We discuss a strategy for polychotomous classification that involves coupling the estimating class probabilities for each pair of classes, and estimates together. The coupling model is similar to the Bradley-Terry method for paired comparisons. We study the nature of the class probability estimates that arise, and examine the performance of the procedure in real and simulated data sets. Classifiers used include linear discriminants, nearest neighbors, adaptive nonlinear methods and the support vector machine.

Samples of directional or axial measurements arise in geophysical, biological and econometric contexts. We represent the rotational difference between two mean directions (or two mean axes) as a direction (or axis). We then construct nonparametric simultaneous confidence sets for all pair-wise rotational differences among the mean directions or mean axes of $s$ samples. By specialization, this methodology yields nonparametric simultaneous tests for pairwise equality of directional means or axes.

Some new probability inequalities involving the ordered components of an $MTP_2$ random vector are derived, which provide an analytical proof of an important conjecture in the field of multiple hypothesis testing. This conjecture has been mostly validated so far using simulation.

In this paper interest is focused on some theoretical investigations concerning the comparison of two popular multiple test procedures, so-called step-down and step-up procedures, in terms of their defining critical values. Such procedures can be applied, for example, to multiple comparisons with a control. In the definition of the critical values for these procedures order statistics play a central role. For $k \epsilon N_0$ fixed we consider the joint cumulative distribution function (cdf) $P(Y_{1:n} \leq c_1,\dots, Y_{n-k:n} \leq c_{n-k})$ of the first $n - k$ order statistics and the cdf$P(Y_{n-k:n} \leq c_{n-k})$ of the ($k + 1$)th largest order statistic $Y_{n-k:n}$ of $n$ random variables $Y_1,\dots, Y_n$ belonging to a sequence of exchangeable real-valued random variables. Our interest is focused on the asymptotic behavior of these cdf's and their interrelation if $n$ tends to $\infty$. It turns out that they sometimes behave completely differently compared with the iid case treated in Finner and Roters so that positive results are only possible under additional assumptions concerning the underlying distribution. We consider different sets of assumptions which then allow analogous results for the exchangeable case. Recently, Dalal and Mallows derived a result concerning the monotonicity of a certain set of critical values in connection with the joint cdf of order statistics in the iid case. We give a counterexample for the exchangeable case underlining the difficulties occurring in this situation. As an application we consider the comparison of certain step-down and step-up procedures in multiple comparisons with a control. The results of this paper yield a more theoretical explanation of the superiority of the step-up procedure which has been observed earlier by comparing tables of critical values. As a by-product we are able to quantify the tightness of the Bonferroni inequality in connection with maximum statistics.

A class of multivariate normal models with symmetry restrictions given by a finite group and conditional independence restrictions given by a finite distributive lattice is defined and studied. The statistical properties of these models including maximum likelihood inference, invariance and hypothesis testing are discussed.

If $U$ and $V$ are independent random variables which are gamma distributed with the same scale parameter, then there exist $a$ and $b$ in $\mathbb{R}$ such that $$\mathbb{E}(U|U + V) = a(U + V)$$ and $$\mathbb{E}(U^2|U + V) = b(U + V)^2$$. This, in fact, is characteristic of gamma distributions. Our paper extends this property to the Wishart distributions in a suitable way, by replacing the real number $U^2$ by a pair of quadratic functions of the symmetric matrix $U$. This leads to a new characterization of the Wishart distributions, and to a shorter proof of the 1962 characterization given by Olkin and Rubin. Similarly, if $\mathbb{E}(U^{-1})$ exists, there exists $c$ in $\mathbb{R}$ such that $$\mathbb{E}(U^{-1}|U + V) = c(U + V)^{-1}$$. Wesołowski has proved that this also is characteristic of gamma distributions. We extend it to the Wishart distributions. Finally, things are explained in the modern framework of symmetric cones and simple Euclidean Jordan algebras.

Nonlinear principal components for an absolutely continuous random vector X with positive bounded density are defined as the solution of a variational problem in a suitable function space. In this way transformations depending on all the components of X are obtained. Some properties of nonlinear principal components are proved: in particular, it is shown that the set of nonlinear principal transformations of X is an orthonormal basis for the function space associated with the optimal problem. The spectral decomposition of X and its covariance matrix with respect to this basis are given. A notion of marginal nonlinear principal components is sketched and the relations with nonlinear principal components are shown. Finally, treating the case of random vectors distributed on unbounded domains, the existence problem is shown to be related to the global existence of the moment generating function of X. Since it is not restrictive, definitions and results are stated in terms of a uniformly distributed random vector.

We prove that the posterior distribution in a possibly incorrect parametric model a.s. concentrates in a strong sense on the set of pseudotrue parameters determined by the true distribution. As a consequence, we obtain in the case of a unique pseudotrue parameter the strong consistency of pseudo-Bayes estimators w.r.t. general loss functions.

Further, we present a simple example based on normal distributions and having two different pseudotrue parameters, where pseudo-Bayes estimators have an essentially different asymptotic behavior than the pseudomaximum likelihood estimator. While the MLE is strongly consistent, the sequence of posterior means is strongly inconsistent and a.s. almost all its accumulation points are not pseudotrue. Finally, we give conditions under which a pseudo-Bayes estimator for a unique pseudotrue parameter has an asymptotic normal distribution.

Selection models arise when the data are selected to enter the sample only if they occur in a certain region of the sample space. When this selection occurs according to some probability distribution, the resulting model is often instead called a weighted distribution model. In either case the "original" density becomes multiplied by a "weight function" $w(x)$. Often there is considerable uncertainty concerning this weight function; for instance, it may be known only that $w$ lies between two specified weight functions. We consider robust Bayesian analysis for this situation, finding the range of posterior quantities of interest, such as the posterior mean or posterior probability of a set, as $w$ ranges over the class of weight functions. The variational analysis utilizes concepts from variation diminishing transformations.

Bayes estimation of the mean of a multivariate normal distribution is considered under quadratic loss. We show that, when particular spherical priors are used, the superharmonicity of the square root of the marginal density provides a viable method for constructing (possibly proper) Bayes (and admissible) minimax estimators. Examples illustrate the theory; most notably it is shown that a multivariate Student-$t$ prior yields a proper Bayes minimax estimate.

Necessary and sufficient conditions for the weak consistency of the sample median of independent, but not identically distributed random variables are given and discussed.

Given $n$ iid copies $X_1,\dots, X_n$ of a random element $X$ in some arbitrary measurable space $S$, we are only interested in those observations that fall into some subset $D$ having but a small probability of occurrence. It is assumed that the distribution $P_X$ of $X$ belongs on $D$ to a parametric family $P_X(\cdot\capD) = P_{\vartheta}, \vartheta \epsilon \Theta \subset \mathbb{R}^d$. Nonlinear regression analysis and the peaks-over-threshold (POT) approach in extreme value analysis are prominent examples. For the POT approach on $S = \mathbb{R}$ and $P_{\vartheta}$ being a generalized Pareto distribution, it is known that the complete information about the underlying parameter $\vartheta_0$ is asymptotically contained in the number $\tau(n)$ of observations in $D$ among $X_1,\dots, X_n$, but not in their actual values. This result is formulated in terms of local asymptotic normality of the log-likelihood ratio of the point process of exceedances with $\tau(n)$ being the central sequence.

In this paper we establish a necessary and sufficient condition such that $\tau(n)$ has this property for a general truncated empirical process in an arbitrary sample space and for an arbitrary parametric family. The known results are then consequences of this result. We can, moreover, characterize the influence of the actual observations in $D$ on the central sequence, if this condition is violated.

Immediate applications are asymptotically optimal tests for testing $\vartheta_0$ and, if $\Theta \subset \mathbb{R}$, asymptotic efficiency of the ML-estimator $\hat{\vartheta}_n$ satisfying $P_{\hat{\vartheta}_n}(D) = \tau(n)/n$, where these statistics are based on $\tau(n)$ only.

This paper develops a state space modeling for long-range dependent data. Although a long-range dependent process has an infinite-dimensional state space representation, it is shown that by using the Kalman filter, the exact likelihood function can be computed recursively in a finite number of steps. Furthermore, an approximation to the likelihood function based on the truncated state space equation is considered. Asymptotic properties of these approximate maximum likelihood estimates are established for a class of long-range dependent models, namely, the fractional autoregressive moving average models. Simulation studies show rapid converging properties of the approximate maximum likelihood approach.

This paper establishes the weak convergence of the sequential empirical process $\hat{K}_n$ of the estimated residuals in nonstationary autoregressive models. Under some regular conditions, it is shown that $\hat{K}_n$ converges weakly to a Kiefer process when the characteristic polynomial does not include the unit root 1; otherwise $\hat{K}_n$ converges weakly to a Kiefer process plus a functional of stochastic integrals in terms of the standard Brownian motion. The latter differs not only from that given by Koul and Levental for an explosive AR(1) model but also from that given by Bai for a stationary ARMA model.

It is well known that $L_1$-estimators of regression parameters are asymptotically normal if the distribution function has a positive derivative at 0. In this paper, we derive the asymptotic distributions under more general conditions on the behavior of the distribution function near 0.

We take a unified approach to asymptotic properties of $M_m$ estimates based on i.i.d. observations defined through the minimization of a real-valued criterion function of one or more variables. Our results are applicable to a host of location and scale estimators found in the literature.

A new test is proposed in order to verify that a regression function, say $g$, has a prescribed (linear) parametric form. This procedure is based on the large sample behavior of an empirical $L^2$-distance between $g$ and the subspace $U$ spanned by the regression functions to be verified. The asymptotic distribution of the test statistic is shown to be normal with parameters depending only on the variance of the observations and the $L^2$-distance between the regression function $g$ and the model space $U$. Based on this result, a test is proposed for the hypothesis that "$g$ is not in a preassigned $L^2$-neighborhood of $U$," whichallows the "verification" of the model $U$ at a controlled type I error rate. The suggested procedure is very easy to apply because of its asymptotic normal law and the simple form of the test statistic. In particular, it does not require nonparametric estimators of the regression function and hence, the test does not depend on the subjective choice of smoothing parameters.