Let x₍₁₎ denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x₍₁₎ is the largest principal component variance of the covariance matrix $X'X$, or the largest eigenvalue of a p­variate Wishart distribution on n degrees of freedom with identity covariance.

Consider the limit of large p and n with $n/p = \gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled by $\sigma_p = (\sqrt{n-1} + \sqrt{p})(1/\sqrt{n-1} + 1/\sqrt{p}^{1/3}$, the distribution of x₍₁₎ approaches the Tracey-Widom law of order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5.

The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.

Let Z be a k­way array consisting of independent standard normal variables. For column vectors h₁, …, h_k, define a multilinear form of degree k by $(h_1 \otimes \cdots \otimes h_k)' \vec(Z)$. We derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the h_i’s under the condition that the h_i’s are unit vectors, and of its standardized statistic obtained by dividing by the norm of Z. We also give formulas for the maximum of a symmetric multilinear form $(h_1 \otimes \cdots \otimes h_k)' \sym(Z)$, where sym (Z) denotes the symmetrization of Z with respect to indices. These classes of statistics are used for testing hypotheses in the analysis of variance of multiway layout data and for testing multivariate normality. In order to derive the tail probabilities we employ a geometric approach developed by Hotelling, Weyl and Sun. Upper and lower bounds for the tail probabilities are given by reexamining Sun's results. Some numerical examples are given to illustrate the practical usefulness of the obtained formulas, including the upper and lower bounds.

We develop a nonparametric estimation theory in a nonstationary environment, more precisely in the framework of null recurrent Markov chains. An essential tool is the split chain, which makes it possible to decompose the times series under consideration into independent and identical parts. A tail condition on the distribution of the recurrence time is introduced. This condition makes it possible to prove weak convergence results for sums of functions of the process depending on a smoothing parameter. These limit results are subsequently used to obtain consistency and asymptotic normality for local density estimators and for estimators of the conditional mean and the conditional variance. In contradistinction to the parametric case, the convergence rate is slower than in the stationary case, and it is directly linked to the tail behavior of the recurrence time. Applications to econometric, and in particular to cointegration models, are indicated.

The subject of this paper is autoregressive (AR) modeling of a stationary, Gaussian discrete time process, based on a finite sequence of observations. The process is assumed to admit an AR($\infty$) representation with exponentially decaying coefficients. We adopt the nonparametric minimax framework and study how well the process can be approximated by a finite­order AR model. A lower bound on the accuracy of AR approximations is derived, and a nonasymptotic upper bound on the accuracy of the regularized least squares estimator is established. It is shown that with a “proper” choice of the model order, this estimator is minimax optimal in order. These considerations lead also to a nonasymptotic upper bound on the mean squared error of the associated one­step predictor. A numerical study compares the common model selection procedures to the minimax optimal order choice.

We derive the asymptotic distribution of the sequential empirical process of the squared residuals of an ARCH(p) sequence. Unlike the residuals of an ARMA process, these residuals do not behave in this context like asymptotically independent random variables, and the asymptotic distribution involves a term depending on the parameters of the model. We show that in certain applications, including the detection of changes in the distribution of the unobservable innovations, our result leads to asymptotically distribution free statistics.

Scatterplot smoothers estimate a regression function y = f(x) by local averaging of the observed data points (x_i, y_i). In using a smoother, the statistician must choose a “window width,” a crucial smoothing parameter that says just how locally the averaging is done. This paper concerns the data­based choice of a smoothing parameter for splinelike smoothers, focusing on the comparison of two popular methods, C_p and generalized maximum likelihood. The latter is the MLE within a normal­theory empirical Bayes model. We show that C_p is also maximum likelihood within a closely related nonnormal family, both methods being examples of a class of selection criteria. Each member of the class is the MLE within its own one­parameter curved exponential family. Exponential family theory facilitates a finite­sample nonasymptotic comparison of the criteria. In particular it explains the eccentric behavior of C_p, which even in favorable circumstances can easily select small window widths and wiggly estimates of f(x). The theory leads to simple geometric pictures of both C_p and MLE that are valid whether or not one believes in the probability models.

We describe a hierarchy of exponential families which is useful for distinguishing types of graphical models. Undirected graphical models with no hidden variables are linear exponential families (LEFs). Directed acyclic graphical (DAG) models and chain graphs with no hidden variables, includ­ ing DAG models with several families of local distributions, are curved exponential families (CEFs). Graphical models with hidden variables are what we term stratified exponential families (SEFs). A SEF is a finite union of CEFs of various dimensions satisfying some regularity conditions. We also show that this hierarchy of exponential families is noncollapsing with respect to graphical models by providing a graphical model which is a CEF but not a LEF and a graphical model that is a SEF but not a CEF. Finally, we show how to compute the dimension of a stratified exponential family. These results are discussed in the context of model selection of graphical models.

In this paper, the problem of constructing optimal blocked regular fractional factorial designs is considered. The concept of minimum aberration due to Fries and Hunter is a well­accepted criterion for selecting good unblocked fractional factorial designs. Cheng, Steinberg and Sun showed that a minimum aberration design of resolution three or higher maximizes the number of two­factor interactions which are not aliases of main effects and also tends to distribute these interactions over the alias sets very uniformly. We extend this to construct block designs in which (i) no main effect is aliased with any other main effect not confounded with blocks, (ii) the number of two­factor interactions that are neither aliased with main effects nor confounded with blocks is as large as possible and (iii) these interactions are distributed over the alias sets as uniformly as possible. Such designs perform well under the criterion of maximum estimation capacity, a criterion of model robustness which has a direct statistical meaning. Some general results on the construction of blocked regular fractional factorial designs with maximum estimation capacity are obtained by using a finite projective geometric approach.

By studying treatment contrasts and ANOVA models, we propose a generalized minimum aberration criterion for comparing asymmetrical fractional factorial designs. The criterion is independent of the choice of treatment contrasts and thus model­free. It works for symmetrical and asymmetrical designs, regular and nonregular designs. In particular, it reduces to the minimum aberration criterion for regular designs and the minimum G₂ ­aberration criterion for two­level nonregular designs. In addition, by exploring the connection between factorial design theory and coding theory, we develop a complementary design theory for general symmetrical designs, which covers many existing results as special cases.

This paper is concerned with nonsequential optimal designs for a class of nonlinear growth models, which includes the asymptotic regression model. This design problem is intimately related to the problem of finding optimal designs for polynomial regression models with only partially known heteroscedastic structure. In each case, a straightforward application of the usual D­optimality criterion would lead to designs which depend on the unknown underlying parameters. To overcome this undesirable dependence a maximin approach is adopted. The theorem of Perron and Frobenius on primitive matrices plays a crucial role in the analysis.

We find a sufficient condition on the spectrum of a partial geometric design d* such that, when d* satisfies this condition, it is better (with respect to all convex decreasing optimality criteria) than all unequally replicated designs (binary or not) with the same parameters b, v, k as d*.

Combining this with existing results, we obtain the following results:

(i) For any q \ge 3, a linked block design with parameters b = q², v = q² + q, k = q² -1 is optimal with respect to all convex decreasing optimality criteria in the unrestricted class of all connected designs with the same parameters.

(ii) A large class of strongly regular graph designs are optimal w.r.t. all type 1 optimality criteria in the class of all binary designs (with the given parameters). For instance, all connected singular group divisible (GD) designs with \lambda_1 = \lambda_2 + 1 (with one possible exception) and many semiregular GD designs satisfy this optimality property.

Specializing these general ideas to the A­criterion, we find a large class of linked block designs which are A­optimal in the un­restricted class. We find an even larger class of regular partial geometric designs (including, for instance, the complements of a large number of partial geometries) which are A­optimal among all binary designs.