Mathematical tools are developed for describing low-dimensional projections of high-dimensional data. Theorems are given to show that under suitable conditions, most projections are approximately Gaussian.

A geometric approach is described for the detection of influential points in nonlinear regression. This work extends and gives geometric interpretation to recent results in logistic regression diagnostics. An application of this approach to matched case-control studies is discussed.

For models similar to those used in econometric work, under suitable regularity conditions, the bootstrap is shown to give asymptotically valid approximations to the distribution of errors in coefficient estimates.

We define the influence function and construct optimal robust estimators for autoregressive processes. We show that the asymptotic bias caused by small contaminations of the marginals can be written as the integral of a certain function with respect to the contamination. This function is called the influence function. It is unique only up to an equivalence relation, but there is a natural unique version which describes the limiting influence of an additional observation given the previous observations. Moreover, with this version the asymptotic variance at the true model can be expressed in a simple form. Optimal robust estimators minimize this asymptotic variance under a constraint on the influence function. As in the i.i.d case, they are found by truncating a multiple of the influence function of the maximum likelihood estimator.

We study estimation of the parameters of a Gaussian linear model $\mathscr{M}_0$ when we entertain the possibility that $\mathscr{M}_0$ is invalid and a larger model $\mathscr{M}_1$ should be assumed. Estimates are robust if their maximum risk over $\mathscr{M}_1$ is finite and the most robust estimate is the least squares estimate under $\mathscr{M}_1$. We apply notions of Hodges and Lehmann (1952) and Efron and Morris (1971) to obtain (biased) estimates which do well under $\mathscr{M}_0$ at a small price in robustness. Extensions to confidence intervals, simultaneous estimation of several parameters and large sample approximations applying to nested parametric models are also discussed.

With a wide variety of approaches to density estimation, it is profitable to perturb the data so as to make 2nd order derivatives of their density vanish. An adaptive transformation to local uniformity for instance will (for unchanged variance) lower bias to a vanishing fraction of what a Rosenblatt-Parzen or nearest neighbor estimator on the raw data yields; fractional pilot sampling, a common technical device of little practical appeal, can be shown by an embedding argument to be dispensable. An upshot is that MSE can be lowered by attacking the variance directly through extra smoothing, without the usual penalty from inflated bias.

The data smoothing aspect of Stein estimates is explored in the nonparametric regression settings. We show that appropriately shrinking the raw data towards any linear smoother will provide a robust "smoother" (which dominates the raw data and hence has a bounded maximum risk when the average squared error loss is concerned).

The spline smoothing approach to nonparametric regression and curve estimation is considered. It is shown that, in a certain sense, spline smoothing corresponds approximately to smoothing by a kernel method with bandwidth depending on the local density of design points. Some exact calculations demonstrate that the approximation is extremely close in practice. Consideration of kernel smoothing methods demonstrates that the way in which the effective local bandwidth behaves in spline smoothing has desirable properties. Finally, the main result of the paper is applied to the related topic of penalized maximum likelihood probability density estimates; a heuristic discussion shows that these estimates should adapt well in the tails of the distribution.

Let $(X, Y)$ be a random vector in the plane. We show that a smoothed N.N. estimate of the regression function $m(x) = \mathbb{E}(Y\mid X = x)$ is asymptotically normal under conditions much weaker than needed for the Nadaraya-Watson estimate. It also turns out that N.N. estimates are more efficient than kernel-type estimates if (in the mean) there are few observations in neighborhoods of $x$.

Suppose $\mathbf{x} = (x_1, \cdots, x_m)^t$ is an $m$-variate normal random variable with mean vector $\mathbf{\theta} = (\theta_1, \cdots, \theta_m)^t$ and identity dispersion matrix. We consider the control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ with respect to the loss $L(\theta, \delta) = (1 - \theta^t\delta)^2,$ where $\delta(x) = (\delta_1(x), \cdots, \delta_m(x))^t$. A necessary and sufficient condition for the admissibility of spherically symmetric generalized Bayes $\delta(x)$ is given in terms of a Dirichlet problem. This condition is also equivalent to recurrence of a diffusion process and insolubility of certain elliptic boundary value problems.

Optimal fixed size confidence procedures are derived for the mean of a normal random variable with known variance, when the mean is restricted to a compact interval. These confidence procedures are, in turn, based on the solution of a related minimax decision problem which is characterized by a zero-one loss function and a compact interval parameter space. The minimax rules obtained are nonrandomized, admissible, Bayes procedures. The decision-theoretic results are extended in two ways: (i) structurally similar (admissible) Bayes minimax rules are also obtained when the sampling distribution has a density function which is unimodal, symmetric about the location parameter and possesses a (strictly) monotone likelihood ratio; (ii) structurally similar minimax rules (minimax within the class of nonrandomized, odd, monotone procedures) are again obtained when the assumption of a monotone likelihood ratio is relaxed.

In this paper we develop a decision theoretic formulation for the problem of deriving posterior distributions for a parameter $\theta$, when the prior information is vague. Let $\pi(d\theta)$ be the true but unknown prior, $Q_\pi(d\theta\mid X)$ the corresponding posterior and $\delta(d\theta\mid X)$ an estimate of the posterior based on an observation $X$. The loss function is specified as a measure of distance between $Q_\pi(\cdot\mid X)$ and $\delta(\cdot\mid X)$, and the risk is the expected value of the loss with respect to the marginal distribution of $X$. When $\theta$ is a location parameter, the best invariant procedure (under translations in $R^n$) specifies the posterior which is obtained from the uniform prior on $\theta$. We show that this procedure is admissible in dimension 1 or 2 but it is inadmissible in all higher dimensions. The results reported here concern a broad class of location families, which includes the normal.

Minimum discriminant information adjustment has primarily been used in the analysis of multinomial data; however, no such restriction is necessary. Let $P$ be a distribution on $R^a$, and let $\mathscr{C}$ be a convex set of distributions on $R^a$. Let $\mathbf{X}_i, 1 \leq i \leq n$, be independent and identically distributed observations with common distribution $P$. The minimum discriminant information adjustment (MDIA) of $P$ relative to $\mathscr{C}$ is the element $Q$ of $\mathscr{C}$ that is closest to $P$ in the sense of Kullback-Leibler discriminant information. If $\bar{P}_n$ is the empirical distribution of the $X_i, 1 \leq i \leq n$, and $\bar{Q}_n$ is the MDIA of $\bar{P}_n$ relative to $\mathscr{C}$, then $\bar{Q}_n$ is the maximum likelihood estimate in $\mathscr{C}$. Let $\mathscr{C}$ consist of distributions $A$ on $R^a$ such that $\int T dA = t$, where $T$ is a measurable transformation from $R^a$ to $R^b$ and $t \in R^b$. It is shown that under mild regularity conditions $\bar{Q}_n$ converges weakly to $Q$, the MDIA of the true $P$, with probability 1 and that $\bar{E}_n(D) = \int Dd\bar{Q}_n$ is an asymptotically normal and asymptotically unbiased estimate of $E(D) = \int D dQ$.

Probability distributions, especially in applications, are generally specified via density functions; alternative representations, including the characteristic function, moment generating function, and sequence of moments, are most commonly encountered in theoretical settings. These alternative means of specification do, however, give rise to the construction of certain approximations that can facilitate the implementation of likelihood methods or the calculation of probabilities, even when the density is not available in closed form. The feasibility of a unified treatment of this topic stems from a number of properties shared by probability transforms in general.

This paper continues the work of Hedayat and Afsarinejad (1978) and of Cheng and Wu (1980) on the optimality of balanced uniform repeated measurements designs. In addition to their well known optimality results over the class of designs which have no pairs of consecutive identical treatments, we compare the balanced uniform designs to all possible designs. Instances are given where they fail and where they succeed in being optimum.

Martingale methods and the martingale invariance principle are used to derive central limit theorems and related results for biased coin designs of the kind previously studied by Efron, Wei and many others. The results are applied to the study of selection bias. The method is developed for the simplest two-treatment case and then extended, first to the case of several treatments, and secondly to the case of two treatments with prognostic factors.

Bayes double sample procedures are given for estimating the mean of exponential family distributions. The distribution can be multiparameter and so the case of normal mean with unknown variance is included. Linear combination loss functions are usually assumed. Stein's double sample procedure, and other intuitive double sample procedures for estimating a binomial parameter, are studied. Recommendations are given, including guidelines for the size of the initial samples.

This paper introduces several notions of symmetry for the joint distribution of two dependent unit vectors. Bivariate generalizations of $\mathscr{L}$-symmetry (Rivest, 1984) and rotational symmetry are introduced. If the joint distribution of two unit vectors is at least $\mathscr{L}$-symmetric the information matrix for the parameters indexing it is shown to have a simple shape.

The problem of selecting a random nonempty subset from $k$ populatinos, characterized by $\theta_1, \cdots, \theta_k$ with possible nuisance parameters $\sigma$, is considered using a decision-theoretic approach. The concept of asymptotic consistency is defined as the property that the risk of a procedure at $(\theta, \sigma)$ tends to the minimum loss at $(\theta, \sigma)$. Necessary and sufficient conditions for both pointwise and uniform (on compact sets) consistency for permutation-invariant procedures are derived with general loss functions. Various loss functions when the goal is to select populations with $\theta_i$ close to $\max \theta_j$ are considered. Applications are made to normal populations. It is shown that Gupta's procedure is the only procedure in Seal's class that can be consistent. Other Bayes and admissible procedures are also considered.

We consider modified empirical Bayes problems in which the prior distribution of $\Theta$ at stage $n + 1$ is $G^{(n+1)}(\theta)$. The Bayes optimality criterion is now given by the sequence of functionals $R(G^{(n+1)}$. The observations $X_1, \cdots, X_n$ are no longer i.i.d so decision procedures are constructed based on modified empirical density estimates for $f_G^{(n+1)}(x)$. Asymptotic optimality together with asymptotic convergence rates is established for two action and estimation problems when the observations are drawn from a member of the one-parameter exponential family.

We derive lower bounds to rates of convergence for estimators of shape and scale parameters in distributions with regularly varying tails. We exhibit simple estimators which attain these rates.

A general definition of symmetry for a directional model is given. Then the information matrix for the parameters indexing a symmetric density is shown to be block-diagonal: one block for the location and one block for the shape. This result is used to construct an algorithm for the efficient estimation of the parameters. Examples are given; a new symmetric distribution, the $FB_6$ is introduced; it generalizes the classical distributions on the hypersphere.

Given independent observations $x_1, \cdots, x_n$ drawn uniformly from an unknown compact convex set $D$ in $\mathbb{R}^p$ ($p$ known) it is desired to estimate $D$ from the observations. This problem was first considered, for $p = 2$, by Ripley and Rasson (1977). We consider a decision-theoretic approach where the loss function is $L(D, \hat{D}) = m(D \Delta \hat{D})$. We prove the completeness of the Bayes estimation rules. A form for the nonrandomized Bayes estimation rules is presented and applied, for an a priori law reflecting ignorance, to the cases $p = 1$ and where $D$ is a rectangle in the plane; some comparisons are made with other estimation methods suggested in the literature. Finally, the consistency of the estimation rules is studied.

A contribution is made to the problem of combining the subjective probability density functions $f_1, \cdots, f_n$ of $n$ individuals for some parameter $\theta$. More precisely, the situation is addressed which occurs when the members of a group share a common likelihood for some data and want to ensure that combining their posterior distributions for $\theta$ will yield the same result obtained by applying Bayes' rule to the aggregated prior distribution. Under certain regularity conditions to be discussed below, the logarithmic opinion pool $\prod^n_{i=1} f^{w_i}_i \big/ \int \prod^n_{i=1} f^{w_i}_i d\mu$ with $w_i \geq 0$ and $\sum^n_{i=1} w_i = 1$ is shown to be the only pooling formula which satisfies this criterion of group rationality.

The problem of characterizing the rectangular region of maximum probability content among the rectangles of fixed area is considered. The rectangles are assumed to have their sides parallel to the axes and the bivariate distribution is assumed to have density elliptically contoured and unimodal.

A complete class result is obtained for double sample estimation of a binomial parameter. The complete class leads to some inadmissibility results for some procedures that are likely to be used in practice. It is also shown that the sample proportion as a true double estimator is not unbiased.

Let $x(t)$ be a Wiener process with drift $\mu$ and variance 1 per unit time. The following problem is treated; test $H:\mu \leq 0$ vs. $A:\mu > 0$ with the loss function $|\mu|$ if the wrong decision is made and 0 otherwise, and with $c =$ cost of observation per unit time, where $\mu$ has a prior distribution which is normal with mean 0 and variance $\sigma^2_0$. An idea of Bickel and Yahav is followed to obtain a lower bound for the Bayes risk which is strict as $\sigma_0 \rightarrow \infty$ for all $c$. An upper bound is also derived.

Asymptotic normality of a class of nonlinear rank statistics to test the null hypothesis of total independence of an $m$-variate population is proved. The rank statistics are generated from $2m$-variate square integrable functions such that they are symmetric and nondegenerate. Some results under contiguous alternatives are also given.

Efron (1979, 1982), in his treatment of the bootstrap, discusses its use for estimation of the asymptotic variance of the sample median, in the sampling situation of independent and identically distributed random variables with common distribution function $F$ having a positive derivative continuous in a neighborhood of the true median $\mu$. The natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example to be false unless a tail condition is imposed on $F$. We prove that such strong convergence does hold under the fairly nonrestrictive condition that $E\lbrack\mid X^\alpha\rbrack < \infty$ for some $\alpha > 0$.

For comparing $k$ treatment effects $\theta_1, \theta_2, \cdots \theta_k$, often the parameters of primary interest are $\theta_i - \max_{j\neq i}\theta_j, i = 1, \cdots, k$. In this article, we develop constrained $100P^\ast{\tt\%}$ two-sided simultaneous confidence intervals for $\theta_i - \max_{j \neq i} \theta_j$ which we refer to as (constrained) MCB intervals. It turns out that the lower bounds of the intervals imply Indifference Zone selection inference, and the upper bounds of these intervals imply Subset Selection inference, each given at the same confidence level $100P^\ast{\tt\%}$ as MCB. We also extend our method to give $100P^\ast{\tt\%}$ simultaneous confidence intervals for $\theta_i - \theta^{(i)}_{(k-t)}, i = 1, \cdots, k$, where $\theta^{(i)}_{(k-t)}$ is the $t$th largest among the $\theta$'s excluding $\theta_i$.