We establish Talagrand’s T₁ and T₂ inequalities for the law of the solution of a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H > 1/2. We use the L² metric and the uniform metric on the path space of continuous functions on [0, T]. These results are applied to study small-time and large-time asymptotics for the solutions of such equations by means of a Hoeffding-type inequality.

In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter H > 1/2. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.

The class of multivariate Lévy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the class of continuous-time state space models. The linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular (β-mixing) under a mild continuity assumption on the driving Lévy process. It is verified that this continuity assumption is satisfied in most practically relevant situations, including the case where the driving Lévy process has a non-singular Gaussian component, is compound Poisson with an absolutely continuous jump size distribution or has an infinite Lévy measure admitting a density around zero.

GARCH is one of the most prominent nonlinear time series models, both widely applied and thoroughly studied. Recently, it has been shown that the COGARCH model (which was introduced a few years ago by Klüppelberg, Lindner and Maller) and Nelson’s diffusion limit are the only functional continuous-time limits of GARCH in distribution. In contrast to Nelson’s diffusion limit, COGARCH reproduces most of the stylized facts of financial time series. Since it has been proven that Nelson’s diffusion is not asymptotically equivalent to GARCH in deficiency, in the present paper, we investigate the relation between GARCH and COGARCH in Le Cam’s framework of statistical equivalence. We show that GARCH converges generically to COGARCH, even in deficiency, provided that the volatility processes are observed. Hence, from a theoretical point of view, COGARCH can indeed be considered as a continuous-time equivalent to GARCH. Otherwise, when the observations are incomplete, GARCH still has a limiting experiment, which we call MCOGARCH, which is not equivalent, but nevertheless quite similar, to COGARCH. In the COGARCH model, the jump times can be more random than for the MCOGARCH, a fact practitioners may see as an advantage of COGARCH.

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on ℝ^d are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. A statistical version of the CLT is considered as well. Some numerical results are also discussed.

We define Letac–Wesolowski–Matsumoto–Yor (LWMY) functions as decreasing functions from (0, ∞) onto (0, ∞) with the following property: there exist independent, positive random variables X and Y such that the variables f(X + Y) and f(X) − f(X + Y) are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is f(x) = 1/x. In this case, referred to in the literature as the Matsumoto–Yor property, the law of X is generalized inverse Gaussian while Y is gamma distributed. In the three other cases, the associated densities are provided. As a consequence, we obtain a new relation of convolution involving gamma distributions and Kummer distributions of type 2.

We consider data-adaptive wavelet estimation of a trend function in a time series model with strongly dependent Gaussian residuals. Asymptotic expressions for the optimal mean integrated squared error and corresponding optimal smoothing and resolution parameters are derived. Due to adaptation to the properties of the underlying trend function, the approach shows very good performance for smooth trend functions while remaining competitive with minimax wavelet estimation for functions with discontinuities. Simulations illustrate the asymptotic results and finite-sample behavior.

In this paper we introduce new estimators of the coefficient functions in the varying coefficient regression model. The proposed estimators are obtained by projecting the vector of the full-dimensional kernel-weighted local polynomial estimators of the coefficient functions onto a Hilbert space with a suitable norm. We provide a backfitting algorithm to compute the estimators. We show that the algorithm converges at a geometric rate under weak conditions. We derive the asymptotic distributions of the estimators and show that the estimators have the oracle properties. This is done for the general order of local polynomial fitting and for the estimation of the derivatives of the coefficient functions, as well as the coefficient functions themselves. The estimators turn out to have several theoretical and numerical advantages over the marginal integration estimators studied by Yang, Park, Xue and Härdle [J. Amer. Statist. Assoc. 101 (2006) 1212–1227].

In the linear random effects model, when distributional assumptions such as normality of the error variables cannot be justified, moments may serve as alternatives to describe relevant distributions in neighborhoods of their means. Generally, estimators may be obtained as solutions of estimating equations. It turns out that there may be several equations, each of them leading to consistent estimators, in which case finding the efficient estimator becomes a crucial problem. In this paper, we systematically study estimation of moments of the errors and random effects in linear mixed models.

Although unbiasedness is a basic property of a good test, many tests on vector parameters or scalar parameters against two-sided alternatives are not finite-sample unbiased. This was already noticed by Sugiura [Ann. Inst. Statist. Math. 17 (1965) 261–263]; he found an alternative against which the Wilcoxon test is not unbiased. The problem is even more serious in multivariate models. When testing the hypothesis against an alternative which fits well with the experiment, it should be verified whether the power of the test under this alternative cannot be smaller than the significance level. Surprisingly, this serious problem is not frequently considered in the literature.

The present paper considers the two-sample multivariate testing problem. We construct several rank tests which are finite-sample unbiased against a broad class of location/scale alternatives and are finite-sample distribution-free under the hypothesis and alternatives. Each of them is locally most powerful against a specific alternative of the Lehmann type. Their powers against some alternatives are numerically compared with each other and with other rank and classical tests. The question of affine invariance of two-sample multivariate tests is also discussed.

Sequential Monte Carlo (SMC) methods are a class of techniques to sample approximately from any sequence of probability distributions using a combination of importance sampling and resampling steps. This paper is concerned with the convergence analysis of a class of SMC methods where the times at which resampling occurs are computed online using criteria such as the effective sample size. This is a popular approach amongst practitioners but there are very few convergence results available for these methods. By combining semigroup techniques with an original coupling argument, we obtain functional central limit theorems and uniform exponential concentration estimates for these algorithms.

Comparison results are obtained for the inclusion probabilities in some unequal probability sampling plans without replacement. For either successive sampling or Hájek’s rejective sampling, the larger the sample size, the more uniform the inclusion probabilities in the sense of majorization. In particular, the inclusion probabilities are more uniform than the drawing probabilities. For the same sample size, and given the same set of drawing probabilities, the inclusion probabilities are more uniform for rejective sampling than for successive sampling. This last result confirms a conjecture of Hájek (Sampling from a Finite Population (1981) Dekker). Results are also presented in terms of the Kullback–Leibler divergence, showing that the inclusion probabilities for successive sampling are more proportional to the drawing probabilities.

In this paper we investigate undirected discrete graphical tree models when all the variables in the system are binary, where leaves represent the observable variables and where all the inner nodes are unobserved. A novel approach based on the theory of partially ordered sets allows us to obtain a convenient parametrization of this model class. The construction of the proposed coordinate system mirrors the combinatorial definition of cumulants. A simple product-like form of the resulting parametrization gives insight into identifiability issues associated with this model class. In particular, we provide necessary and sufficient conditions for such a model to be identified up to the switching of labels of the inner nodes. When these conditions hold, we give explicit formulas for the parameters of the model. Whenever the model fails to be identified, we use the new parametrization to describe the geometry of the unidentified parameter space. We illustrate these results using a simple example.

The distributions of the largest and the smallest eigenvalues of a p-variate sample covariance matrix S are of great importance in statistics. Focusing on the null case where nS follows the standard Wishart distribution W_p(I, n), we study the accuracy of their scaling limits under the setting: n/p → γ ∈ (0, ∞) as n → ∞. The limits here are the orthogonal Tracy–Widom law and its reflection about the origin.

With carefully chosen rescaling constants, the approximation to the rescaled largest eigenvalue distribution by the limit attains accuracy of order O(min(n, p)^−2/3). If γ > 1, the same order of accuracy is obtained for the smallest eigenvalue after incorporating an additional log transform. Numerical results show that the relative error of approximation at conventional significance levels is reduced by over 50% in rectangular and over 75% in ‘thin’ data matrix settings, even with min(n, p) as small as 2.

Let P be a probability distribution on ℝ^d (equipped with an Euclidean norm |⋅|). Let r > 0 and let (α_n)_n≥1 be an (asymptotically) L^r(P)-optimal sequence of n-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence (α_n)_n≥1 defined for every n ≥ 1 by ρ(α_n) = max{|a|, a ∈ α_n}. When card(supp(P)) is infinite, the maximal radius sequence goes to sup {|x|, x ∈ supp(P)} as n goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.