We consider the empirical Bayes estimation of a distribution using binary data via the Dirichlet process. Let $\mathscr{D}(\alpha)$ denote a Dirichlet process with $\alpha$ being a finite measure on Instead of having direct samples from an unknown random distribution F from $\mathscr{D}(\alpha)$, we assume that only indirect binomial data are observable. This paper presents a new interpretation of Lo's formula, and thereby relates the predictive density of the observations based on a Dirichlet process model to likelihoods of much simpler models. As a consequence, the log-likelihood surface, as well as the maximum likelihood estimate of $c = \alpha([0, 1])$, is found when the shape of $\alpha$ a is assumed known, together with a formula for the Fisher information evaluated at the estimate. The sequential imputation method of Kong, Liu and Wong is recommended for overcoming computational difficulties commonly encountered in this area. The related approximation formulas are provided. An analysis of the tack data of Beckett and Diaconis, which motivated this study, is supplemented to illustrate our methods.

In hierarchical Bayesian modeling of normal means, it is common to complete the prior specification by choosing a constant prior density for unmodeled hyperparameters (e.g., variances and highest-level means). This common practice often results in an inadequate overall prior, inadequate in the sense that estimators resulting from its use can be inadmissible under quadratic loss. In this paper, hierarchical priors for normal means are categorized in terms of admissibility and inadmissibility of resulting estimators for a quite general scenario. The Jeffreys prior for the hypervariance and a shrinkage prior for the hypermeans are recommended as admissible alternatives. Incidental to this analysis is presentation of the conditions under which the (generally improper) priors result in proper posteriors.

A concept of asymptotically efficient estimation is presented when a misspecified parametric time series model is fitted to a stationary process. Efficiency of several minimum distance estimates is proved and the behavior of the Gaussian maximum likelihood estimate is studied. Furthermore, the behavior of estimates that minimize the h-step prediction error is discussed briefly. The paper answers to some extent the question what happens when a misspecified model is fitted to time series data and one acts as if the model were true.

This investigation is concerned with recovering a regression function $g(x_i)$ on the basis of noisy observations taken at uniformly spaced design points $x_i$. It is presumed that the corresponding observations are corrupted by additive dependent noise, and that the noise is, in fact, induced by a general linear process in which the summand law can be discrete, as well as continuously distributed. Discreteness induces a complication because such noise is not known to be strong mixing, the postulate by which regression estimates are often shown to be asymptotically normal. In fact, as cited, there are processes of this character which have been proven not to be strong mixing. The main analytic result of this study is that, in general circumstances which include the non-strong mixing example, the smoothers we propose are asymptotically normal. Some motivation is offered, and a simple illustrative example calculation concludes this investigation. The innovative elements of this work, mainly, consist of compassing models with discrete noise, important in practical applications, and in dispensing with mixing assumptions. The ensuing mathematical difficulties are overcome by sharpening standard arguments.

Let $X_n = \Sigma_{i=1}^{\infty} a_i \varepsilon_{n-i}$, where the $\varepsilon_i$ are iid with mean 0 finite fourth moment and the $a_i$ are regularly varying with index $-\beta$ where $\beta \epsilon (1/2, 1)$ so that ${X_n}$ has long-range dependence. This covers an important class of the fractional ARIMA process. For $r \geq 0$, let $Y_{N, r} = \sum_{n=1}^N \sum_{1\leq j_1 < \dots < j_r} \Pi_{s=1}^r a_{j_s}, Y_{N, 0} = N, \sigma_{N, r}^2 = \Var(Y_{N, r})$ and $F^{(r)} =$ the rth derivative of the distribution function of $X_n$. The $Y_{N, r}$ are uncorrelated and are stochastically decreasing in r. For any positive integer $p < (2\beta - 1)^{-1}$, it is shown under mild regularity conditions that, with probability 1, $$\sum_{n=1}^N I(X_n \leq x) = \sum_{r=0}^p (-1)^r F^{(r)} (x) Y_{N,r} + o(N^{-\lambda} \sigma_{N,p})\\ \text{uniformly for all $x \epsilon\Re \forall 0 < \lambda < (\beta - 1/2) \wedge (1/2 - p(\beta - 1/2))$}.$$ This generalizes a host of existing results and provides the vehicle for a number of statistical applications.

This paper proves the local asymptotic normality of a stationary and ergodic first order random coefficient autoregressive model in a semiparametric setting. This result is used to show that Stein's necessary condition for adaptive estimation of the mean of the random coefficient is satisfied if the distributions of the innovations and the errors in the random coefficients are symmetric around zero. Under these symmetry assumptions, a locally asymptotically minimax adaptive estimator of the mean of the random coefficient is constructed. The paper also proves the asymptotic normality of generalized M-estimators of the parameter of interest. These estimators are used as preliminary estimators in the above construction.

To explore nonlinear structures hidden in high-dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression, Li and Duan proposed the sliced inverse regression (SIR) method which is simple to use. In this paper, the asymptotic properties of the kernel estimate of sliced inverse regression are investigated. It turns out that regardless of the kernel function, the asymptotic distribution remains the same for a wide range of smoothing parameters.

The most popular technique for reducing the dimensionality in comparing two multidimensional samples of $\mathbf{X} \sim F$ and $\mathbf{Y}\sim G$ is to analyze distributions of interpoint comparisons based on a univariate function h (e.g. the interpoint distances). We provide a theoretical foundation for this technique, by showing that having both i) the equality of the distributions of within sample comparisons $(h(\mathbf{X}_1, \mathbf{X}_2) =_L h(\mathbf{Y}_1, \mathbf{Y}_2))$ and ii) the equality of these with the distribution of between sample comparisons $((h(\mathbf{X}_1, \mathbf{X}_2) =_L h(\mathbf{X}_3, \mathbf{Y}_3))$ is equivalent to the equality of the multivariate distributions $(F = G)$.

A large-deviations criterion is proposed for optimality of nonparametric regression estimators. The criterion is one of minimaxity of the large-deviations probabilities. We study the case where the underlying class of regression functions is either Lipschitz or Hölder, and when the loss function involves estimation at a point or in supremum norm. Exact minimax asymptotics are found in the Gaussian case.

We establish general sufficient conditions for the $L_2$-consistency of multivariate histogram regression estimates based on data-dependent partitions. These same conditions insure the consistency of partitioning regression estimates based on local polynomial fits, and, with an additional regularity assumption, the consistency of histogram estimates for conditional medians.

Our conditions require shrinking cells, subexponential growth of a combinatorial complexity measure and sublinear growth of restricted cell counts. It is not assumed that the cells of every partition be rectangles with sides parallel to the coordinate axis or that each cell contain a minimum number of points. Response variables are assumed to be bounded throughout.

Our results may be applied to a variety of partitioning schemes. We established the consistency of histograms regression estimates based on cubic partitions with data-dependent offsets, k-thresholding in one dimension and empirically optimal nearest-neighbor clustering schemes. In addition, it is shown that empirically optimal regression trees are consistent when the size of the trees grows with the number of samples at an appropriate rate.

Minimax mean-squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes. In contrast to many nonparametric function estimation problems there are both regular and irregular cases. In the regular cases the minimax mean-squared error converges at a rate proportional to the inverse of the sample size, whereas in the irregular case much slower rates are the rule.

We investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known. It is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case. A lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case. On the other hand, we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case. Moreover, we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small.

We consider the problem of sharp-optimal estimation of a response function $f(x)$ in a random design nonparametric regression under a general model where a pair of observations $(Y, X)$ has a joint density $p(y, x) = p(y|f(x)) \pi(x)$. We wish to estimate the response function with optimal minimax mean integrated squared error convergence as the sample size tends to $\infty$. Traditional regularity assumptions on the conditional density $p(y| \theta)$ assumed for parameter $\theta$ estimation are sufficient for sharp-optimal nonparametric risk convergence as well as for the existence of the best constant and rate of risk convergence. This best constant is a nonparametric analog of Fisher information. Many examples are sketched including location and scale families, censored data, mixture models and some well-known applied examples. A sequential approach and some aspects of experimental design are considered as well.

Previously, Huggins and Staudte examined robust estimators for a variance components formulation of the bifurcating autoregressive model for cell lineage data. They gave asymptotic properties of the estimators if a large number of trees were observed. However, for single trees the derivation of these asymptotic properties is more complex. Here the asymptotic distributions of robust estimators of parameters associated with the stationary bifurcating autoregressive process as a single tree becomes large are obtained. These results follow from the formulation of the estimating functions as the product of a nonrandom matrix and the sum of vectors of functions of an infinite sequence of exchangeable random variables.

Birth and death on a flow refers to a particle system on a Brownian flow. Particles are born in a point process and move on the flow subject to position-dependent killing. They die eventually and leave the flow. The particle process is a measure-valued Markov process tracking these motions. Its law depends on the distribution of births, the coefficients of the flow and the rate of killing. We treat asymptotic likelihood estimation of these parameters from chronicles of the particle process as observed over a long period of time.

Optimal design is studied for factorial experiments in the nested row and column setting. The approach is analogous to that of orthogonal Latin squares: main effects plans are found by the superimposition of one nested row and column design upon another. Conditions are stated for statistical orthogonality of the superimposed components, resulting in orthogonal main effects plans, and a number of constructions are given. Orthogonal collections of sets of Latin squares are introduced. All of the constructed designs are also optimal main effects plans for the row-column and the unstructured block design settings. Further applications are as optimal multidimensional incomplete block designs and as optimal designs for multistage experimentation.

Let X be a random vector uniformly distributed on the unit cube and $f: [0, 1]^3 \to \mathsf{R}$ be a measurable function. An objective of many computer experiments is to estimate $\mu = E(f \circ X)$ by computing f at a set of points in $[0, 1]^3$. There is a design issue in choosing these points. Recently Owen and Tang independently suggested using randomized orthogonal arrays in the choice of such a set. This paper investigates the convergence rate to normality of the distribution of the average of a set of f values taken from one of these designs.

We present a version of Elfving's theorem for the Bayesian D-optimality criterion in nonlinear regression models. The Bayesian optimal design can be characterized as a design which allows a representation of a (uniquely determined) boundary point of a convex subset of $L^2$-integrable functions. A similar characterization is given for the Bayesian c-optimality criterion where a (possible) nonlinear function of the unknown parameters has to be estimated. The results are illustrated in the example of an exponential growth model using a gamma prior distribution.

4 If ${p_{\theta}$ is a $\sigma$-additive statistical model and $\pi$ a finitely additive prior, then any statistic T is sufficient, with respect to a suitable inference consistent with ${p_{\theta}$ and $\pi$, provided only that $p_{\theta}(T = t) = 0$ for all $\theta$ and t. Here, sufficiency is to be intended in the Bayesian sense, and consistency in the sense of Lane and Sudderth. As a corollary, if ${p_{\theta}$ is $\sigma$-additive and diffuse, then, whatever the prior $\pi$, there is an inference which is consistent with ${p_{\theta}$ and $\pi$. Two versions of the main result are also obtained for predictive problems.

Choquet capacities are a generalization of probability measures that arise in robustness, decision theory and game theory. Many capacities that arise in robustness are symmetric or can be transformed into symmetric capacities. We characterize the extreme points of the set of upper distribution functions corresponding to coherent, symmetric Choquet capacities on [0, 1]. We also show that the set of 2-alternating capacities is a simplex and we give a Choquet representation of this set.

For a given statistical model $\mathsf{P}$ it may happen that the order statistic is complete for each IID model based on $\mathsf{P}$. After reviewing known relevant results for large nonparametric models and pointing out generalizations to small nonparametric models, we essentially prove that this happens generically even in smooth parametric models.

As a consequence it may be argued that any statistic depending symmetrically on the observations can be regarded as an optimal unbiased estimator of its expectation.

In particular, the sample mean $\overline{X}_n$ is generically an optimal unbiased estimator, but, as it turns out, also generically asymptotically inefficient.

The consistency of estimating equations has been studied, in the main, along the lines of Cramér's classical argument, which only asserts the existence of consistent solutions. The statement similar to that of Doob and Wald, which identifies the consistent solutions, has not yet been established. The obstacle is that the solutions of estimating equations cannot in general be defined as the maximum of likelihood functions. In this paper we demonstrate that the consistent solutions can be identified as the minimax of a function R, whose properties resemble those of a log likelihood ratio, but which exists in a much wider context. Furthermore, since we do not need R to be differentiable, the minimax is consistent even when the estimating equation does not exist. In this respect, the minimax is a new estimator. We first convey the idea by focusing on the quasi-likelihood estimate, and then indicate its full generality by providing a set of sufficient conditions for consistency and studying a number of important cases. Efficiency will also be verified.

Using the variational properties of epi-convergence together with suitable results on the measurability of multifunctions and integrands, we prove a strong law of large numbers for sequences of integrands from which we deduce a general theorem of almost sure convergence (strong consistency) for the maximum likelihood estimator.

This paper is devoted to studying the asymptotic behavior of LS-estimators in constrained nonlinear regression problems. Here the constraints are given by nonlinear equalities and inequalities. Thus this is a very general setting. Essentially this kind of estimation problem is a stochastic optimization problem. So we make use of methods in optimization to overcome the difficulty caused by nonlinearity in the regression model and given constraints.

For the problem of robust estimation of multivariate location and shape, defining S-estimators using scale transformations of a fixed $\rho$ function regardless of the dimension, as is usually done, leads to a perverse outcome: estimators in high dimension can have a breakdown point approaching 50%, but still fail to reject as outliers points that are large distances from the main mass of points. This leads to a form of nonrobustness that has important practical consequences. In this paper, estimators are defined that improve on known S-estimators in having all of the following properties: (1) maximal breakdown for the given sample size and dimension; (2) ability completely to reject as outliers points that are far from the main mass of points; (3) convergence to good solutions with a modest amount of computation from a nonrobust starting point for large (though not near 50%) contamination. However, to attain maximal breakdown, these estimates, like other known maximal breakdown estimators, require large amounts of computational effort. This greater ability of the new estimators to reject outliers comes at a modest cost in efficiency and gross error sensitivity and at a greater, but finite, cost in local shift sensitivity.

Consider the problem of estimating the location vector and scatter matrix from a set of multivariate data. Two standard classes of robust estimates are M-estimates and S-estimates. The M-estimates can be tuned to give good local robustness properties, such as good efficiency and a good bound on the influence function at an underlying distribution such as the multivariate normal. However, M-estimates suffer from poor breakdown properties in high dimensions. On the other hand, S-estimates can be tuned to have good breakdown properties, but when tuned in this way, they tend to suffer from poor local robustness properties. In this paper a hybrid estimate called a constrained M-estimate is proposed which combines both good local and good global robustness properties.

Asymptotic normality and a representation of all possible subsequential limiting distributions of a simple linear rank statistic are obtained. This is then applied to finite sampling and permutation tests for slope coefficients. The effects of Winsorizing in these situations are considered carefully. Of particular interest regarding slope coefficients is that either using normal score regression constants or Winsorizing slowly increasing numbers of the population values will guarantee asymptotic normality.

In this paper the asymptotic normality of a class of statistics, including Gini's index of cograduation and Spearman's rank correlation coefficient, is proved. The asymptotic normality is stated under a large class of alternatives including the bivariate distributions corresponding to a condition of lack of association introduced in Section 3. The problems of testing the hypothesis of lack of association and of constructing confidence intervals for the population index of cograduation are also considered.

We consider the distributional and the almost sure pointwise Bahadur-Kiefer representation for U-quantiles. We show that the order of this representation depends on the order of the local variance of the empirical process of U-statistic structure at the U-quantile. Our results indicate that U-quantiles can be smoother than quantiles. U-quantiles can either be as unsmooth as quantiles or can behave as differentiable statistical functionals.

It is shown that the location of the set of $m + 1$ observations with minimal diameter, within local data, is a robust estimator of the location of a vertical tangent in a distribution function. The rate of consistency of these estimators is shown to be the same as that of asymptotically efficient estimators for the same model. Robustness means (1) only properties of the distribution local to the vertical tangent play a role in the asymptotics, and (2) these asymptotics can be proven given approximate information about just two parameters, the shape and quantile of the vertical tangent.