This paper addresses two closely related questions, "What is a statistical model?" and "What is a parameter?" The notions that a model must "make sense," and that a parameter must "have a well-defined meaning" are deeply ingrained in applied statistical work, reasonably well understood at an instinctive level, but absent from most formal theories of modelling and inference. In this paper, these concepts are defined in algebraic terms, using morphisms, functors and natural transformations. It is argued that inference on the basis of a model is not possible unless the model admits a natural extension that includes the domain for which inference is required. For example, prediction requires that the domain include all future units, subjects or time points. Although it is usually not made explicit, every sensible statistical model admits such an extension. Examples are given to show why such an extension is necessary and why a formal theory is required. In the definition of a subparameter, it is shown that certain parameter functions are natural and others are not. Inference is meaningful only for natural parameters. This distinction has important consequences for the construction of prior distributions and also helps to resolve a controversy concerning the Box-Cox model.

Let $\Theta$ be a parameter space included in a finite-dimensional Euclidean space and let $A$ be a half-space. Suppose that the maximum likelihood estimate $\theta_n$ of $\theta$ is not in $A$ (otherwise, replace $A$ by its complement) and let $\Delta$ be the maximum log likelihood (at $\theta_n$) minus the maximum log likelihood over the boundary $\partial A$. It is shown that under some conditions, uniformly over all half-spaces $A$, either the posterior probability of $A$ is asymptotic to $\Phi(-\sqrt{2\Delta}\,)$ where $\Phi$ is the standard normal distribution function, or both the posterior probability and its approximant go to 0 exponentially in $n$. Sharper approximations depending on the prior are also defined.

This paper studies a class of default priors, which we call single observation unbiased priors (SOUP). A prior for a parameter is a SOUP if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter. We prove that, under mild regularity conditions, a default prior for a convolution parameter is "noninformative" in the sense of yielding a posterior inference invariant under amalgamation only if it is a SOUP. Therefore, when amalgamation invariance is desirable, as in our motivating example of performing imputation for census undercount, SOUP is the only possible coherent "noninformative" prior for Bayesian predictions that will be utilized under aggregation. The use of SOUP also mutually calibrates Bayesian and frequentist inferences for aggregates of convolution parameters across many small areas. We describe approaches that identify SOUPs in many common models, in particular a constructive duality method that identifies SOUPs in pairs of distribution families. We introduce O-completeness, a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean. Uniqueness of the SOUP is determined by the O-completeness of the dual family. O-completeness of a natural exponential family is implied by its completeness. Hence, the Diaconis-Ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families. For most of the examples we have examined, the inverse of the variance function is the SOUP for the mean parameter of the corresponding family, suggesting that Hartigan's results on asymptotic unbiasedness can be generalized to some families with discrete parameters. We also discuss a possible extension of Berger's result on the inadmissibility of unbiased estimators, as the nonexistence of SOUP can be a first step in establishing such inadmissibility.

The distribution of a mean or, more generally, of a vector of means of a Dirichlet process is considered. Some characterizing aspects of this paper are: (i) a review of a few basic results, providing new formulations free from many of the extra assumptions considered to date in the literature, and giving essentially new, simpler and more direct proofs; (ii) new numerical evaluations, with any prescribed error of approximation, of the distribution at issue; (iii) a new form for the law of a vector of means. Moreover, as applications of these results, we give: (iv) the sharpest condition sufficient for the distribution of a mean to be symmetric; (v) forms for the probability distribution of the variance of the Dirichlet random measure; (vi) some hints for determining the finite-dimensional distributions of a random function connected with Bayesian methods for queuing models.

We develop simple methods for constructing parameter priors for model choice among directed acyclic graphical (DAG) models. In particular, we introduce several assumptions that permit the construction of parameter priors for a large number of DAG models from a small set of assessments. We then present a method for directly computing the marginal likelihood of every DAG model given a random sample with no missing observations. We apply this methodology to Gaussian DAG models which consist of a recursive set of linear regression models. We show that the only parameter prior for complete Gaussian DAG models that satisfies our assumptions is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let $W$ be an $n \times n$, $n \ge 3$, positive definite symmetric matrix of random variables and $f(W)$ be a pdf of $W$. Then, $f(W)$ is a Wishart distribution if and only if $W_{11} - W_{12} W_{22}^{-1} W'_{12}$ is independent of $\{W_{12},W_{22}\}$ for every block partitioning $W_{11},W_{12}, W'_{12}, W_{22}$ of $W$. Similar characterizations of the normal and normal-Wishart distributions are provided as well.

In this paper we provide a detailed characterization of the asymptotic behavior of kernel density estimators for one-sided linear processes. The conjecture that asymptotic normality for the kernel density estimator holds under short-range dependence is proved under minimal assumptions on bandwidths. We also depict the dichotomous and trichotomous phenomena for various choices of bandwidths when the process is long-range dependent.

We suggest two new, translation-based methods for estimating and correcting for bias when estimating the edge of a distribution. The first uses an empirical translation applied to the argument of the kernel, in order to remove the main effects of the asymmetries that are inherent when constructing estimators at boundaries. Placing the translation inside the kernel is in marked contrast to traditional approaches, such as the use of high-order kernels, which are related to the jackknife and, in effect, apply the translation outside the kernel. Our approach has the advantage of producing bias estimators that, while enjoying a high order of accuracy, are guaranteed to respect the sign of bias. Our second method is a new bootstrap technique. It involves translating an initial boundary estimate toward the body of the dataset, constructing repeated boundary estimates from data that lie below the respective translations, and employing averages of the resulting empirical bias approximations to estimate the bias of the original estimator. The first of the two methods is most appropriate in univariate cases, and is studied there; the second approach may be used to bias-correct estimates of boundaries of multivariate distributions, and is explored in the bivariate case.

This paper considers a class of local likelihood methods introduced by Eguchi and Copas. Unified asymptotic results are presented in the usual smoothing context of the bandwidth, h, tending to zero as the sample size tends to infinity. We present our results pointwise in the univariate case, but then go on to extend them to global properties and to indicate how to cope with the multivariate case. Specific members of the class due to Copas, and Hjort and Jones are seen to be members of a subset of the whole class with the same, and best, small h behavior. Further comparisons between members of the class are alluded to based on the complementary large h asymptotic results of Eguchi and Copas.

Wu and Hamada recommend selecting resolution IV designs with the maximum number of clear two-factor interactions (2FIs), called MaxC2 designs. In this paper, we develop a method by using graphical representations, combinatorial and group-theoretic arguments to prove if a given design is a MaxC2 design. In particular, we show that all known minimum aberration designs with resolution IV are MaxC2 designs (except in six cases) and that the second $2^{9-4}$, $2^{13-7}$, $2^{16-10}$ and $2^{17-11}$ designs given in Wu and Hamada are MaxC2 designs. The method also enables us to identify new MaxC2 designs that are too large to be verified by computer search.

Finite projective geometry is used to obtain fractional factorial plans for m-level symmetrical factorial experiments, where m is a prime or a prime power. Under a model that includes the mean, all main effects and a specified set of two-factor interactions, the plans are shown to be universally optimal within the class of all plans involving the same number of runs.

Consider an NEF $F$ on the real line parametrized by $\theta \in \Theta $. Also let $\theta _0$ be a specified value of $\theta $. Consider the test of size $\alpha$ for a simple hypothesis $H_0\dvtx \theta =\theta _0$ versus two sided alternative $H_1\dvtx \theta \neq \theta _0$. A~UMPU test of size~$\alpha $ then exists for any given $\alpha$. Suppose that $F$ is continuous. Therefore the UMPU test is nonrandomized and then becomes comparable with the generalized likelihood ratio test (GLR). Under mild conditions we show that the two tests coincide iff $F$ is either a normal or inverse Gaussian or gamma family. This provides a new global characterization of this set of NEFs. The proof involves a differential equation obtained by the cancelling of a determinant of order 6.