We investigate the problem of fiducial prediction for unobserved quantities within the framework of the functional model described previously by Dawid and Stone. It is supposed that these are related to a completely unknown parameter by means of a regular functional model, and that the observations are either given as known functions of the predictands, or are themselves related to them by means of a functional model. We develop algebraic conditions which allow the application of fiducial logic to the prediction problem, and explore the consequences of such an application--some of which appear unacceptable unless still stronger conditions are imposed. A reinterpretation of the fiducial prediction problem is given which can be applied to yield an inferential distribution for the unknown parameter in the presence of partial prior information, expressible as a functional hypermodel for the parameter, governed by a completely unknown hyperparameter. This solution agrees with the fiducial distribution when the hypermodel is vacuous and with the Bayes posterior distribution when the hyperparameter is fully known, but allows in addition for intermediate levels of partial prior knowledge.

Suppose that a probability measure $P$ is known to lie in a set of probability measures $M$. Upper and lower bounds on the probability of any event may then be computed. Sometimes, the bounds on the probability of an event $A$ conditional on an event $B$ may strictly contain the bounds on the unconditional probability of $A$. Surprisingly, this might happen for every $B$ in a partition $\mathscr{B}$. If so, we say that dilation has occurred. In addition to being an interesting statistical curiosity, this counterintuitive phenomenon has important implications in robust Bayesian inference and in the theory of upper and lower probabilities. We investigate conditions under which dilation occurs and we study some of its implications. We characterize dilation immune neighborhoods of the uniform measure.

We show that when dynamic sampling is feasible, there exist surveillance schemes for which the probability of a false alarm is bounded and which have a bounded expected delay when detecting a (true) change. In the case of detecting a change of a normal mean, we probe optimality and suggest procedures. These procedures compare favorably to those having a fixed sampling rate which have been developed for an expectation constraint on the average run length until a false alarm.

Standard empirical likelihood confidence intervals for quantiles are identical to sign-test intervals. They have relatively large coverage error, of size $n^{-1/2}$, even though they are two-sided intervals. We show that smoothed empirical likelihood confidence intervals for quantiles have coverage error of order $n^{-1}$, and may be Bartlett-corrected to produce intervals with an error of order only $n^{-2}$. Necessary and sufficient conditions on the smoothing parameter, in order for these sizes of error to be attained, are derived. The effects of smoothing on the positions of endpoints of the intervals are analysed, and shown to be only of second order.

It is well known that we can use the likelihood ratio statistic to test hypotheses and to construct confidence intervals in full parametric models. Recently, Owen introduced the empirical likelihood method in nonparametric models. In this paper, we generalize his results to biased sample problems. A Wilks theorem leading to a likelihood ratio confidence interval for the mean is given. Some extensions, discussion and simulations are presented.

A new mathematical object called a preferred point geometry is introduced in order to (a) provide a natural geometric framework in which to do statistical inference and (b) reflect the distinction between homogeneous aspects (e.g., any point $\theta$ may be the true parameter) and preferred point ones (e.g., when $\theta_0$ is the true parameter). Although preferred point geometry is applicable generally in statistics, we focus here on its relationship to statistical manifolds, in particular to Amari's expected geometry. A symmetry condition characterises when a preferred point geometry both subsumes a statistical manifold and, simultaneously, generalises it to arbitrary order. There are corresponding links with Barndorff-Nielsen's strings. The rather unnatural mixing of metric and nonmetric connections in statistical manifolds is avoided since all connections used are shown to be metric. An interpretation of duality of statistical manifolds is given in terms of the relation between the score vector and the maximum likelihood estimate.

The Riemannian metric structure of the shape space $\sum^k_m$ for $k$ labelled points in $\mathbb{R}^m$ was given by Kendall for the atypically simple situations in which $m = 1$ or 2 and $k \geq 2$. Here we deal with the general case $(m \geq 1, k \geq 2)$ by using the properties of Riemannian submersions and warped products as studied by O'Neill. The approach is via the associated size-and-shape space that is the warped product of the shape space and the half-line $\mathbb{R}_+$ (carrying size), the warping function being equal to the square of the size. When combined with parallel studies by Le of the corresponding global geodesic geometry, the results obtained here determine the environment in which shape-statistical calculations have to be acted out. Finally three different applications are discussed that illustrate the theory and its use in practice.

This paper introduces and investigates the notion of a hyper Markov law, which is a probability distribution over the set of probability measures on a multivariate space that (i) is concentrated on the set of Markov probabilities over some decomposable graph, and (ii) satisfies certain conditional independence restrictions related to that graph. A stronger version of this hyper Markov property is also studied. Our analysis starts by reconsidering the properties of Markov probabilities, using an abstract approach which thereafter proves equally applicable to the hyper Markov case. Next, it is shown constructively that hyper Markov laws exist, that they appear as sampling distributions of maximum likelihood estimators in decomposable graphical models, and also that they form natural conjugate prior distributions for a Bayesian analysis of these models. As examples we construct a range of specific hyper Markov laws, including the hyper multinomial, hyper Dirichlet and the hyper Wishart and inverse Wishart laws. These laws occur naturally in connection with the analysis of decomposable log-linear and covariance selection models.

The lattice conditional independence model $\mathbf{N}(\mathscr{K})$ is defined to be the set of all normal distributions on $\mathbb{R}^I$ such that for every pair $L, M \in \mathscr{K}, x_L$ and $x_M$ are conditionally independent given $x_{L \cap M}$. Here $\mathscr{K}$ is a ring of subsets of the finite index set $I$ and, for $K \in \mathscr{K}, x_K$ is the coordinate projection of $x \in \mathbb{R}^I$ onto $\mathbb{R}^K$. Statistical properties of $\mathbf{N}(\mathscr{K})$ may be studied, for example, maximum likelihood inference, invariance and the problem of testing $H_0: \mathbf{N}(\mathscr{K})$ vs. $H: \mathbf{N}(\mathscr{M})$ when $\mathscr{M}$ is a subring of $\mathscr{K}$. The set $J(\mathscr{K})$ of join-irreducible elements of $\mathscr{K}$ plays a central role in the analysis of $\mathbf{N}(\mathscr{K})$. This class of statistical models occurs in the analysis of nonnested multivariate missing data patterns and nonnested dependent linear regression models.

We consider two recent approaches to characterizing the manifest probabilities of a strictly unidimensional latent variable representation (one satisfying local independence and response curve monotonicity with respect to a unidimensional latent variable) for binary response variables, such as those arising from the dichotomous scoring of items on standardized achievement and aptitude tests. Holland and Rosenbaum showed that conditional association is a necessary condition for strict unidimensionality; and Stout treated the class of essentially unidimensional models, in which the latent variable may be consistently estimated as the length of the response sequence grows using the proportion of positive responses. Of particular concern are strictly unidimensional representations that are minimally useful in the sense that: (1) the latent variable can be consistently estimated from the responses; (2) the regression of proportion of positive responses on the latent variable is monotone; and (3) the latent variable is not constant in the population. We introduce two new conditions, a negative association condition and a natural monotonicity condition on the empirical response curves, that help link strict unidimensionality with the conditional association and essential unidimensionality approaches. These conditions are illustrated with a partial characterization of useful, strictly unidimensional representations.

Let $\mathbf{H}$ and $\mathbf{G}$ be independently distributed according to the Wishart distributions $W_m(M,\Phi)$ and $W_m(N,\Psi)$, respectively. We derive the limiting null distributions of the likelihood ratio criteria for testing $H_0: \Phi = \Psi$ against $H_1 - H_0$ with $H_1: \Phi \geq \Psi$, and for testing $H^{(R)}_0: \Phi \geq \Psi, \operatorname{rank}(\Phi - \Psi) \leq R$ (for given $R$) against $H_1 - H^{(R)}_0$. They are particular cases of the chi-bar-squared distributions.

Consistency is shown for the minimum covariance determinant (MCD) estimators of multivariate location and scale and asymptotic normality is shown for the former. The proofs are made possible by showing a separating ellipsoid property for the MCD subset of observations. An analogous property is shown for the MCD subset computed from the population distribution.

Exploratory graphical methods for fully and partially ranked data are proposed. In fully ranked data, $n$ items are ranked in order of preference by a group of judges. In partially ranked data, the judges do not completely specify their ranking of the $n$ items. The resulting set of frequencies is a function on the symmetric group of permutations if the data is fully ranked, and a function on a coset space of the symmetric group if the data is partially ranked. Because neither the symmetric group nor its coset spaces have a natural linear ordering, traditional graphical methods such as histograms and bar graphs are inappropriate for displaying fully or partially ranked data. For fully ranked data, frequencies can be plotted naturally on the vertices of a permutation polytope. A permutation polytope is the convex hull of the $n$! points in $\mathbb{R}^n$ whose coordinates are the permutations of $n$ distinct numbers. The metrics Spearman's $\rho$ and Kendall's $\tau$ are easily interpreted on permutation polytopes. For partially ranked data, the concept of a permutation polytope must be generalized to include permutations of nondistinct values. Thus, a generalized permutation polytope is defined as the convex hull of the points in $\mathbb{R}^n$ whose coordinates are permutations of $n$ not necessarily distinct values. The frequencies with which partial rankings are chosen can be plotted in a natural way on the vertices of a generalized permutation polytope. Generalized permutation polytopes induce a new extension of Kendall's $\tau$ for partially ranked data. Also, the fixed vector version of Spearman's $\rho$ for partially ranked data is easily interpreted on generalized permutation polytopes. The problem of visualizing data plotted on polytopes in $\mathbb{R}^n$ is addressed by developing the theory needed to define all the faces, especially the three and four dimensional faces, of any generalized permutation polytope. This requires writing a generalized permutation polytope as the intersection of a system of linear equations, and extending results for permutation polytopes to generalized permutation polytopes. The proposed graphical methods is illustrated on five different data sets.

We introduce a nonparametric estimator for the renewal function and discuss its properties, including consistency, asymptotic normality and asymptotic validity of bootstrap confidence regions. The underlying theme is that stochastic models can be regarded as functionals or nonlinear operators. This view leads to nonparametric estimators in a natural way and statistical properties of the estimators can be related to the local behaviour of the functionals.

This paper considers estimators of the distribution function of a variable over a finite population, when a sample of units is available and values of a related auxiliary variable are known for the whole population. Theory is offered for several estimators which rely on nonparametric regression, and for calibration estimators which require a parametric model.

In a recent paper, C. F. J. Wu showed that the jackknife estimator of a distribution function has optimal convergence rate $O(n^{-1/2})$, where $n$ denotes the sample size. This rate is achieved by retaining $O(n)$ data values from the original sample during the jackknife algorithm. Wu's result is particularly important since it permits a direct comparison of jackknife and bootstrap methods for distribution estimation. In the present paper we show that a very simple, nonempirical modification of the jackknife estimator improves the convergence rate from $O(n^{-1/2})$ to $O(n^{-5/6})$, and that this rate may be achieved by retaining only $O(n^{2/3})$ data values from the original sample. Our technique consists of mixing the jackknife distribution estimator with the standard normal distribution in an appropriate proportion. The convergence rate of $O(n^{-5/6})$ makes the jackknife significantly more competitive with the bootstrap, which enjoys a convergence rate of $O(n^{-1})$ in this particular problem.

In this paper we consider the regression model with smooth regression function and smooth error and covariate distributions. We study how well one can estimate functionals of the regression function which may also depend on the distribution of the covariate. This is done by deriving the efficient influence functions of least dispersed regular estimators of such functionals under various assumptions on the parameters of our model. Then we demonstrate how efficient estimates can be constructed. We provide a general procedure for constructing efficient estimates that relies on appropriate auxiliary estimates. We illustrate the usefulness of this procedure by constructing efficient estimates for various parametric, nonparametric and semiparametric models.

Smoothing splines of a fixed order are commonly used as nonparametric regression estimates. The only parameter, then, that needs to be estimated is the smoothing parameter, which is often estimated using some form of cross validation. This work allows the order of the smoothing spline to be estimated using a model in which the order parameter is continuous. Within this setting, generalized cross validation and modified maximum likelihood estimates of the order and smoothing parameters are compared. I show that there are both stochastic and fixed regression functions for which modified maximum likelihood yields asymptotically better estimates of the regression function than generalized cross validation. These results are supported by a small simulation study, although there are functions for which the asymptotic results can be misleading even for fairly large sample sizes.

In the fixed-design nonparametric regression model, kernel-type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed. These kernel-type estimators are analyzed with almost sure results and limiting distributions. Using the limiting distributions, we are able to test the number of jump points and give asymptotic confidence intervals for the sizes of jump values of the regression function. Simulation studies demonstrate that the asymptotic results hold for reasonable sample sizes.

We study in detail asymptotic properties of maximum likelihood estimators of parameters when observations are taken from a two-dimensional Gaussian random field with a multiplicative Ornstein-Uhlenbeck covariance function. Under the complete lattice sampling plan, it is shown that the maximum likelihood estimators are strongly consistent and asymptotically normal. The asymptotic normality here is normalized by the fourth root of the sample size and is obtained through higher order expansions of the likelihood score equations. Extensions of these results to higher-dimensional processes are also obtained, showing that the convergence rate becomes better as the dimension gets higher.

Let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with d.f. $F$. We observe $Z_i = \min(X_i,Y_i)$ and $\delta_i = 1_{\{X_i \leq Y_i\}}$, where $Y_1, Y_2, \ldots$ is a sequence of i.i.d. censoring random variables. Denote by $\hat{F}_n$ the Kaplan-Meier estimator of $F$. We show that for any $F$-integrable function $\varphi, \int\varphi d\hat{F}_n$ converges almost surely and in the mean. The result may be applied to yield consistency of many estimators under random censorship.

Large deviation properties of the Kaplan-Meier estimator are studied and applied to obtain the rate of exponential convergence of the estimator to the underlying survival curve.

We consider general chi-square goodness-of-fit test statistics for randomly censored data--call these generalized Pearson statistics--which are nonnegative definite quadratic forms in the cell frequencies obtained from the product-limit estimator, allowing random cells and general estimators of nuisance parameters. This class of statistics generalizes the class studied by Moore and Spruill in the no censoring case. The large sample behavior of these statistics under the null hypothesis and local alternatives is presented. The chi-square type statistics based on the observed cell frequencies obtained from the product-limit estimator are members of this class for which the quadratic form is selected to produce a chi-square asymptotic null distribution. The generalized Pearson statistic and the statistic by Akritas for a simple hypothesis are compared on the basis of asymptotic relative Pitman efficiency. It is shown that neither statistic dominates the other. The efficiencies are shown to depend on the degree of censoring and the number of cells. For heavily censored data, the Akritas statistic is superior to the generalized Pearson statistic. In the uncensored case, the Akritas statistic, which does not reduce to the Pearson statistic, is not as good as the Pearson statistic in the sense of Pitman efficiency.

Dufour gives a conjecture concerning a characterization of the exponential distribution based on type 2 right censored samples. This conjecture, if true, generalizes the characterization based on complete samples of Seshadri, Csorgo and Stephens (1969) and Dufour, Maag and van Eeden (1984). In this paper it is shown that Dufour's conjecture is true if the number of censored observations is no larger than $(1/3)n - 1$, where $n$ is the sample size. The result has implications for testing fit of censored data to the exponential distribution.

Clayton's model for association in bivariate survival data is both of intrinsic importance and an interesting example of a semiparametric estimation problem, that is, a problem where inference about a parameter is required in the presence of nuisance functions. The joint distribution of the two survival times in this model is absolutely continuous and a single parameter governs the association between the two survival times. In this paper we describe an algorithm to derive the asymptotic lower bound for the information of the parameter governing the association. We discuss the construction of one-step estimators and compare their performance to that of other estimators in a Monte Carlo study.