Pattern theory offers concepts for modelling images and methods for making inferences from observed images. This will be described briefly and illustrated by examples. We shall present limit theorems for the Markov processes on graphs (that are basic to pattern theory) motivated by computational considerations. They will yield approximations that have been exploited to make the inference algorithms computationally feasible. We shall also consider the problem of estimating parameters in the prior measures encountered in pattern theory. These parameters are high dimensional, not automatically identifiable and notoriously difficult to estimate by standard methods. We therefore present a standardization technique for dealing with them and show how, after standardization, the remaining free parameters can be estimated by different methods. The estimation methods are examined in terms of their asymptotic efficiencies.

We define and investigate classes of statistical models for the analysis of associations between variables, some of which are qualitative and some quantitative. In the cases where only one kind of variables is present, the models are well-known models for either contingency tables or covariance structures. We characterize the subclass of decomposable models where the statistical theory is especially simple. All models can be represented by a graph with one vertex for each variable. The vertices are possibly connected with arrows or lines corresponding to directional or symmetric associations being present. Pairs of vertices that are not connected are conditionally independent given some of the remaining variables according to specific rules.

Projection pursuit regression and kernel regression are methods for estimating a smooth function of several variables from noisy data obtained at scattered sites. Methods based on local averaging can perform poorly in high dimensions (curse of dimensionality). Intuition and examples have suggested that projection based approaches can provide better fits. For what sorts of regression functions is this true? When and by how much do projection methods reduce the curse of dimensionality? We make a start by focusing on the two-dimensional problem and study the $L^2$ approximation error (bias) of the two procedures with respect to Gaussian measure. Let RA stand for a certain PPR-type approximation and KA for a particular kernel-type approximation. Building on a simple but striking duality for polynomials, we show that RA behaves significantly better than the minimax rate of approximation for radial functions, while KA performs significantly better than the minimax rate for harmonic functions. In fact, the rate improvements carry over to large classes, RA behaving very well for functions with enough angular smoothness (oscillating slowly with angle), while KA behaves very well for functions with enough Laplacian smoothness, (oscillations averaging out locally). The rate improvements matter: They are equivalent to lowering the dimensionality of the problem. For example, for functions with nice tail behavior, RA behaves as if the dimensionality of the problem were 1.5 rather than its nominal value 2. Also, RA and KA are complementary: For a given function, if one method offers a dimensionality reduction, the other does not.

The simple hypothesis is tested that the distribution of independent random variables $X_1, X_2, \cdots, X_n$ is a given probability measure $P_0$. Let $\pi_n$ be any sequence of partitions. The alternative hypothesis is the set of probability measures $P$ with $\sum_{A\in \pi_n}|P(A) - P_0(A)| \geq \delta$, where $\delta > 0$. Note the dependence of this set of alternatives on the sample size. It is shown that if the effective cardinality of the partitions is of the same order as the sample size, then sequences of tests exist with uniformly exponentially small probabilities of error. Conversely, if the effective cardinality is of larger order than the sample size, then no such sequence of tests exists. The effective cardinality is the number of sets in the partition which exhaust all but a negligible portion of the probability under the null hypothesis.

Stochastic procedures are randomized statistical procedures which are functions of the observed sample and of one or more artificially constructed auxiliary samples. As the size of the auxiliary samples increases, a stochastic procedure becomes nearly nonrandomized. The stochastic test of this paper arises as a numerically feasible approximation to a natural minimum distance goodness-of-fit test for multivariate parametric models. The distance being minimized here is the half-space metric for probabilities on a Euclidean space. It is shown that the various approximations used in constructing the stochastic test and its critical values do not detract from its first-order asymptotic performance.

In this paper, the asymptotic behavior of some nonparametric tests is studied in situations where both bootstrap tests and randomization tests are applicable. Under fairly general conditions, the tests are asymptotically equivalent in the sense that the resulting critical values and power functions are appropriately close. This implies, among other things, that the difference in the critical functions of the tests, evaluated at the observed data, tends to 0 in probability. Randomization tests may be preferable since an exact desired level of the test may be obtained for finite samples. Examples considered are: testing independence, testing for spherical symmetry, testing for exchangeability, testing for homogeneity, and testing for a change point.

The bootstrap estimate of distribution functions of studentized statistics is shown to be more accurate than even the two-term empirical Edgeworth expansion, thus strengthening the claim of superiority of the bootstrap over the normal approximation method. The two methods are compared not only with respect to bounded bowl-shaped loss functions but also with respect to squared error loss and, more generally, in $L^p$-norms.

The rate of convergence in the central limit theorem for functions of independent random variables is studied in a unifying approach. The basic result sharpens and extends a theorem of van Zwet. Applications to $U$-, $L$- and $R$-statistics are also given, improving or extending the results of Helmers and van Zwet, Helmers and Huskova, Does and van Es and Helmers.

Two new derivations of the Hotelling-Naiman results on the volume of tubes about curves in spheres are presented. The first involves simple differential inequalities. The second is probabilistic, using the concept of upcrossing borrowed from the theory of Gaussian processes. The upcrossings method is extended to an harmonic regression problem not covered by the Hotelling-Naiman formulation.

In this paper we characterize transformation models by means of the functional form of the densities. We discuss sufficiency of the pair $(t, \pi)$ where $t$ is an equivariant estimator and $\pi$ is a maximal invariant. Furthermore, we introduce and discuss the algebraic concept of structural sufficiency. This gives rise to an example of a transformation model where $(t, \pi)$ is nonsufficient.

Hypothesis testing problems in which the null hypothesis is simple, the parameter space is finite dimensional and the supports of the probability measures are independent of the parameter are considered. Essentially complete class results are obtained for characterizing the limits of Bayes tests. Conditions for tests to be admissible and the class to be complete are given. Results are then specialized to exponential families, along with some illustrative examples.

Consider the one-way fixed and balanced analysis of variance model under the assumptions of independence and normality. The problem is to test for homogeneity of variances. A necessary and sufficient condition for admissibility of a test among the class of scale invariant tests is given. Hartley's test and Cochran's test are not in the class and therefore are inadmissible. Various scale invariant tests are examined for parameter consistency. Parameter consistency in this case means the following: Consider a sequence of values of the maximal invariant parameter in the alternative space. If the Kullback-Leibler distance from this sequence to the null point tends to infinity then the power of the test tends to 1. Several well known test are shown to be parameter consistent (PC) for all significance levels. Some well known tests however may not be PC or may be PC only for certain significance levels. Extensions of PC results to nonnormal cases are indicated.

Assume the standard linear model $X_{n \times 1} = A_{n \times p} \theta_{p \times 1} + \varepsilon_{n \times 1},$ where $\varepsilon$ has an $n$-variate normal distribution with zero mean vector and identity covariance matrix. The least squares estimator for the coefficient $\theta$ is $\hat{\theta} \equiv (A'A)^{-1}A'X$. It is well known that $\hat{\theta}$ is dominated by James-Stein type estimators under the sum of squared error loss $|\theta - \hat{\theta}|^2$ when $p \geq 3$. In this article we discuss the possibility of improving upon $\hat{\theta}$, simultaneously under the "universal" class of losses: $\{L(|\theta - \hat{\theta}|): L(\cdot) \text{any nondecreasing function}\}.$ An estimator that can be so improved is called universally inadmissible ($U$-inadmissible). Otherwise it is called $U$-admissible. We prove that $\hat{\theta}$ is $U$-admissible for any $p$ when $A'A = I$. Furthermore, if $A'A \neq I$, then $\hat{\theta}$ is $U$-inadmissible if $p$ is "large enough." In a special case, $p \geq 4$ is large enough. The results are surprising. Implications are discussed.

The paper is devoted to the problem of simultaneous estimation of scale and natural parameters of the multiparameter gamma distribution under a quadratic loss. The vector of the scale parameters is assumed to range over a certain subset of the Cartesian product $\mathscr{R}^n_+$ of $n$ positive half lines. We identify the class of all linear admissible estimators for the scale parameters and show that all linear estimators of the natural parameters are inadmissible. Since the problem of invariant quadratic estimation of variance components in balanced random effects normal models leads to a problem of linear estimation of parametric functions of gamma scale parameters restricted to subsets of $\mathscr{R}^n_+$ being considered in this paper, some results on admissibility of invariant quadratic estimators of variance components are also established.

Multivariate errors-in-variables regression models with normal errors are considered and residuals, similar to those calculated from ordinary least squares regressions, are defined for these models. It is shown that under the assumption of a $n^{1/2}$-consistent estimator of the vector of regression coefficients, certain empirical processes based upon the residuals converge to the same Gaussian process as that of an infinite sequence of normal random variables standardized by their sample mean sample variance.

Suppose $u_1, \cdots, u_n, v_1, \cdots, v_n$ are random points on the sphere such that for unknown points $\xi_1, \cdots, \xi_n$ and unknown rotation $A_0$, the distribution of $u_i$ depends only on $u^t_i\xi_i$ and that of $v_i$ on $v^t_iA_0\xi_i$. This paper provides asymptotic tests and confidence regions for $A_0$ and for its axis of rotation. Results are given in arbitrary dimension.

Spherical regression studies models which postulate that the unit vector $v$ is equal to an unknown rotation $P$ of the unit vector $u$ "plus" an experimental error. The case where the experimental errors follow a Fisher-von Mises distribution with a large concentration parameter $\kappa$ is considered in this work. Asymptotic $(\kappa \rightarrow \infty)$ inferential procedures for $P$ are proposed when $n$, the sample size, is fixed. Diagnostic methods for spherical regression are suggested. The key for their derivation is the fact that spherical regression is "locally" identical to ordinary least square regression. The results are presented in an arbitrary dimension. For the three-dimensional case, asymptotic tests and confidence regions for the axis and the angle of $P$ are obtained. The data from a plate tectonic analysis of the Gulf of Aden, presented by Cochran, illustrate the proposed methodology.

A class of invariant hypothesis tests is considered for the purpose of testing a variance ratio arising in mixed models. Members of the class are most powerful for specific alternatives and limiting members of the class are Wald's test and the locally most powerful test. It is demonstrated that the locally most powerful test has the highest and Wald's test has the lowest asymptotic power when an asymptotically unbalanced sequence of ANOVA designs is considered under Pitman alternatives.

In the context of Huber's theory of robust estimation of a location parameter, the literature on minimax properties of $M$-, $R$- and $L$-estimators is surveyed. New results are obtained for the model in which the unknown error distribution is assumed to lie in a Levy neighbourhood of a symmetric distribution $G: \mathscr{P}_{\varepsilon,\delta}(G) = \{F\mid G(x - \delta) - \varepsilon \leq F(x) \leq G(x + \delta) + \varepsilon \text{for all} x\}$. Under reasonably general conditions on $G$, the distribution $F_0$ in $\mathscr{P}_{\varepsilon,\delta}(G)$ which minimizes Fisher information for location is found. Huber's minimax property for $M$-estimators is shown to hold for $R$-estimators but to fail for $L$-estimators in Levy neighbourhoods. The latter is proved by constructing a subneighbourhood of distributions $\mathscr{F}_0$, with $F_0 \in \mathscr{F}_0 \subset \mathscr{P}_{\varepsilon\delta}(G)$, such that the asymptotic variance of the $L$-estimator which is asymptotically efficient at $F_0$ is minimized over $\mathscr{F}_0$ at $F_0$.

We relate the asymptotic behavior of $M$-estimators of the regression parameter in a linear model in which the dimension of the regression parameter may increase with the sample size to the stochastic equicontinuity of an associated $M$-process. The approach synthesises a number of results for the dimensionally fixed regression model and then extends these results in a direct unified way. The resulting theorems require only mild conditions on the $\psi$-function and the underlying distribution function. In particular, the results do not require $\psi$ to be smooth and hence can be applied to such estimators as the least absolute deviations estimator. We also treat one-step $M$-estimation.

Asymptotically efficient (adaptive) estimators for the slope parameters of the linear regression model are constructed based upon the "regression quantile" statistics suggested by Koenker and Bassett. The estimators are natural analogues of the adaptive $L$-estimators of location of Sacks, but employ kernel-density type estimators of the optimal $L$-estimator weight function.

A stochastic expansion for $M$-estimates in linear models with many parameters is derived under the weak condition $\kappa n^{1/3}(\log n)^{2/3} \rightarrow 0$, where $n$ is the sample size and $\kappa$ the maximal diagonal element of the hat matrix. The expansion is used to study the asymptotic distribution of linear contrasts and the consistency of the bootstrap. In particular, it turns out that bootstrap works in cases where the usual asymptotic approach fails.

This paper aims at improvement in regularity conditions on spectral densities for the limit theorems of the quasi maximum-likelihood estimator and the quasi likelihood-ratio statistic. The approach parallels the Daniels-Huber-Pollard proofs of central limit theorems under nonstandard conditions for i.i.d. situations. The results of the paper enable one to dispense with excessive regularity conditions on the spectral density.

The characteristic equation of a multiple autoregressive time series involves the eigenvalues of a matrix equation which determine if the series is stationary. Suppose one eigenvalue is 1 and the rest are less than 1 in magnitude. We show that ordinary least squares may be used to estimate the matrices involved and that the largest estimated eigenvalue has distributional properties that allow us to test this unit root hypothesis using critical values tabulated by Dickey (1976). See also Fountis (1983). If a single unit root is suspected, a model can be fit whose parameters are constrained to produce an exact unit root. This is the vector analog of differencing in the univariate case. In the fitting process, canonical series can be computed thus extending the work of Box and Tiao (1977) to the unit root case.

This paper concerns a partial ordering of positive dependence of a class of random variables which have a common marginal distribution and are not necessarily exchangeable. The main theorem is obtained by applying a moment inequality via majorization. Inequalities for exchangeable random variables, for random variables whose marginal densities possess the semigroup property and for the multivariate normal distribution are then obtained as special cases.

Each individual in a population of size $n$ is assigned a positive number as its weight. $k$ of the $n$ are sampled without replacement, with the individuals remaining at the time of each selection chosen with probabilities proportional to their weights. It is shown that for two fixed individuals, the events that each is in the sample can be positively correlated.

Let $\phi$ be a parameter for which there is no unbiased estimator. This note shows that for an arbitrary sequence of estimators $T^{(k)}$, if the biases of $T^{(k)}$ tend to 0 then their variances must tend to $\infty$.

Let $X$ be a random vector with at least one marginal having a lattice distribution. For a wide class of statistics which can be written as a function of means of independent copies of $X$, it is established in this article that the one-term Edgeworth expansion is typically the same as the usual one-term expansion in the pure nonlattice case.

We show that Bush's bound for the maximum number of constraints in an orthogonal array of index unity is uniformly better than Rao's bound.