This work aims at estimating inverse autocovariance matrices of long memory processes admitting a linear representation. A modified Cholesky decomposition is used in conjunction with an increasing order autoregressive model to achieve this goal. The spectral norm consistency of the proposed estimate is established. We then extend this result to linear regression models with long-memory time series errors. In particular, we show that when the objective is to consistently estimate the inverse autocovariance matrix of the error process, the same approach still works well if the estimated (by least squares) errors are used in place of the unobservable ones. Applications of this result to estimating unknown parameters in the aforementioned regression model are also given. Finally, a simulation study is performed to illustrate our theoretical findings.

Instead of defining goodness of fit (GOF) tests in terms of their test statistics, we present an alternative method by introducing the concept of local levels, which indicate high or low local sensitivity of a test. Local levels can act as a starting point for the construction of new GOF tests. We study the behavior of local levels when applied to some well-known GOF tests such as Kolmogorov–Smirnov (KS) tests, higher criticism (HC) tests and tests based on phi-divergences. The main focus is on a rigorous characterization of the asymptotic behavior of local levels of the original HC tests which leads to several further asymptotic results for local levels of other GOF tests including GOF tests with equal local levels. While local levels of KS tests, which are related to the central range, are asymptotically strictly larger than zero, all local levels of HC tests converge to zero as the sample size increases. Consequently, there exists no asymptotic level $\alpha$ GOF test such that all local levels are asymptotically bounded away from zero. Finally, by means of numerical computations we compare classical KS and HC tests to a GOF test with equal local levels.

For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas.

We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an Itô formula. Moreover, we derive an alternative definition using a random-field approach and relate both concepts. We present examples related to fundamental solutions to partial differential equations.

New results and improvements in the study of nonparametric exponential and mixture models are proposed. In particular, different equivalent characterizations of maximal exponential models, in terms of open exponential arcs and Orlicz spaces, are given. Our theoretical results are supported by several examples and counterexamples and provide an answer to some open questions in the literature.

Stochastic volatility processes with heavy-tailed innovations are a well-known model for financial time series. In these models, the extremes of the log returns are mainly driven by the extremes of the i.i.d. innovation sequence which leads to a very strong form of asymptotic independence, that is, the coefficient of tail dependence is equal to $1/2$ for all positive lags. We propose an alternative class of stochastic volatility models with heavy-tailed volatilities and examine their extreme value behavior. In particular, it is shown that, while lagged extreme observations are typically asymptotically independent, their coefficient of tail dependence can take on any value between $1/2$ (corresponding to exact independence) and 1 (related to asymptotic dependence). Hence, this class allows for a much more flexible extremal dependence between consecutive observations than classical SV models and can thus describe the observed clustering of financial returns more realistically.

The extremal dependence structure of lagged observations is analyzed in the framework of regular variation on the cone $(0,\infty)^{d}$. As two auxiliary results which are of interest on their own we derive a new Breiman-type theorem about regular variation on $(0,\infty)^{d}$ for products of a random matrix and a regularly varying random vector and a statement about the joint extremal behavior of products of i.i.d. regularly varying random variables.

We consider the passage time problem for Lévy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in.

We study conditions under which, given a dictionary $F=\{f_{1},\ldots,f_{M}\}$ and an i.i.d. sample $(X_{i},Y_{i})_{i=1}^{N}$, the empirical minimizer in $\operatorname{span}(F)$ relative to the squared loss, satisfies that with high probability

\[R(\tilde{f}^{\mathrm{ERM}})\leq\inf_{f\in\operatorname{span}(F)}R(f)+r_{N}(M),\] where $R(\cdot)$ is the squared risk and $r_{N}(M)$ is of the order of $M/N$.

Among other results, we prove that a uniform small-ball estimate for functions in $\operatorname{span}(F)$ is enough to achieve that goal when the noise is independent of the design.

This paper studies posterior concentration behavior of the base probability measure of a Dirichlet measure, given observations associated with the sampled Dirichlet processes, as the number of observations tends to infinity. The base measure itself is endowed with another Dirichlet prior, a construction known as the hierarchical Dirichlet processes (Teh et al. [J. Amer. Statist. Assoc. 101 (2006) 1566–1581]). Convergence rates are established in transportation distances (i.e., Wasserstein metrics) under various conditions on the geometry of the support of the true base measure. As a consequence of the theory, we demonstrate the benefit of “borrowing strength” in the inference of multiple groups of data – a powerful insight often invoked to motivate hierarchical modeling. In certain settings, the gain in efficiency due to the latent hierarchy can be dramatic, improving from a standard nonparametric rate to a parametric rate of convergence. Tools developed include transportation distances for nonparametric Bayesian hierarchies of random measures, the existence of tests for Dirichlet measures, and geometric properties of the support of Dirichlet measures.

This paper is a continuation of (Bernoulli 20 (2014) 2169–2216) where we prove a characterization of the support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable and null initial conditions. Here, we allow for non-null initial conditions and, therefore, the solution does not possess a stationary property in space. As in (Bernoulli 20 (2014) 2169–2216), the support theorem is a consequence of an approximation result, in the convergence of probability, of a sequence of evolution equations driven by a family of regularizations of the driving noise. However, the method of the proof differs from (Bernoulli 20 (2014) 2169–2216) since arguments based on the stationarity property of the solution cannot be used.

Coupling by reflection mixed with synchronous coupling is constructed for a class of stochastic differential equations (SDEs) driven by Lévy noises. As an application, we establish the exponential contractivity of the associated semigroups $(P_{t})_{t\ge0}$ with respect to the standard $L^{p}$-Wasserstein distance for all $p\in[1,\infty)$. In particular, consider the following SDE:

\[\mathrm{d}X_{t}=\mathrm{d}Z_{t}+b(X_{t})\,\mathrm{d}t,\] where $(Z_{t})_{t\ge0}$ is a symmetric $\alpha$-stable process on $\mathbb{R}^{d}$ with $\alpha\in(1,2)$. We show that if the drift term $b$ satisfies that for any $x,y\in\mathbb{R}^{d}$,

\[\langle b(x)-b(y),x-y\rangle\le\cases{K_{1}|x-y|^{2},\quad\phantom{-} |x-y|\le L_{0};\cr-K_{2}|x-y|^{\theta},\quad |x-y|>L_{0}}\] holds with some positive constants $K_{1}$, $K_{2}$, $L_{0}>0$ and $\theta\ge2$, then there is a constant $\lambda:=\lambda(\theta,K_{1},K_{2},L_{0})>0$ such that for all $p\in[1,\infty)$, $t>0$ and $x,y\in\mathbb{R}^{d}$,

\[W_{p}(\delta_{x}P_{t},\delta_{y}P_{t})\le C(p,\theta,K_{1},K_{2},L_{0})\mathrm{e}^{-\lambda t/p}[\frac{|x-y|^{1/p}\vee|x-y|}{1+|x-y|{\mathbf{1}}_{(1,\infty)\times(2,\infty)}(t,\theta)}].\]

We are interested in the problem of robust parametric estimation of a density from $n$ i.i.d. observations. By using a practice-oriented procedure based on robust tests, we build an estimator for which we establish non-asymptotic risk bounds with respect to the Hellinger distance under mild assumptions on the parametric model. We show that the estimator is robust even for models for which the maximum likelihood method is bound to fail. A numerical simulation illustrates its robustness properties. When the model is true and regular enough, we prove that the estimator is very close to the maximum likelihood one, at least when the number of observations $n$ is large. In particular, it inherits its efficiency. Simulations show that these two estimators are almost equal with large probability, even for small values of $n$ when the model is regular enough and contains the true density.

We prove functional central and non-central limit theorems for generalized variations of the anisotropic $d$-parameter fractional Brownian sheet (fBs) for any natural number $d$. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.

We propose a convergence mode for positive Radon measures which allows a sequence of probability measures to have an improper limiting measure. We define a sequence of vague priors as a sequence of probability measures that converges to an improper prior. We consider some cases where vague priors have necessarily large variances and other cases where they have not. We study the consequences of the convergence of prior distributions on the posterior analysis. Then we give some constructions of vague priors that approximate the Haar measures or the Jeffreys priors. We also revisit the Jeffreys–Lindley paradox.

We establish minimax convergence rates for classification of functional data and for nonparametric regression with functional design variables. The optimal rates are of logarithmic type under smoothness constraints on the functional density and the regression mapping, respectively. These asymptotic properties are attainable by conventional kernel procedures. The bandwidth selector does not require knowledge of the smoothness level of the target mapping. In this work, the functional data are considered as realisations of random variables which take their values in a general Polish metric space. We impose certain metric entropy constraints on this space; but no algebraic properties are required.

Sign tests are among the most successful procedures in multivariate nonparametric statistics. In this paper, we consider several testing problems in multivariate analysis, directional statistics and multivariate time series analysis, and we show that, under appropriate symmetry assumptions, the fixed-$p$ multivariate sign tests remain valid in the high-dimensional case. Remarkably, our asymptotic results are universal, in the sense that, unlike in most previous works in high-dimensional statistics, $p$ may go to infinity in an arbitrary way as $n$ does. We conduct simulations that (i) confirm our asymptotic results, (ii) reveal that, even for relatively large $p$, chi-square critical values are to be favoured over the (asymptotically equivalent) Gaussian ones and (iii) show that, for testing i.i.d.-ness against serial dependence in the high-dimensional case, Portmanteau sign tests outperform their competitors in terms of validity-robustness.

Quantile- and copula-related spectral concepts recently have been considered by various authors. Those spectra, in their most general form, provide a full characterization of the copulas associated with the pairs $(X_{t},X_{t-k})$ in a process $(X_{t})_{t\in\mathbb{Z}}$, and account for important dynamic features, such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes that their traditional counterparts cannot capture. Despite various proposals for estimation strategies, only quite incomplete asymptotic distributional results are available so far for the proposed estimators, which constitutes an important obstacle for their practical application. In this paper, we provide a detailed asymptotic analysis of a class of smoothed rank-based cross-periodograms associated with the copula spectral density kernels introduced in Dette et al. [Bernoulli 21 (2015) 781–831]. We show that, for a very general class of (possibly nonlinear) processes, properly scaled and centered smoothed versions of those cross-periodograms, indexed by couples of quantile levels, converge weakly, as stochastic processes, to Gaussian processes. A first application of those results is the construction of asymptotic confidence intervals for copula spectral density kernels. The same convergence results also provide asymptotic distributions (under serially dependent observations) for a new class of rank-based spectral methods involving the Fourier transforms of rank-based serial statistics such as the Spearman, Blomqvist or Gini autocovariance coefficients.

For a reversible and ergodic Markov chain $\{X_{n},n\geq0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi(h)$ can be constructed whenever the asymptotic variance $\sigma^{2}_{P}(h)$ is finite and positive. We do not impose any additional condition on the convergence rate of the Markov chain. The confidence interval is derived using the so-called fixed-b lag-window estimator of $\sigma_{P}^{2}(h)$. We also derive a result that suggests that the proposed confidence interval procedure converges faster than classical confidence interval procedures based on the Gaussian distribution and standard central limit theorems for Markov chains.

Given random samples drawn i.i.d. from a probability measure $\mathbb{P}$ (defined on say, $\mathbb{R}^{d}$), it is well-known that the empirical estimator is an optimal estimator of $\mathbb{P}$ in weak topology but not even a consistent estimator of its density (if it exists) in the strong topology (induced by the total variation distance). On the other hand, various popular density estimators such as kernel and wavelet density estimators are optimal in the strong topology in the sense of achieving the minimax rate over all estimators for a Sobolev ball of densities. Recently, it has been shown in a series of papers by Giné and Nickl that these density estimators on $\mathbb{R}$ that are optimal in strong topology are also optimal in $\Vert\cdot\Vert_{\mathcal{F} }$ for certain choices of $\mathcal{F}$ such that $\Vert\cdot\Vert_{\mathcal{F} }$ metrizes the weak topology, where $\Vert\mathbb{P} \Vert_{\mathcal{F} }:=\sup\{\int f\,\mathrm{d}\mathbb{P} \colon\ f\in\mathcal{F} \}$. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing $\mathcal{F}$ to be a unit ball in a reproducing kernel Hilbert space (say $\mathcal{F}_{H}$ defined over $\mathbb{R}^{d}$), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that $\Vert\cdot\Vert_{\mathcal{F}_{H}}$ metrizes the weak topology and the kernel density estimator (with $L^{1}$ optimal bandwidth) estimates $\mathbb{P}$ at dimension independent optimal rate of $n^{-1/2}$ in $\Vert\cdot\Vert_{\mathcal{F}_{H}}$ along with providing a uniform central limit theorem for the kernel density estimator.

This paper presents a central limit theorem for a pre-averaged version of the realized covariance estimator for the quadratic covariation of a discretely observed semimartingale with noise. The semimartingale possibly has jumps, while the observation times show irregularity, non-synchronicity, and some dependence on the observed process. It is shown that the observation times’ effect on the asymptotic distribution of the estimator is only through two characteristics: the observation frequency and the covariance structure of the noise. This is completely different from the case of the realized covariance in a pure semimartingale setting.

An important consideration for variable selection in interaction models is to design an appropriate penalty that respects hierarchy of the importance of the variables. A common theme is to include an interaction term only after the corresponding main effects are present. In this paper, we study several recently proposed approaches and present a unified analysis on the convergence rate for a class of estimators, when the design satisfies the restricted eigenvalue condition. In particular, we show that with probability tending to one, the resulting estimates have a rate of convergence $s\sqrt{\log p_{1}/n}$ in the $\ell_{1}$ error, where $p_{1}$ is the ambient dimension, $s$ is the true dimension and $n$ is the sample size. We give a new proof that the restricted eigenvalue condition holds with high probability, when the variables in the main effects and the errors follow sub-Gaussian distributions. Under this setup, the interactions no longer follow Gaussian or sub-Gaussian distributions even if the main effects follow Gaussian, and thus existing works are not applicable. This result is of independent interest.