Mean density of lower dimensional random closed sets, as well as the mean boundary density of full dimensional random sets, and their estimation are of great interest in many real applications. Only partial results are available so far in current literature, under the assumption that the random set is either stationary, or it is a Boolean model, or it has convex grains. We consider here non-stationary random closed sets (not necessarily Boolean models), whose grains have to satisfy some general regularity conditions, extending previous results. We address the open problem posed in (Bernoulli 15 (2009) 1222–1242) about the approximation of the mean density of lower dimensional random sets by a pointwise limit, and to the open problem posed by Matheron in (Random Sets and Integral Geometry (1975) Wiley) about the existence (and its value) of the so-called specific area of full dimensional random closed sets. The relationship with the spherical contact distribution function, as well as some examples and applications are also discussed.

We study the weak convergence (in the high-frequency limit) of the parameter estimators of power spectrum coefficients associated with Gaussian, spherical and isotropic random fields. In particular, we introduce a Whittle-type approximate maximum likelihood estimator and we investigate its asympotic weak consistency and Gaussianity, in both parametric and semiparametric cases.

We investigate the behavior of the Generalized Likelihood Ratio Test (GLRT) (Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153–193]) for time varying coefficient models where the regressors and errors are non-stationary time series and can be cross correlated. It is found that the GLRT retains the minimax rate of local alternative detection under weak dependence and non-stationarity. However, in general, the Wilks phenomenon as well as the classic residual bootstrap are sensitive to either conditional heteroscedasticity of the errors, non-stationarity or temporal dependence. An averaged test is suggested to alleviate the sensitivity of the test to the choice of bandwidth and is shown to be more powerful than tests based on a single bandwidth. An alternative wild bootstrap method is proposed and shown to be consistent when making inference of time varying coefficient models for non-stationary time series.

We study the nonparametric calibration of exponential Lévy models with infinite jump activity. In particular our analysis applies to self-decomposable processes whose jump density can be characterized by the $k$-function, which is typically nonsmooth at zero. On the one hand the estimation of the drift, of the activity measure $\alpha:=k(0+)+k(0-)$ and of analogous parameters for the derivatives of the $k$-function are considered and on the other hand we estimate nonparametrically the $k$-function. Minimax convergence rates are derived. Since the rates depend on $\alpha$, we construct estimators adapting to this unknown parameter. Our estimation method is based on spectral representations of the observed option prices and on a regularization by cutting off high frequencies. Finally, the procedure is applied to simulations and real data.

We consider efficient estimation of the Euclidean parameters in a generalized partially linear additive models for longitudinal/clustered data when multiple covariates need to be modeled nonparametrically, and propose an estimation procedure based on a spline approximation of the nonparametric part of the model and the generalized estimating equations (GEE). Although the model in consideration is natural and useful in many practical applications, the literature on this model is very limited because of challenges in dealing with dependent data for nonparametric additive models. We show that the proposed estimators are consistent and asymptotically normal even if the covariance structure is misspecified. An explicit consistent estimate of the asymptotic variance is also provided. Moreover, we derive the semiparametric efficiency score and information bound under general moment conditions. By showing that our estimators achieve the semiparametric information bound, we effectively establish their efficiency in a stronger sense than what is typically considered for GEE. The derivation of our asymptotic results relies heavily on the empirical processes tools that we develop for the longitudinal/clustered data. Numerical results are used to illustrate the finite sample performance of the proposed estimators.

This paper considers an estimation of semiparametric functional (varying)-coefficient quantile regression with spatial data. A general robust framework is developed that treats quantile regression for spatial data in a natural semiparametric way. The local M-estimators of the unknown functional-coefficient functions are proposed by using local linear approximation, and their asymptotic distributions are then established under weak spatial mixing conditions allowing the data processes to be either stationary or nonstationary with spatial trends. Application to a soil data set is demonstrated with interesting findings that go beyond traditional analysis.

We consider Kallenberg’s hypothesis on the characteristic function of a Lévy process and show that it allows the construction of weakly continuous bridges of the Lévy process conditioned to stay positive. We therefore provide a notion of normalized excursions Lévy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Lévy process conditioned to stay positive by using the weakly continuous bridges of the Lévy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper semi-continuous versions. Using the bridges of the conditioned Lévy process, the Durrett–Iglehart theorem stating that the Brownian bridge from $0$ to $0$ conditioned to remain above $-\varepsilon $ converges weakly to the Brownian excursion as $\varepsilon \to0$, is extended to Lévy processes. We also extend the Denisov decomposition of Brownian motion to Lévy processes and their bridges, as well as Vervaat’s classical result stating the equivalence in law of the Vervaat transform of a Brownian bridge and the normalized Brownian excursion.

For a class of martingales, this paper provides a framework on the uniform consistency with broad applicability. The main condition imposed is only related to the conditional variance of the martingale, which holds true for stationary mixing time series, stationary iterated random function, Harris recurrent Markov chains and $I(1)$ processes with innovations being a linear process. Using the established results, this paper investigates the uniform convergence of the Nadaraya–Watson estimator in a non-linear cointegrating regression model. Our results not only provide sharp convergence rate, but also the optimal range for the uniform convergence to be held. This paper also considers the uniform upper and lower bound estimates for a functional of Harris recurrent Markov chain, which are of independent interests.

We show that the density of $Z=\mathop{\operatorname{argmax}}\{W(t)-t^{2}\}$, sometimes known as Chernoff’s density, is log-concave. We conjecture that Chernoff’s density is strongly log-concave or “super-Gaussian”, and provide evidence in support of the conjecture.

Let $X$ be an absolutely continuous random variable from the integrated Pearson family and assume that $X$ has finite moments of any order. Using some properties of the associated orthonormal polynomial system, we provide a class of strengthened Chernoff-type variance bounds.

In this paper, we study the asymptotic behaviour of one-dimensional integrated Ornstein–Uhlenbeck processes driven by $\alpha$-stable Lévy processes of small amplitude. We prove that the integrated Ornstein–Uhlenbeck process converges weakly to the underlying $\alpha$-stable Lévy process in the Skorokhod $M_{1}$-topology which secures the weak convergence of first passage times. This result follows from a more general result about approximations of an arbitrary Lévy process by continuous integrated Ornstein–Uhlenbeck processes in the $M_{1}$-topology.

In the present paper, we consider the problem of matrix completion with noise. Unlike previous works, we consider quite general sampling distribution and we do not need to know or to estimate the variance of the noise. Two new nuclear-norm penalized estimators are proposed, one of them of “square-root” type. We analyse their performance under high-dimensional scaling and provide non-asymptotic bounds on the Frobenius norm error. Up to a logarithmic factor, these performance guarantees are minimax optimal in a number of circumstances.

We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding $p$-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the $p$-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of $p$-values, or under the assumption that the finite dimensional distributions of the $p$-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables.

In this paper, we prove central limit theorems for bias reduced estimators of the structure function of several multifractal processes, namely mutiplicative cascades, multifractal random measures, multifractal random walk and multifractal fractional random walk as defined by Ludeña [Ann. Appl. Probab. 18 (2008) 1138–1163]. Previous estimators of the structure functions considered in the literature were severely biased with a logarithmic rate of convergence, whereas the estimators considered here have a polynomial rate of convergence.

We consider a supercritical Galton–Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit ${ \mathcal{W}}$ for the normalized population size. Small value probabilities for ${ \mathcal{W}}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized.