We introduce some new nonparametic tests of independence which are functionals of the modified multivariate empirical copula process. In this work we extend the modified empirical process of Nikitin and Sporysheva (2009), which is defined by using a family of Gaussian processes of Deheuvels (2007b) to the multivariate case. The limiting behavior of the proposed statistics is established under the null hypothesis. Emphasis is placed on explanation of the strong approximation methodology.

L 1-estimation of a location parameter is studied for the “product type” stochastic volatility models. The asymptotic distribution of the L 1-estimator is established under general conditions on the behavior of the distribution function of the errors near zero.

In this paper we describe a novel approach to the study of U-statistics in the Markovian setup based on the (pseudo-) regenerative properties of Harris Markov chains. Exploiting the fact that any sample path X 1, ..., X n of a general Harris chain X may be divided into asymptotically i.i.d. data blocks $\mathcal{B}_1 , \ldots \mathcal{B}_N$ of random length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative U-statistic Ω N = Σ k≠l ω h $\left( {\mathcal{B}_k ,\mathcal{B}_l } \right)$ /(N(N − 1)) related to a U-statistic U n = Σ i≠j h(X i , X j )/(n(n − 1)). We show that, under mild conditions, these two statistics are asymptotically equivalent up to the order O ℙ(n−1). This result serves as a basis for establishing limit theorems related to statistics of the same form as U n . Beyond its use as a technical tool for proving results of a theoretical nature, the regenerative method is also employed here in a constructive fashion for estimating the limiting variance or the sampling distribution of certain U-statistics through resampling. The proof of the asymptotic validity of this statistical methodology is provided, together with an illustrative simulation result.

In this paper we consider a (p × q)-matrix X = (X 1, ..., X q ), where a pq-vector vec (X) = (X 1 , ...,X ) T is assumed to be distributed normally with mean vector vec (M) = (M 1 , ...,M ) T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ⊗Σ, where Φ = (ϕ ij ) is a (q × q)-matrix and Σ = (σ ij ) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.

Consider a class of M-estimators indexed by a criterion function ψ. When the function ψ is taken to be in a class of functions \( \mathcal{F} \), a family of processes indexed by the class \( \mathcal{F} \) is obtained and called M-processes. Pooling the M-estimators in such class may be used to define new kind of estimators. In order to get the asymptotic properties of these pooled estimators, the convergence in probability of the corresponding M-process is studied uniformly on \( \mathcal{F} \) together with their weak convergence towards a Gaussian process. An application to location estimation is presented and discussed.

We present a statistical process depending on a continuous time parameter τ whose each margin provides a Generalized Hill’s estimator. In this paper, the asymptotic normality of the finite-dimensional distributions of this family are completely characterized for τ > 1/2 when the underlying distribution function lies on the maximum domain of attraction. The ratio of two different margins of the statistical process characterizes entirely the whole domain of attraction. Its asymptotic normality is also studied. The results permit in general to build a new family of estimators for the extreme value index whose asymptotic properties can be easily derived. For example, we give a new estimate of the Weibull extreme value index and we study its consistency and its asymptotic normality.

We consider the linear regression model with Gaussian error. We estimate the unknown parameters by a procedure inspired by the Group Lasso estimator introduced in [22]. We show that this estimator satisfies a sparsity inequality, i.e., a bound in terms of the number of non-zero components of the oracle regression vector. We prove that this bound is better, in some cases, than the one achieved by the Lasso and the Dantzig selector.

In this paper, a general autoregressive model with Markov switching is considered, where the autoregression may be of an infinite order. The consistency of the maximum likelihood estimators for this model is obtained under regularity assumptions. Examples of finite and infinite order autoregressive models with Markov switching are discussed. Simulation studies with these examples illustrate the consistency and asymptotic normality of the estimators.

In this paper, we prove large deviations principle for the Nadaraya-Watson estimator and for the semi-recursive kernel estimator of the regression in the multidimensional case. Under suitable conditions, we show that the rate function is a good rate function. We thus generalize the results already obtained in the one-dimensional case for the Nadaraya-Watson estimator. Moreover, we give a moderate deviations principle for these two estimators. It turns out that the rate function obtained in the moderate deviations principle for the semi-recursive estimator is larger than the one obtained for the Nadaraya-Watson estimator.