Functional data analysis in a mixed-effects model framework is done using operator calculus. In this approach the functional parameters are treated as serially correlated effects giving an alternative to the penalized likelihood approach, where the functional parameters are treated as fixed effects. Operator approximations for the necessary matrix computations are proposed, and semi-explicit and numerically stable formulae of linear computational complexity are derived for likelihood analysis. The operator approach renders the usage of a functional basis unnecessary and clarifies the role of the boundary conditions.

With the progress of measurement apparatus and the development of automatic sensors, it is not unusual anymore to get large samples of observations taking values in high-dimension spaces, such as functional spaces. In such large samples of high-dimensional data, outlying curves may not be uncommon, and even a few individuals may corrupt simple statistical indicators, such as the mean trajectory. We focus here on the estimation of the geometric median which is a direct generalization of the real median in metric spaces and has nice robustness properties. It is possible to estimate the geometric median, being defined as the minimizer of a simple convex functional that is differentiable everywhere when the distribution has no atom, with online gradient algorithms. Such algorithms are very fast and can deal with large samples. Furthermore, they also can be simply updated when the data arrive sequentially. We state the almost sure consistency and the $L^{2}$ rates of convergence of the stochastic gradient estimator as well as the asymptotic normality of its averaged version. We get that the asymptotic distribution of the averaged version of the algorithm is the same as the classic estimators, which are based on the minimization of the empirical loss function. The performances of our averaged sequential estimator, both in terms of computation speed and accuracy of the estimations, are evaluated with a small simulation study. Our approach is also illustrated on a sample of more than 5000 individual television audiences measured every second over a period of 24 hours.

We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.

We propose a flexible stochastic framework for modeling the market share dynamics over time in a multiple markets setting, where firms interact within and between markets. Firms undergo stochastic idiosyncratic shocks, which contract their shares, and compete to consolidate their position by acquiring new ones in both the market where they operate and in new markets. The model parameters can meaningfully account for phenomena such as barriers to entry and exit, fixed and sunk costs, costs of expanding to new sectors with different technologies and competitive advantage among firms. The construction is obtained in a Bayesian framework by means of a collection of nonparametric hierarchical mixtures, which induce the dependence between markets and provide a generalization of the Blackwell–MacQueen Pólya urn scheme, which in turn is used to generate a partially exchangeable dynamical particle system. A Markov Chain Monte Carlo algorithm is provided for simulating trajectories of the system, by means of which we perform a simulation study for transitions to different economic regimes. Moreover, it is shown that the infinite-dimensional properties of the system, when appropriately transformed and rescaled, are those of a collection of interacting Fleming–Viot diffusions.

In this paper we propose a numerical scheme for the class of backward doubly stochastic differential equations (BDSDEs) with possible path-dependent terminal values. We prove that our scheme converges in the strong $L^{2}$-sense and derives its rate of convergence. As an intermediate step we derive an $L^{2}$-type regularity of the solution to such BDSDEs. Such a notion of regularity, which can be thought of as the modulus of continuity of the paths in an $L^{2}$-sense, is new.

We prove Chung-type laws of the iterated logarithm for general Lévy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated logarithm.

This reveals laws of the iterated logarithm for Lévy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.

Donsker’s theorem shows that random walks behave like Brownian motion in an asymptotic sense. This result can be used to approximate expectations associated with the time and location of a random walk when it first crosses a nonlinear boundary. In this paper, correction terms are derived to improve the accuracy of these approximations.

We introduce a four-parameter extended family of distributions related to the wrapped Cauchy distribution on the circle. The proposed family can be derived by altering the settings of a problem in Brownian motion which generates the wrapped Cauchy. The densities of this family have a closed form and can be symmetric or asymmetric depending on the choice of the parameters. Trigonometric moments are available, and they are shown to have a simple form. Further tractable properties of the model are obtained, many by utilizing the trigonometric moments. Other topics related to the model, including alternative derivations and Möbius transformation, are considered. Discussion of the symmetric submodels is given. Finally, generalization to a family of distributions on the sphere is briefly made.

In this paper, we study robust estimators of the memory parameter $d$ of a (possibly) non-stationary Gaussian time series with generalized spectral density $f$. This generalized spectral density is characterized by the memory parameter $d$ and by a function $f^{\ast}$ which specifies the short-range dependence structure of the process. Our setting is semi-parametric since both $f^{\ast}$ and $d$ are unknown, and $d$ is the only parameter of interest. The memory parameter $d$ is estimated by regressing the logarithm of the estimated variance of the wavelet coefficients at different scales. The two estimators of $d$ that we consider are based on robust estimators of the variance of the wavelet coefficients, namely the square of the scale estimator proposed by Rousseeuw and Croux [J. Amer. Statist. Assoc. 88 (1993) 1273–1283] and the median of the square of the wavelet coefficients. We establish a central limit theorem, for these robust estimators as well as for the estimator of $d$, based on the classical estimator of the variance proposed by Moulines, Roueff and Taqqu [Fractals 15 (2007) 301–313]. Some Monte-Carlo experiments are presented to illustrate our claims and compare the performance of the different estimators. The properties of the three estimators are also compared to the Nile river data and the Internet traffic packet counts data. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the memory parameter $d$ of Gaussian time series.

We study statistical inferences for a class of modulated stationary processes with time-dependent variances. Due to non-stationarity and the large number of unknown parameters, existing methods for stationary, or locally stationary, time series are not applicable. Based on a self-normalization technique, we address several inference problems, including a self-normalized central limit theorem, a self-normalized cumulative sum test for the change-point problem, a long-run variance estimation through blockwise self-normalization, and a self-normalization-based wild bootstrap. Monte Carlo simulation studies show that the proposed self-normalization-based methods outperform stationarity-based alternatives. We demonstrate the proposed methodology using two real data sets: annual mean precipitation rates in Seoul from 1771–2000, and quarterly U.S. Gross National Product growth rates from 1947–2002.

Markov processes are used in a wide range of disciplines, including finance. The transition densities of these processes are often unknown. However, the conditional characteristic functions are more likely to be available, especially for Lévy-driven processes. We propose an empirical likelihood approach, for both parameter estimation and model specification testing, based on the conditional characteristic function for processes with either continuous or discontinuous sample paths. Theoretical properties of the empirical likelihood estimator for parameters and a smoothed empirical likelihood ratio test for a parametric specification of the process are provided. Simulations and empirical case studies are carried out to confirm the effectiveness of the proposed estimator and test.

We consider the problem of simultaneous variable selection and estimation in additive, partially linear models for longitudinal/clustered data. We propose an estimation procedure via polynomial splines to estimate the nonparametric components and apply proper penalty functions to achieve sparsity in the linear part. Under reasonable conditions, we obtain the asymptotic normality of the estimators for the linear components and the consistency of the estimators for the nonparametric components. We further demonstrate that, with proper choice of the regularization parameter, the penalized estimators of the non-zero coefficients achieve the asymptotic oracle property. The finite sample behavior of the penalized estimators is evaluated with simulation studies and illustrated by a longitudinal CD4 cell count data set.

We investigate the maximal size of distinguished submatrices of a Gaussian random matrix. Of interest are submatrices whose entries have an average greater than or equal to a positive constant, and submatrices whose entries are well fit by a two-way ANOVA model. We identify size thresholds and associated (asymptotic) probability bounds for both large-average and ANOVA-fit submatrices. Probability bounds are obtained when the matrix and submatrices of interest are square and, in rectangular cases, when the matrix and submatrices of interest have fixed aspect ratios. Our principal result is an almost sure interval concentration result for the size of large average submatrices in the square case.

This paper develops an empirical likelihood approach to testing for the presence of stochastic ordering among univariate distributions based on independent random samples from each distribution. The proposed test statistic is formed by integrating a localized empirical likelihood statistic with respect to the empirical distribution of the pooled sample. The asymptotic null distribution of this test statistic is found to have a simple distribution-free representation in terms of standard Brownian bridge processes. The approach is used to compare the lengths of rule of Roman Emperors over various historical periods, including the “decline and fall” phase of the empire. In a simulation study, the power of the proposed test is found to improve substantially upon that of a competing test due to El Barmi and Mukerjee.

Exploratory data analysis is often used to test the goodness-of-fit of sample observations to specific target distributions. A few such graphical tools have been extensively used to detect subexponential or heavy-tailed behavior in observed data. In this paper we discuss asymptotic limit behavior of two such plotting tools: the quantile–quantile plot and the mean excess plot. The weak consistency of these plots to fixed limit sets in an appropriate topology of $\mathbb{R}^{2}$ has been shown in Das and Resnick (Stoch. Models 24 (2008) 103–132) and Ghosh and Resnick (Stochastic Process. Appl. 120 (2010) 1492–1517). In this paper we find asymptotic distributional limits for these plots when the underlying distributions have regularly varying right-tails. As an application we construct confidence bounds around the plots which enable us to statistically test whether the underlying distribution is heavy-tailed or not.

We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.