The paper develops a class of extensions of the univariate quantile function to the multivariate case (M-quantiles), related in a certain way to M-parameters of a probability distribution and their M-estimators. The spatial (geometric) quantiles, recently introduced by Koltchinskii and Dudley and by Chaudhuri as well as the regression quantiles of Koenker and Basset, are the examples of the M-quantile function discussed in the paper. We study the main properties of M-quantiles and develop the asymptotic theory of empirical M-quantiles. We useM-quantiles to extend L-parameters and L-estimators to the multivariate case; to introduce a bootstrap test for spherical symmetry of a multivariate distribution, and to extend the notion of regression quantiles to multiresponse linear regression models.

In this paper we introduce a depth function for distributions on graphs that is analogous to recent multivariate definitions. Using the property of geodesic convexity on graphs, a median-like center for distributions on graphs is constructed and applied to ranking data as well as multivariate data spanned by the minimal spanning tree.

Contours of depth often provide a good geometrical understanding of the structure of a multivariate dataset. They are also useful in robust statistics in connection with generalized medians and data ordering. If the data constitute a random sample from a spherical or elliptic distribution, the depth contours are generally required to converge to spherical or elliptical shapes. We consider contour constructions based on a notion of data depth and prove a uniform contour convergence theorem under verifiable conditions on the depth measure. Applications to several existing depth measures discussed in the literature are also considered.

Undirected graphs and acyclic digraphs (ADG's), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multiviarate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may be many ADG's that determine the same dependence (i.e., Markov) model. Thus, the family of all ADG's with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection of model averaging, that fail to take into account these equivalence classes may incur substantial computational or other inefficiences. Here it is show that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADG's in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical questions are described.

Asymptotic Pitman efficiencies of multivariate spatial sign and rank methods are considered in the one-sample location case. Limiting distributions of the spatial sign and signed-rank tests under the null hypothesis as well as under contiguous sequences of alternatives are given. Formulae for asymptotic relative efficiencies are found and, under multivariate t distributions, relative efficiencies with respect to Hotelling's $T^2$ test are calculated.

A class of procedures based on "impartial trimming" (self-determined by the data) is introduced with the aim of robustifying k-means, hence the associated clustering analysis. We include a detailed study of optimal regions, showing that only nonpathological regions can arise from impartial trimming procedures. The asymptotic results provided in the paper focus on strong consistency of the suggested methods under widely general conditions. A section is devoted to exploring the performance of the procedure to detect anomalous data in simulated data sets.

It is not uncommon to find nonlinear patterns in the scatterplots of regressor variables. But how such findings affect standard regression analysis remains largely unexplored. This article offers a theory on nonlinear confounding, a term for describing the situation where a certain nonlinear relationship in regressors leads to difficulties in modeling and related analysis of the data. The theory begins with a measure of nonlinearity between two regressor variables. It is then used to assess nonlinearity between any two projections from the high-dimensional regressor and a method of finding most nonlinear projections is given. Nonlinear confounding is addressed by taking a fresh new look at fundamental issues such as the validity of prediction and inference, diagnostics, regression surface approximation, model uncertainty and Fisher information loss.

In this paper we study a marked empirical process based on residuals. Results on its large-sample behavior may be used to provide nonparametric full-model checks for regression. Their decomposition into principal components gives new insight into the question: which kind of departure from a hypothetical model may be well detected by residual-based goodness-of-fit methods? The work also contains a small simulation study on straight-line regression.

We describe models for infectious disease attack rates outside Aalen's multiplicative class and incorporating heterogeneity and interactions between subjects. Large-sample theory for the Nelson-Aalen estimator is developed, and its relevance examined in a simulation study. Planning for randomized, controlled clinical trials of prophylactic HIV vaccines partly motivated this work.

A general asymptotic theory is established for the two-parameter Cox score process with staggered entry data. It extends in several directions the existing theory developed by Sellke and Siegmund, Slud and Gu and Lai. An essential tool employed here is a modern empirical process theory, as elucidated in a recent monograph by Pollard.

In the paper a general class of stochastic processes based on the sums of weighted martingale-transform residuals for goodness-of-fit inference in general Cox's type regression models is studied. Their form makes the inference robust to covariate outliers. A weak convergence result for such processes is obtained giving the possibility of establishing the randomness of their graphs together with the construction of the formal $\chi^2$-type goodness-of-fit tests. By using the Khmaladze innovation approach, a modified version of the initial class of processes is also defined. Weak convergence results for the processes are derived. This leads to the main application which concerns the formal construction of the Kolmogorov-Smirnov and Cramér-von Mises-type goodness-of-fit tests. This is done within the general situation considered.

For fixed $\alpha \epsilon (0, 1)$, the quantile regression function gives the $\alpha$th quantile $\theta_{\alpha}(\mathbf{x})$ in the conditional distribution of a response variable Y given the value $\mathbf{X} = \mathbf{x}$ of a vector of covariates. It can be used to measure the effect of covariates not only in the center of a population, but also in the upper and lower tails. A functional that summarizes key features of the quantile specific relationship between X and Y is the vector $\beta_{\alpha}$ of weighted expected values of the vector of partial derivatives of the quantile function $\theta_{\alpha}(\mathbf{x})$. In a nonparametric setting, $\beta_{\alpha}$ can be regarded as a vector of quantile specific nonparametric regression coefficients. In survival analysis models (e.g., Cox's proportional hazard model, proportional odds rate model, accelerated failure time model) and in monotone transformation models used in regression analysis, $\beta_{\alpha}$ gives the direction of the parameter vector in the parametric part of the model. $\beta_{\alpha}$ can also be used to estimate the direction of the parameter vector in semiparametric single index models popular in econometrics. We show that, under suitable regularity conditions, the estimate of $\beta_{\alpha}$ obtained by using the locally polynomial quantile estimate of Chaudhuri (1991a) is $n^{1/2}$-consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional $\beta_{\alpha}$. We discuss how the estimate of $\beta_{\alpha}$ can be used for model diagnostics and in the construction of a link function estimate in general single index models.

.Linear estimation of $f(x)$ at a point in a white noise model is considered. The exact linear minimax estimator of $f(0)$ is found for the family of $f(x)$ in which $f'(x)$ is Lip (M). The resulting estimator is then used to verify a conjecture of Sacks and Ylvisaker concerning the near optimality of the Epanechnikov kernel.

In problems where a high-dimensional design is projected into a lower number of dimensions, the density of the new design is typically not bounded away from zero over its support, even if the original one was. Contexts where this problem arises include projection pursuit regression, estimation in single index models and application of the projection-slice method of Radon transform inversion. Theoretical work in these settings typically involves ignoring data toward the ends of the support of the projected design, but in practice that waste of information is not an attractive option. Motivated by these difficulties, we analyze the way in which local linear smoothing is affected by unboundedly sparse design and apply the conclusions of that study to develop empirical, adaptive bandwidth choice methods. Our results even add to knowledge in the familiar case of a design density that is bounded away from zero, where they provide adaptive bandwidth selectors that are optimal right to the ends of the design interval.

In this paper we establish the basic asymptotic theory for periodic moving averages of i.i.d. random variables with regularly varying tails. The moving average coefficients are allowed to vary according to the season. A simple reformulation yields the corresponding results for moving averages of random vectors. Our main result is that when the underlying random variables have finite variance but infinite fourth moment, the sample autocorrelations are asymptotically stable. It is well known in this case that sample autocorrelations in the classical stationary moving average model are asymptotically normal.

In a framework particularly suited for many time-series models we obtain a LAN result under quite natural and economical conditions. This enables us to construct adaptive estimators for (part of) the Euclidean parameter in these semiparametric models. Special attention is directed to group models in time series with the important subclass of models with time varying location and scale. Our set-up is confronted with the existing literature and, as examples, we reconsider linear regression and ARMA, TAR and ARCH models.

This paper obtains a higher-order asymptotic expansion of a class of M-estimators of the one-sample location parameter when the errors form a long-memory moving average. A suitably standardized difference between an M-estimator and the sample mean is shown to have a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter $\theta$. If, for example, $1/3 < \theta < 1$, then a suitably standardized difference between the sample median and the sample mean converges weakly to a normal distribution provided the common error distribution is symmetric. If $0 < \theta < 1/3$, then the corresponding limiting distribution is nonnormal. This paper thus goes beyond that of Beran who observed, in the case of long-memory Gaussian errors, that M-estimators $T_n$ of the one-sample location parameter are asymptotically equivalent to the sample mean in the sense that $\Var(T_n)/\Var(\bar{X}_n) \to 1$ and $T_n = \bar{X}_n + o_P(\sqrt{\Var(\bar{X}_n)}).$

We consider asymptotically normal statistics which are symmetric functions of N i.i.d. random variables. For these statistics we prove the validity of an Edgeworth expansion with remainder $O(N^{-1})$ under Cramér's condition on the linear part of the statistic and moment assumptions for all parts of the statistic. By means of a counterexample we show that it is generally not possible to obtain an Edgeworth expansion with remainder $o(N^{-1})$ without imposing additional assumptions on the structure of the nonlinear part of the statistic.

In this paper, we analyze the statistic which is the difference in the values of an estimating function evaluated at its local maxima on two different subsets of the parameter space, assuming that the true parameter is in each subset, but possibly on the boundary. Our results extend known methods by covering a large class of estimation problems which allow sampling from nonidentically distributed random variables. Specifically, the existence and consistency of the local maximum estimators and asymptotic properties of useful hypothesis tests are obtained under certain law of large number and central limit-type assumptions. Other models covered include those with general log-likelihoods and/or covariates. As an example, the large sample theory of two-way nested random variance components models with covariates is derived from our main results.