a + b\mathbf{x}$, (2) the expected criterion function $E\lbrack L(a + b\mathbf{x}, y)\rbrack$ has a proper minimizer and (3) the regressor variable $\mathbf{x}$ is sampled randomly from a probability distribution such that $E(b\mathbf{x}\mid\beta\mathbf{x})$ is linear in $\beta\mathbf{x}$ for all linear combinations $b\mathbf{x}$. The least squares estimate, the GLM estimates and the $M$-estimates for robust regression are discussed in detail. These estimates are asymptotically normal. With the assumption that the regressor variable has an elliptically symmetric distribution, we show that under a scale-invariant null hypothesis of the form $H_0: \beta W = 0$, the asymptotic covariance matrix for $\hat{\beta}W$ is proportional to the one derived by treating the assumed link function as being true. The Wald test as well as the likelihood ratio test for a scale-invariant null hypothesis has the correct asymptotic null distribution after an appropriate rescaling of the test statistic to account for the proportionality constant between the two asymptotic covariance matrices. For normally distributed $\mathbf{x}$, the rescaling factor for $M$-estimates is the same as the one used in robust regression, while the rescaling factor for GLM estimates is related to adjustment for overdispersion. Confidence sets can be constructed by inverting Wald's tests. The impact of the violation of linear conditional expectation condition 3 is discussed. A new dimension is added to the regression diagnostics by exploring the elliptical symmetry of the design distribution. A connection between this work and adaptive estimation is briefly discussed.

It is shown that consistent estimates of the optimal bandwidths for kernel estimators of location and size of a peak of a regression function are available. Such estimates yield the same joint asymptotic distribution of location and size of a peak as the optimal bandwidths themselves. Therefore data-adaptive efficient estimation of peaks is possible. In order to prove this result, the weak convergence of a two-dimensional stochastic process with appropriately scaled bandwidths as arguments to a Gaussian limiting process is shown. A practical method which leads to consistent estimates of the optimal bandwidths and is therefore asymptotically efficient is proposed and its finite sample properties are investigated by simulation.

Let $f$ be a density on the real line and let $f_n$ be the kernel estimate of $f$ in which the smoothing factor is obtained by maximizing the cross-validated likelihood product according to the method of Duin and Habbema, Hermans and Vandenbroek. Under mild regularity conditions on the kernel and $f$, we show, among other things that $\int|f_n - f| \rightarrow 0$ almost surely if and only if the sample extremes of $f$ are strongly stable.

In sequentially designed experiments with linear models, each design variable may depend on previous responses. The use of such sequential designs does not affect the likelihood function or the functional form of the maximum likelihood estimator, but it may affect sampling distributions. In this paper, asymptotic expansions for sampling distributions are obtained. The expansions are very weak ones in which a confidence curve (a function of the unknown parameters) is replaced by a confidence functional defined on a class of prior distributions. The proofs use a version of Stein's identity.

An Edgeworth expansion of first order is established for general linear rank statistics under the null hypothesis with a remainder term that is usually of order $n^{-1}$. Furthermore, corresponding results for the second order are formulated, but not proved here. The proof for the first order is based on Stein's method and on an extension of a combinatorial method of Bolthausen. It is also shown that conditions of van Zwet imply up to a small factor our conditions for the validity of Edgeworth expansions. Moreover, our proof for the first order also provides us with a result about Edgeworth expansions for smooth functions.

This paper deals with the distribution of $\mathbf{X} = \sum^{1/2}\mathbf{Z}$, where $\mathbf{Z}: p \times 1$ is distributed as $N_p(0, I_p), \sum$ is a positive definite random matrix and $\mathbf{Z}$ and $\sum$ are independent. Assuming that $\sum = I_p + BB'$, we obtain an asymptotic expansion of the distribution function of $\mathbf{X}$ and its error bound, which is useful in the situation where $\sum$ tends to $I_p$. A stronger version of the expansion is also given. The results are applied to the asymptotic distribution of the MLE in a general MANOVA model.

Statistical models are considered for (partial) observation of independent, identically distributed failure times, subjected to censoring from the right. A minimal class of censoring patterns is determined under the assumption that the failure intensities are as they would have been without censoring. This class is then used to discuss the statistical models for right censored survival data, where, e.g., the Kaplan-Meier estimator exploits all available information about the failure time distribution. The counting process description of survival data is used throughout.

We consider a class of nonparametric regression estimates introduced by Beran to estimate conditional survival functions in the presence of right censoring. An exponential probability bound for the tails of distributions of kernel estimates of conditional survival functions is derived. This inequality is next used to prove weak and strong uniform consistency results. The developments rest on sharp exponential bounds for the oscillation modulus of multivariate empirical processes obtained by Stute.

Athreya showed that the bootstrap distribution of a sum of infinite variance random variables did not (with probability 1) tend weakly to a fixed distribution but instead tended in distribution to a random distribution. In this paper, we give a different proof of Athreya's result motivated by a heuristic large sample representation of the bootstrap distribution.

The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded $d$ are consistent and asymptotically unbiased and (ii) for "nonsmooth" estimators, $d$ has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness. Improved results are obtained for several classes of estimators. In particular, for the sample $p$-quantiles, the jackknife variance estimators with $d$ satisfying $n^{1/2}/d \rightarrow 0$ and $n - d \rightarrow \infty$ are consistent and asymptotically unbiased.

The notion of a bootstrap estimator of the distribution of the maximum likelihood estimator in log-linear models is defined for common sampling models. It is shown that the bootstrap estimator is consistent under assumptions which allow the dimension of the model to increase to infinity. Such an approach allows treatment of large, sparse contingency tables.

We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$. General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.

Robust nonparametric estimators for regression and autoregression are proposed for $\varphi$- and $\alpha$-mixing processes. Two families of $M$-type robust equivariant estimators are considered: (i) estimators based on kernel methods and (ii) estimators based on $k$-nearest neighbor kernel methods. Strong consistency of both families is proved under mild conditions. For the first class the result is true under no assumptions whatsoever on the distribution of the observations.

We give sufficient conditions for strong consistency of estimators for the order of general nonstationary autoregressive models based on the minimization of an information criterion a la Akaike's (1969) AIC. The case of a time-dependent error variance is also covered by the analysis. Furthermore, the more general case of regressor selection in stochastic regression models is treated.

Least squares estimators of the coefficients of an autoregression of known, finite order are biased to order $1/T$, where $T$ is the sample length, unless the observed time series is generated by a unique model for that order. The coefficients of this special model are the fixed point of a linear mapping defined by the bias of the least squares estimator. Separate results are given for models with known mean and unknown mean. The "fixed point models" for different orders of autoregression are least squares approximations to an infinite-order autoregression which is unique but for arbitrary scaling. Explicit expressions are given for the coefficients of the fixed point models at each order. The autocorrelation function and spectral density of the underlying infinite-order process are also presented. Numerical calculations suggest similar properties hold for Yule-Walker estimators. Implications for bootstrapping autoregressive models are discussed.

In this paper we consider generalized chi-square goodness-of-fit tests based on increasingly finer partitions (as the sample size increases) for models with location-scale nuisance parameters. The asymptotic distributions are derived both under the null hypothesis and under local alternatives, obtained by taking contamination families of densities between the null hypothesis and fixed alternative hypotheses. If the number of random cells increases to infinity, the Rao-Robson-Nikulin test statistic is shown to be superior to the Watson-Roy and Dzhaparidze-Nikulin statistics. Conditions are derived under which it is optimal to let the number of classes tend to infinity.

Suppose that real numbers $y_i$ are associated with the units $i = 1, 2, \ldots, N$ of a population $U$ and that the vector $y = (y_1, y_2, \ldots, y_N)$ is known to be an element of the parameter space $\Theta$. The statistician has to select a sample $s \subset U$ of $n$ units and to employ $y_i, i \in s,$ to estimate $\bar{y} = \sum y_i/N.$ We propose to base this decision on an asymptotic version of the minimax principle. The asymptotically minimax principle is applied to three parameter spaces, including the parameter space considered by Scott and Smith and a space discussed by Cheng and Li. It turns out that stratified sampling is asymptotically minimax if the allocation is adapted to the parameter space. In addition we show that the commonly used ratio strategy [i.e., simple random sampling (srs) together with ratio estimation] and the RHC-strategy (see Rao, Hartley and Cochran) are asymptotically minimax with respect to parameter spaces chosen appropriately.

A strategy of randomizing a design that is $L$-optimal for an ideal model is shown to possess a minimax property under a larger, more realistic model. This generalizes a result of Li.

Optimal-partitioning and minimax risk inequalities are obtained for the classification and multi-hypotheses testing problems. Best possible bounds are derived for the minimax risk for location parameter families, based on the tail concentrations and Levy concentrations of the distributions. Special attention is given to continuous distributions with the maximum likelihood ratio property and to symmetric unimodal continuous distributions. Bounds for general (including discontinuous) distributions are also obtained.

Let $\Omega$ be an interval and let $F_\omega, \omega \in \Omega,$ denote a one-parameter exponential family of probability distributions on $\mathscr{R} = (-\infty, \infty),$ each of which has a finite mean $\theta,$ depending on some unknown parameter $\omega \in \Omega.$ The main results of this paper determine an asymptotic lower bound for the local minimax regret, under a general smooth loss function and for a general class of estimators of $\theta.$ This bound is obtained by first determining the limit of the Bayes regret and then maximizing with respect to the prior distribution of $\omega.$

Consider the classical invariant decision problem of estimating an unknown continuous distribution function $F,$ with the loss function $L(F, a) = \int(F(t) - a(t))^2\lbrack F(t) \rbrack^\alpha \lbrack 1 - F(t) \rbrack^\beta dF(t),$ and a random sample of size $n$ from $F.$ It is proved that the best invariant estimator is inadmissible when: 1. $ n > 0, - 1 < \alpha, \beta \leq 0 \text{and} -1 \leq \alpha + \beta.$ 2. $ n > 0, -1 < \alpha = \beta \leq - \frac{1}{2}.$ 3. $ n > 1, (\mathrm{i}) \alpha = -1 \text{and} \beta = 0, \text{or} (\mathrm{ii}) \alpha = 0 \text{and} \beta = -1.$ 4. $ n > 2, \alpha = \beta = -1.$ Thus the empirical distribution function, which is the best invariant estimator when $\alpha = \beta = -1,$ is inadmissible when $n \geq 3.$ This extends some results of Brown.

A distribution-free inadmissibility theorem is proved for estimating under quadratic loss nonnegative functionals that are allowed to depend on the unknown c.d.f. as well as the data. It follows from the theorem that, subject to finiteness of the risk, some natural estimators of the eigenvalues of many commonly occurring matrices in multivariate problems are inadmissible, typically from dimension 2, when samples are drawn from any multivariate elliptically symmetric distribution. As an example, if $X_1, \ldots, X_{k + 1}$ are i.i.d. observations from such a distribution with dispersion matrix $\Sigma,$ then the eigenvalues of $S/k$ are inadmissible for the eigenvalues of $\Sigma$ if a certain conditions holds where $S$ if the sample sum of squares and products matrix. It also follows from the theorem that in the general scale-parameter family, the best equivariant estimator of the scale-parameters is inadmissible for $p \geq 2,$ and some natural estimators of the losses of the best equivariant estimators are inadmissible, usually for $p \geq 2.$ The theorem also has certain applications to the Ferguson family of distributions, the multivariate $F$ distribution and for unbiased estimators in some families of distributions.

The dominance and related optimality properties of the usual Stein-rule or shrinkage estimators are typically developed for quadratic error loss functions. It is shown that under the classical Pitman closeness criterion the Stein-rule estimators possess a similar dominance property when the "closeness" measure is based on suitable quadratic norms.

Likelihood regions are shown to be robust in the sense that their posterior probability content is relatively insensitive to contaminations of the prior. This provides a Bayesian interpretation of regions that are commonly used by frequentists to construct confidence intervals and whose use are also advocated by the pure likelihood approach.

A theory of coherence is formulated for rates of exchange between events. The theory can be viewed as a generalization of de Finetti's theory of coherence as well as the theory of conditional coherence. Coherent rates of exchange on a fixed Boolean algebra are in one-to-one correspondence with finitely additive conditional probability measures on the algebra. Results of Renyi and Krauss on conditional probability spaces are used to show that coherent rates of exchange are generated by ordered families of finitely additive measures, possibly infinite measures. This provides an interpretation of improper prior distributions in terms of coherence. An extension theorem is proved and gives a generalization of extension theorems for finitely additive probability measures.

Darroch and Ratcliff's iterative algorithm for minimizing $I$-divergence subject to linear constraints is equivalent to a cyclic iteration of explicitly performable $I$-projection operations.