We develop a nonparametric Bayes factor for testing the fit of a parametric model. We begin with a nominal parametric family which we then embed into an infinite-dimensional exponential family. The new model then has a parametric and nonparametric component. We give the log density of the nonparametric component a Gaussian process prior. An asymptotic consistency requirement puts a restriction on the form of the prior, leaving us with a single hyperparameter for which we suggest a default value based on simulation experience. Then we construct a Bayes factor to test the nominal model versus the semiparametric alternative. Finally, we show that the Bayes factor is consistent. The proof of the consistency is based on approximating the model by a sequence of exponential families.

This article introduces and develops a constructive method for generating random probability measures with a prescribed mean or distribution of the means. The method involves sequentially generating an array of barycenters which uniquely defines a probability measure. Basic properties of the generated measures are presented, including conditions under which almost all the generated measures are continuous or almost all are purely discrete or almost all have finite support. Applications are given to models for average-optimal control problems and to experimental approximation of universal constants.

.Measures of association between two random variables (r.v.) that are symmetric nondecreasing functions of the canonical coefficients provided by the nonlinear canonical analysis of the r.v.’s are studied. These measures can be used to characterize independence. Their estimators are obtained by estimating a suitable approximation to nonlinear canonical analysis. When some conditions are satisfied, asymptotic distributions of the estimators, both under the independence hypothesis and under dependence, are given. A class of tests of independence with asymptotic level of significance can be investigated.

The two-way analysis of variance with interactions is a well established and integral part of statistics. In spite of its long standing, it is shown that the standard definition of interactions is counterintuitive and obfuscates rather than clarifies. A different definition of interaction is given which among other advantages allows the detection of interactions even in the case of one observation per cell. A characterization of unconditionally identifiable interaction patterns is given and it is proved that such patterns can be identified by the $L^1$ functional. The unconditionally identifiable interaction patterns describe the optimal breakdown behavior of any equivariant location functional from which it follows that the $L^1$ functional has optimal breakdown behavior. Possible lack of uniqueness of the $L^1$ functional can be overcome using an $M$ functional with an external scale derived independently from the observations. The resulting procedures are applied to some data sets including one describing the results of an interlaboratory test.

This paper concerns linear regression with interval censored data. $M$-estimators for the regression coefficients are derived. Asymptotic consistency and normality of the $M$-estimators are obtained via an exponential inequality for $U$-statistics. Asymptotically efficient estimators are provided under mild conditions.

The longitudinal regression model $Y_{i,j} = m(V_\tau_{i,j}^i) + \varepsilon_{i,j}$ where $Y_{i,j}$, is the $j$th measurement of the $i$th subject at random time $\tau_{i,j}$, $m$ is the regression function, $V_\tau_{i, j}$ is a predictable covariate process observed at time $\tau_{i,j}$ and $\varepsilon_{i,j}$ is noise, often provides an adequate framework for modeling and comparing groups of data. The proposed longitudinal regression model is based on marked point process theory, and allows a quite general dependency structure among the observations.

In this paper we find the asymptotic distribution of the cumulative regression function (CRF), and present a nonparametric test to compare the regression functions for two groups of longitudinal data. The proposed test, denoted the CRF test, is based on the cumulative regression function (CRF) and is the regression equivalent of the log-rank test in survival analysis. We show as a special case that the CRF test is valid for groups of independent identically distributed regression data. Apart from the CRF test, we also consider a maximal deviation statistic that may be used when the CRF test is inefficient.

We propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or change-points of the underlying function $f$ or its derivatives. For instance, in the case of a jump of a regression function, beyond the intervals of length (in order) $n^{-1} \log n$ around change-points the quality of estimation is essentially the same as if locations of jumps were known. The method is fully adaptive and no assumptions are imposed on the design, number and size of jumps. The results are formulated in a nonasymptotic way and can therefore be applied for an arbitrary sample size.

We define the sharp change point problem as an extension of earlier problems in change point analysis related to nonparametric regression. As particular cases, these include estimation of jump points in smooth curves. More generally, we give a systematic treatment of the correct rate of convergence for estimating the position of a “cusp”of an arbitrary order. We propose a test function for the local regularity of a signal that characterizes such a point as a global maximum. In the sample implementation of our method, from observations of the signal at discrete time positions $i/n, i =1 \ldots,n$, we use a wavelet transformation to approximate the position of the change point in the no-noise case. We study the noise effect, in the worst case scenario over a wide class of functions having a unique irregularity of “order $\alpha$” and propose a sequence of estimators which converge at the rate $n_{-1/(1+2\alpha)}$, as $n$ tends to infinity. Finally we analyze the likelihood ration of the problem and show that this is actually the minimaz rate of convertence. Examples of thresholding empirical wavelet coefficients to estimate the position of sharp change points are also presented.

Let a linear regression be given. For detecting change-points, it is usual to consider the sequence of partial sums of least squares residuals whence a partial sums process is defined. Given a sequence of exact experimental designs, we consider for each design the corresponding partial sums process. If the sequence of designs converges to a continuous design, we derive the explicit form of the limit process of the corresponding sequence of partial sums processes. This is a complicated function of the Brownian motion. These results are useful for the problem of testing for change of regression at known or unknown times.

To assess the adequacy of a nonreplicated linear regression model, Christensen introduced the concepts of orthogonal between- and within-cluster lack of fit with corresponding optimal tests. However, the properties of these tests depend on the choice of near-replicate clusters. In this paper, a graph theoretic framework is presented to represent candidate clusterings. A clustering is then selected according to a proposed maximin power criterion from among the clusterings consistent with a specified graph on the predictor settings. Examples are given to illustrate the methodology.

A chain graph (CG) is a graph admitting both directed and undirected edges with (partially) directed cycles forbidden. It generalizes both the concept of undirected graph (UG) and the concept of directed acyclic graph (DAG). A chain graph can be used to describe efficiently the conditional independence structure of a multidimensional discrete probability distribution in the form of a graphoid, that is, in the form of a list of statements “$X$ is independent of $Y$ given $Z$” obeying a set of five properties (axioms). An input list of independency statements for every CG is defined and it is shown that the classic moralization criterion for CGs embraces exactly the graphoid closure of the input list. A new direct separation criterion for reading independency statements from a CG is introduced and shown to be equivalent to the moralization criterion. Using this new criterion, it is proved that for every CG, there exists a strictly positive discrete probability distribution that embodies exactly the independency statements displayed by the graph. Thus, both criteria are shown to be complete and the use of CGs as tools for description of conditional independence structures is justified.

For the Fourier regression model, we determine optimal designs for identifying the order of periodicity. It is shown that the optimal design problem for trigonometric regression models is equivalent to the problem of optimal design for discriminating between certain homo-and heteroscedastic polynomial regression models. These optimization problems are then solved using the theory of canonical moments, and the optimal discriminating designs for the Fourier regression model can be found explicitly. In contrast to many other optimality criteria for the trigonometric regression model, the optimal discriminating designs are not uniformly distributed on equidistant points.

The construction of run orders of two-level factorial designs with extreme (minimum and maximum) numbers of level changes is considered. Minimizing the number of level changes is mainly due to economic considerations, while the problem of maximizing the number of level changes arises from some recent results on trend robust designs. The construction is based on the fact that the $2^k$ runs of a saturated regular fractional factorial design for $2^k -1$ factors can be ordered in such a way that the numbers of level changes of the factors consist of each integer between 1 and $2^k -1$. Among other results, we give a systematic method of constructing designs with minimum and maximum numbers of level changes among all designs of resolution at least three and among those of resolution at least four. It is also shown that among regular fractional factorial designs of resolution at least four, the number of level changes can be maximized and minimized by different run orders of the same fraction.

In this paper the validity of a one-term Edgeworth expansion for Studentized symmetric statistics is proved. We propose jackknife estimates for the unknown constants appearing in the expansion and prove their consistency. As a result we obtain the second-order correctness of the empirical Edgeworth expansion for a very general class of statistics, including $U$-statistics, $L$-statistics and smooth functions of the sample mean. We illustrate the application of the bootstrap in the case of a $U$-statistic of degree two.

We derive a strong approximation of a local polynomial estimator LPE in nonparametric autoregression by an (LPE) in a corresponding nonparametric regression model. This generally suggests the application of regression-typical tools for statistical inference in nonparametric autoregressive models. It provides an important simplification for the boot-strap method to be used: It is enough to mimic the structure of a nonparametric regression model rather than to imitate the more complicated process structure in the autoregressive case. As an example we consider a simple wild bootstrap, which is used for the construction of simultaneous confidence bands and nonparametric supremum-type tests.

Hidden Markov models (HMMs) have during the last decade become a widespread tool for modeling sequences of dependent random variables. Inference for such models is usually based on the maximum-likelihood estimator (MLE), and consistency of the MLE for general HMMs was recently proved by Leroux. In this paper we show that under mild conditions the MLE is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the Fisher information.

In this paper, the computation of the exact Fisher information matrix of a large class of Gaussian time series models is considered. This class, which is often called the single-input–single-output (SISO) model, includes dynamic regression with autocorrelated errors and the transfer function model, with autoregressive moving average errors. The method is based on a combination of two computational procedures: recursions for the covariance matrix of the derivatives of the state vector with respect to the parameters, and the fast Kalman filter recursions used in the evaluation of the likelihood function. It is much faster than existing procedures. An expression for the asymptotic information matrix is also given.