In this paper, a nonparametric Bayesian analysis of queueing models with geometric input and general service time is performed. In particular, statistical inference for the probability generating function p.g.f. of the equilibrium waiting time distribution is considered. The consistency of the posterior distribution for such a p.g.f., as well as the weak convergence to a Gaussian process of a suitable rescaling, are proved. As by-products, results on statistical inference for queueing characteristics are also obtained. Finally, the problem of estimating the probability of a long delay is considered.

Discriminant analysis for two data sets in $\mathbb{R}^d$ with probability densities $f$ and $g$ can be based on the estimation of the set $G = \{x: f(x) \geq g(x)\}$. We consider applications where it is appropriate to assume that the region $G$ has a smooth boundary or belongs to another nonparametric class of sets. In particular, this assumption makes sense if discrimination is used as a data analytic tool. Decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered. Their rates of convergence are obtained. We show that these rules achieve optimal rates for estimation of $G$ and optimal rates of convergence for Bayes risks. An interesting conclusion is that the optimal rates for Bayes risks can be very fast, in particular, faster than the “parametric” root-$n$ rate. These fast rates cannot be guaranteed for plug-in rules.

Given $n$ independent replicates of a jointly distributed pair $(X, Y) \in \mathscr{R}^d \times \mathscr{R}$, we wish to select from a fixed sequence of model classes $\mathscr{F}_1, \mathscr{F}_2,\dots$ a deterministic prediction rule $f: \mathscr{R}^d \to \mathscr{R}$ whose risk is small.We investigate the possibility of empirically assessing the complexity of each model class, that is, the actual difficulty of the estimation problem within each class. The estimated complexities are in turn used to define an adaptive model selection procedure, which is based on complexity penalized empirical risk.

The available data are divided into two parts. The first is used to form an empirical cover of each model class, and the second is used to select a candidate rule from each cover based on empirical risk. The covering radii are determined empirically to optimize a tight upper bound on the estimation error. An estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk. A distinguishing feature of the approach is that the complexity of each model class is assessed empirically, based on the size of its empirical cover.

Finite sample performance bounds are established for the estimates, and these bounds are applied to several nonparametric estimation problems. The estimates are shown to achieve a favorable trade-off between approximation and estimation error and to perform as well as if the distribution-dependent complexities of the model classes were known beforehand. In addition, it is shown that the estimate can be consistent, and even possess near optimal rates of convergence, when each model class has an infinite VC or pseudo dimension.

For regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence.

Sequential randomized prediction of an arbitrary binary sequence is investigated. No assumption is made on the mechanism of generating the bit sequence. The goal of the predictor is to minimize its relative loss (or regret), that is, to make almost as few mistakes as the best “expert” in a fixed, possibly infinite, set of experts. We point out a surprising connection between this prediction problem and empirical process theory. First, in the special case of static (memoryless) expert, we completely characterize the minimax regret in terms of the maximum of an associated Rademacher process. Then we show general upper and lower bounds on the minimax regret in terms of the geometry of the class of experts. As main examples, we determine the exact order of magnitude of the minimax regret for the class of autoregressive linear predictors and for the class of Markov experts.

Observations are generated according to a regression with normal error as a function of time,when the process is in control. The process potentially changes at some unknown point oftime and then the ensuing observations are normal with the same mean function plus an arbitrary function under suitable regularity conditions. The problem is to obtain a stopping rule that is optimal in the sense that the rule minimizes the expected delay in detecting a change subject to a constraint on the average run length to a false alarm. A bound on the expected delay is first obtained. It is then shown that the cusum and Shiryayev–Roberts procedures achieve this bound to first order.

Deng and Tang proposed generalized resolution and minimum aberration criteria for comparing and assessing nonregular fractional factorials, of which Plackett–Burman designs are special cases.A relaxed variant of generalized aberration is proposed and studied in this paper.We show that a best design according to this criterion minimizes the contamination of nonnegligible interactions on the estimation of main effects in the order of importance given by the hierarchical assumption.The new criterion is defined through a set of $B$ values, a generalization of word length pattern. We derive some theoretical results that relate the $B$ values of a nonregular fractional factorial and those of its complementary design. Application of this theory to the construction of the best designs according to the new aberration criterion is discussed. The results in this paper generalize those in Tang and Wu, which characterize a minimum aberration (regular) $2^{m-k}$ design through its complementary design.

A classification model is easiest to analyze when it has a balanced design. Many of the nice features of balanced designs are retained by error-orthogonal designs, which were introduced in a recent paper by the authors. The present paper defines a kind of ‘‘partially balanced’’ design and shows that this partial balance is sufficient to ensure the error-orthogonality of a mixed classification model. Results are provided that make the partial balance condition easy to check. It is shown that, for a maximal-rank error-orthogonal design, the Type I sum of squares for a random effect coincides with the Type II sum of squares.

In this paper, we define the blocking wordlength pattern of a blocked fractional factorial design by combining the wordlength patterns of treatment-defining words and block-defining words. The concept of minimum aberration can be defined in terms of the blocking wordlength pattern and provides a good measure of the estimation capacity of a blocked fractional factorial design. By blending techniques of coding theory and finite projective geometry, we obtain combinatorial identities that govern the relationship between the blocking wordlength pattern of a blocked $2^{n-m}$ design and the split wordlength pattern of its blocked residual design. Based on these identities, we establish general rules for identifying minimum aberration blocked $2^{n-m}$ designs in terms of their blocked residual designs. Using these rules, we study the structures of some blocked $2^{n-m}$ designs with minimum aberration.

We propose a method of inference for generalized linear mixed models (GLMM) that in many ways resembles the method of least squares. We also show that adequate inference about GLMM can be made based on the conditional likelihood on a subset of the random effects. One of the important features of our methods is that they rely on weak distributional assumptions about the random effects. The methods proposed are also computationally feasible. Asymptotic behavior of the estimates is investigated. In particular, consistency is proved under reasonable conditions.

In this paper,we consider a multivariate one-way random effect model with equal replications. We propose nonnegative definite estimators for “between” and “within” components of variance. Under the Stein loss function, it is shown that the proposed estimators of the “within” component dominate the best unbiased estimator. Restricted maximum likelihood, truncated and order-preserving minimax estimators are also proposed. A Monte Carlo simulation is carried out to choose among these estimators. For estimating the “between” component, we consider the Stein loss function for jointly estimating the two positive definite matrices (“within” and “within” plus “between”) and obtain estimators for the “between” component dominating the best unbiased estimator. Other estimators as considered for “within” are also proposed. A Monte Carlo simulation is carried out to choose among these estimators.

The problem of estimating a density $g$ based on a sample $X_1, X_2,\dots, X_n$ from $p = q*g$ is considered. Linear and nonlinear wavelet estimators based on Meyer-type wavelets are constructed. The estimators are asymptotically optimal and adaptive if $g$ belongs to the Sobolev space $H^{\alpha}$ . Moreover, the estimators considered in this paper adjust automatically to the situation when $g$ is supersmooth.

The local asymptotic normality property is established for a regression model with fractional ARIMA($p, d, q$) errors. This result allows for solving, in an asymptotically optimal way, a variety of inference problems in the long-memory context: hypothesis testing, discriminant analysis, rank-based testing, locally asymptotically minimax andadaptive estimation, etc. The problem of testing linear constraints on the parameters, the discriminant analysis problem, and the construction of locally asymptotically minimax adaptive estimators are treated in some detail.