Stable laws are often fit to outlier-prone data and, if the index $\alpha$ is estimated to be much less than two, then the normal law is rejected in favor of an infinite-variance stable law. This paper derives the theoretical properties of such a procedure. When the true distribution is stable, the distribution of the m.l.e. of $\alpha$ is non-regular if $\alpha = 2$. When the true distribution is not stable, the estimate of $\alpha$ is not a robust measure of the rate of decrease of the tail probabilities. A more robust procedure is developed, and a statistic for describing and comparing the tail-shapes of arbitrary samples is proposed.

This work provides the elements of a framework for multi-Bayesian statistical decision theory. Solution concepts and criteria are presented. The relationship to Wald's theory is discussed. And two criteria for assessing group decision procedures are defined. One is based on the idea of subsampling the group, and it is found that among the proposed solution concepts only Nash's solution is optimal under subsampling as well. The other assumes the group is itself a sample from a superpopulation, and this yields an analogue of Wald's theory where the elicitation of the priors becomes part of the experimental process. Results on admissibility, minimaxity and so on found in Wald's classical theory become directly applicable in the new setting.

Suppose you are given the opportunity of guessing whether it will snow or not in Chicago next Christmas. If you guess correctly, you win 1,000; if not, you win nothing. Which event, snow or no snow, would you bet on? It is widely accepted among decision theorists that your answer reveals which of the two events you deem more probable. Furthermore, if your choices over a field of events obey certain postulates of coherency and consistency, then there is a probability measure $P$ on the field that reflects your choices: you regard $A$ as more probable than $B$ if, and only if, $P(A) > P(B)$. Numerous experiments have shown that people often violate those postulates, so they lack full descriptive validity. Moreover, because of systematic and persistent violations of one of the postulates--an independence axiom-the theory has been questioned on its normative adequacy as a guide to well-reasoned judgments and choices. The purpose of the present paper is to examine a weaker set of postulates that avoids the independence axiom as well as the usual assumption of fully transitive preferences. Despite this weakening, the assumptions imply that there is a unique normalized functional $\rho$ on pairs of events that preserves choices in the sense that $A$ is more probable than $B$ if, and only if, $\rho(A, B) > 0$. The functional $\rho$ has several nice mathematical properties, including "conditional additivity," that reflect vestiges of numerical probability, and it is related to the conventional measure $P$ by $\rho(A, B) = P(A) - P(B)$ when the omitted independence axiom is coupled to the other postulates.

The problem of optimal experimental design for estimating parameters in linear regression models is placed in a general convex analysis setting. Duality results are obtained using two approaches, one based on subgradients and the other on Lagrangian theory. The subgradient concept is also used to derive a potentially useful equivalence theorm for establishing the optimality of a singular design and, finally, general versions of the original equivalence theorems of Kiefer and Wolfowitz (1960) are obtained.

The notion of general balance due to Nelder is discussed in relation to the eigenvectors of an information matrix, combinatorial balance and the simple combinability of information from uncorrelated sources in an experiment.

Necessary and sufficient conditions for an invariant test to be admissible among invariant tests in the general multivariate analysis of variance problem are presented. It is shown that in many cases the popular tests based on the likelihood ratio matrix are inadmissible. Other tests are shown admissible. Numerical work suggests that the inadmissibility of the likelihood ratio test is not serious. The results are given for the multivariate analysis of variance problem as a special case.

Suppose that a test of fit to a parametric family of distributions is employed, with critical points determined from the limiting null distribution of the test statistic for IID observations. It is shown that if the observations are in fact a stationary process satisfying a positive dependence condition, the test will reject a true null hypothesis too often. This result is established for a broad class of chi squared and empiric df tests, including the Pearson, Kolmogorov-Smirnov and Cramer-von Mises tests with general estimators of unknown parameters. Furthermore, the method of proof is sufficiently general to apply also to other classes of tests. Confounding of positive dependence with lack of fit is therefore a general phenomenon in the use of omnibus tests of fit.

Consider sampling from an unknown probability distribution on the integers. With a tail-free prior, the posterior distribution is consistent. With a mixture of a tail-free prior and a point mass, however, the posterior may be inconsistent. This is likewise true for a countable mixture of tail-free priors. Similar results are given for Dirichlet priors.

This paper concerns nonparametric inference for hazard rates with censored serial data. The focus is upon "delta sequence" estimators of the form $h_n(x) = \int K_b(x, y) dH_n(y)$ with $K_b$ integrating to 1 and concentrating mass near $x$ as $b \rightarrow 0. H_n$ is the Nelson-Aalen empirical cumulative hazard. Strong approximation and simultaneous confidence bands are derived for Rosenblatt-Parzen estimators, with $K_b(x, y) = w((x - y)/b)/b, b = o(n^{-1}),$ and $w(\cdot)$ a well-behaved density. This work generalizes global deviation and mean square deviation results of Bickel and Rosenblatt and others to censored survival data. Simulations with exponential survival and censoring indicate the effect of censoring on bias, variance, and maximal absolute deviation. Data from a survival experiment with serial sacrifice are analysed.

For the nonparametric regression model $Y(t_i) = \theta(t_i) + \varepsilon(t_i)$ where $\theta$ is a smooth function to be estimated, $t_i$'s are nonrandom, $\varepsilon(t_i)$'s are i.i.d. errors, this paper studies the behavior of the kernel regression estimate $\hat{\theta}(t) = \big\lbrack \sum^n_{j=1}K \big(\frac{t_j - t}{\lambda}\big) Y(t_j) \big\rbrack / \big\lbrack\ sum^n_{j=1} K \big(\frac{t_j - t}{\lambda}\big) \big\rbrack$ when $\lambda$ is chosen by cross-validation on the average squared error. Strong consistency in terms of the average squared error is established for uniform spacing, compact kernel and finite fourth error moment.

Consider the multiclassification (discrimination) problem with known prior probabilities and a multi-dimensional vector of observations. Assume the underlying densities corresponding to the various classes are unknown but a training sample of size $N$ is available from each class. Rates of convergence to Bayes risk are investigated under smoothness conditions on the underlying densities of the type often seen in nonparametric density estimation. These rates can be drastically affected by a small change in the prior probabilities, so the error criterion used here is Bayes risk averaged (uniformly) over all prior probabilities. Then it is shown that a certain rate, $N^{-r}$, is optimal in the sense that no rule can do better (uniformly over the class of smooth densities) and a rule is exhibited which does that well. The optimal value of $r$ depends on the smoothness of the distributions and the dimensionality of the observations in the same way as for nonparametric density estimation with integrated square error loss.

We prove that the method of cross-validation suggested by A. W. Bowman and M. Rudemo achieves its goal of minimising integrated square error, in an asymptotic sense. The tail conditions we impose are only slightly more severe than the hypothesis of finite variance, and so least squares cross-validation does not exhibit the pathological behaviour which has been observed for Kullback-Leibler cross-validation. This is apparently the first time that a cross-validatory procedure for density estimation has been shown to be asymptotically optimal, rather then simply consistent.

In the general two-sample testing problem, $X_1, \cdots, X_m$ i.i.d. with continuous c.d.f. $F, Y_1, \cdots, Y_n$ i.i.d. with continuous c.d.f. $G$, and null hypothesis $H_0: F = G$ versus alternative $H_1: F \leq G, F \neq G$, we construct uniformly consistent and tractable rank estimators of the underlying optimal nonparametric score-function for a large subclass of (fixed) alternatives. Moreover, we prove asymptotic normality of the corresponding adaptive rank statistics under any fixed alternative $(F, G)$ from the same subclass, and compare the results with the corresponding results for the (local) asymptotically optimum linear rank statistic for $H_0$ versus $(F, G)$. In addition we prove some results on the estimation of a density and its derivative in the i.i.d. case if the support is [0,1], which are needed for a comparison argument in the case of rank estimators, but which may be of interest in other situations, too.

Consider the problem of estimating the unknown location parameter $\theta$ based on a random sample from $F(x - \theta)$, where $F$ is an unknown member of the class of distribution functions $\mathscr{F} = \{F: F$ is symmetric about 0 and $\sup_x |F(x) - \Phi(x)| \leq \varepsilon\}$, where $\Phi$ denotes the standard normal distribution function. Huber (1964) showed that $M$-estimation has a minimax property for this model, whereas Sacks and Ylvisaker (1972) showed that $L$-estimation fails to have the minimax property. It is shown here that $R$-estimation does have the minimax property for this model.

Solutions of simultaneous equations of the maximum likelihood type or $M$-estimators can be represented as functionals. Existence and uniqueness of a root in a local region of the parameter space are proved under conditions that are easy to check. If only one root of the equation exists, the resulting statistical functional is Frechet differentiable and robust. When several solutions exists, conditions on the loss criterion used to select the root for the statistic ensure Frechet differentiability. An interesting example of a Frechet differentiable functional is the solution of the maximum likelihood equations for location and scale parameters in a Cauchy distribution. The estimator is robust and asymptotically efficient.

The parameters of a stationary process can be viewed as functions of the spectral distribution function. This work concerns (estimators) parameters defined as integrals of $m (\geq 1)$-dimensional kernel functions with respect to the (sample) spectral distribution function. Conditions for asymptotic normality, almost sure convergence, and probability one bounds are derived for such estimators. The approach taken is based upon the reduction of an $m$-dimensional problem to one-dimension via consideration of a Frechet differential and its linearity. The probability one bound for the estimators is obtained by first establishing it $(O((n^{-1}\log n)^{1/2}))$ for the difference of the sample and true spectral distribution functions in the supnorm and then showing that this rate is transferred to the estimators through integration.

The problem of estimating an event, having positive probability content, based on a sample of $n$ observations is considered. A natural metric is shown to exist on the space of possible values for the event. This leads to the definition of optimal estimators. We derive optimal estimators for events which correspond to quantiles for the univariate exponential model. Further optimal estimators are derived for the events bounded by the ellipsoidal contours of the density function in the multivariate normal model.

For testing a probability distribution on a compact Riemannian manifold for symmetry under the action of a given group of isometries, two classes of invariant tests are proposed and some properties noted. These tests are based on Sobolev norms and generalize Gine's Sobolev tests of uniformity. For general compact manifolds randomization tests analogous to Wellner's tests for the two-sample case are suggested. For the circle, distribution-free tests of symmetry based on uniform scores are provided.

The asymptotic distribution for the principal component roots under local alternatives to multiple population roots is derived. The asymptotic theory assumes the estimate of the population covariance or scatter matrix to be asymptotically normal and to possess certain invariance properties. These assumptions are satisfied for the affine-invariant $M$-estimates of scatter for an elliptical distribution. The local alternative framework is used in deriving a local power function for the test for subsphericity.

In this paper, the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered. A class of invariant asymptotic tests based on the sample covariance matrix is derived. Tests in this class are shown to be consistent and their local power functions are given. The arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix. The results are shown to generalize when the procedures are based on any affine-invariant $M$-estimate of scatter and when the population is elliptical.

The expressions for coverage probability and ASN of the Stein's two-stage confidence interval procedure for estimating the normal mean have been obtained under the assumption that the underlying distribution is, in fact, different from normal but could be approximated by the first four terms of Edgeworth series. The comparison of coverage probabilities with the corresponding probabilities obtained for the normal distribution shows that the procedure is quite insensitive to moderate departures from normality, and skewness of the parent population has very little effect on the coverage probability of the procedure. The Monte Carlo investigations which involve sampling from the gamma population confirm these conclusions.

Wu (1980) defined the consistent directions of the least squares estimator in a linear model as the linear combinations of parameter estimates that are asymptotically consistent. For the polynomial regression model, a characterization of the space of consistent directions $S$ was obtained in terms of the convergence rates of the corresponding design sequence to its limit points. By employing a more general and yet simpler approach, we obtain here a similar result for any regression model with one independent variable and smooth regression function. When $f_i(x)$ is an extended Tchebycheff system, the above characterization is further refined and the consistency region $C$ is shown to be either the set of limit points of the design sequence or the whole real line.

An integral representation of the positive stable distributions is used to give a tabulation of the asymptotic test statistic for Watson's optimal invariant test for uniformity of distribution on a circle.