We consider spatially homogeneous marked point patterns in an unboundedly expanding convex sampling window. Our main objective is to identify the distribution of the typical mark by constructing an asymptotic $\chi^{2}$-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smoothed covariance estimator which turns out to be mean square consistent. Our approach does not require independent marks and allows dependences between the mark field and the point pattern. Instead we impose a suitable $\beta$-mixing condition on the underlying stationary marked point process which can be checked for a number of Poisson-based models and, in particular, in the case of geostatistical marking. In order to study test performance, our test approach is applied to detect anisotropy of specific Boolean models.

The objective of this paper is to study the Gibbs sampling for computing the mean of observable in very high dimension – a powerful Markov chain Monte Carlo method. Under the Dobrushin’s uniqueness condition, we establish some explicit and sharp estimate of the exponential convergence rate and prove some Gaussian concentration inequalities for the empirical mean.

In extreme value statistics, the peaks-over-threshold method is widely used. The method is based on the generalized Pareto distribution characterizing probabilities of exceedances over high thresholds in $\mathbb{R}^{d}$. We present a generalization of this concept in the space of continuous functions. We call this the generalized Pareto process. Differently from earlier papers, our definition is not based on a distribution function but on functional properties, and does not need a reference to a related max-stable process.

As an application, we use the theory to simulate wind fields connected to disastrous storms on the basis of observed extreme but not disastrous storms. We also establish the peaks-over-threshold approach in function space.

Comparisons are made for the amount of agreement of the composite likelihood information criteria and their full likelihood counterparts when making decisions among the fits of different models, and some properties of penalty term for composite likelihood information criteria are obtained. Asymptotic theory is given for the case when a simpler model is nested within a bigger model, and the bigger model approaches the simpler model under a sequence of local alternatives. Composite likelihood can more or less frequently choose the bigger model, depending on the direction of local alternatives; in the former case, composite likelihood has more “power” to choose the bigger model. The behaviors of the information criteria are illustrated via theory and simulation examples of the Gaussian linear mixed-effects model.

A basic result of large deviations theory is Sanov’s theorem, which states that the sequence of empirical measures of independent and identically distributed samples satisfies the large deviation principle with rate function given by relative entropy with respect to the common distribution. Large deviation principles for the empirical measures are also known to hold for broad classes of weakly interacting systems. When the interaction through the empirical measure corresponds to an absolutely continuous change of measure, the rate function can be expressed as relative entropy of a distribution with respect to the law of the McKean–Vlasov limit with measure-variable frozen at that distribution. We discuss situations, beyond that of tilted distributions, in which a large deviation principle holds with rate function in relative entropy form.

This paper deals with minimax rates of convergence for estimation of density functions on the real line. The densities are assumed to be location mixtures of normals, a global regularity requirement that creates subtle difficulties for the application of standard minimax lower bound methods. Using novel Fourier and Hermite polynomial techniques, we determine the minimax optimal rate – slightly larger than the parametric rate – under squared error loss. For Hellinger loss, we provide a minimax lower bound using ideas modified from the squared error loss case.

The purpose of this article is to observe that the zero sets of continuous-state branching processes with immigration (CBI) are infinitely divisible regenerative sets. Indeed, they can be constructed by the procedure of random cutouts introduced by Mandelbrot in 1972. We then show how very precise information about the zero sets of CBI can be obtained in terms of the branching and immigrating mechanism.

We investigate the Large Deviations (LD) properties of bootstrapped empirical measures with exchangeable weights. Our main results show in great generality how the resulting rate functions combine the LD properties of both the sample weights and the observations. As an application, we obtain new LD results and discuss both conditional and unconditional LD-efficiency for many classical choices of entries such as Efron’s, leave-$p$-out, i.i.d. weighted, $k$-blocks bootstraps, etc.

Sequential Monte Carlo (SMC) methods, also known as particle filters, are simulation-based recursive algorithms for the approximation of the a posteriori probability measures generated by state-space dynamical models. At any given time $t$, a SMC method produces a set of samples over the state space of the system of interest (often termed “particles”) that is used to build a discrete and random approximation of the posterior probability distribution of the state variables, conditional on a sequence of available observations. One potential application of the methodology is the estimation of the densities associated to the sequence of a posteriori distributions. While practitioners have rather freely applied such density approximations in the past, the issue has received less attention from a theoretical perspective. In this paper, we address the problem of constructing kernel-based estimates of the posterior probability density function and its derivatives, and obtain asymptotic convergence results for the estimation errors. In particular, we find convergence rates for the approximation errors that hold uniformly on the state space and guarantee that the error vanishes almost surely as the number of particles in the filter grows. Based on this uniform convergence result, we first show how to build continuous measures that converge almost surely (with known rate) toward the posterior measure and then address a few applications. The latter include maximum a posteriori estimation of the system state using the approximate derivatives of the posterior density and the approximation of functionals of it, for example, Shannon’s entropy.

We consider the Random Walk Metropolis algorithm on $\mathbb{R}^{n}$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one-dimensional law. It is well known (see Roberts et al. (Ann. Appl. Probab. 7 (1997) 110–120)) that, in the limit $n\to\infty$, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain. In Jourdain et al. (Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-field limit (2012) Preprint), we generalize this result when the initial distribution is not the target probability measure. The obtained diffusive limit is the solution to a stochastic differential equation nonlinear in the sense of McKean. In the present paper, we prove convergence to equilibrium for this equation. We discuss practical counterparts in order to optimize the variance of the proposal distribution to accelerate convergence to equilibrium. Our analysis confirms the interest of the constant acceptance rate strategy (with acceptance rate between $1/4$ and $1/3$) first suggested in Roberts et al. (Ann. Appl. Probab. 7 (1997) 110–120).

We also address scaling of the Metropolis-Adjusted Langevin Algorithm. When starting at equilibrium, a diffusive limit for an optimal scaling of the variance is obtained in Roberts and Rosenthal (J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 (1998) 255–268). In the transient case, we obtain formally that the optimal variance scales very differently in $n$ depending on the sign of a moment of the distribution, which vanishes at equilibrium. This suggest that it is difficult to derive practical recommendations for MALA from such asymptotic results.

In the random-effects model of meta-analysis a canonical representation of the restricted likelihood function is obtained. This representation relates the mean effect and the heterogeneity variance estimation problems. An explicit form of the variance of weighted means statistics determined by means of a quadratic form is found. The behavior of the mean squared error for large heterogeneity variance is elucidated. It is noted that the sample mean is not admissible nor minimax under a natural risk function for the number of studies exceeding three.

We link optimal filtering for hidden Markov models to the notion of duality for Markov processes. We show that when the signal is dual to a process that has two components, one deterministic and one a pure death process, and with respect to functions that define changes of measure conjugate to the emission density, the filtering distributions evolve in the family of finite mixtures of such measures and the filter can be computed at a cost that is polynomial in the number of observations. Special cases of our framework include the Kalman filter, and computable filters for the Cox–Ingersoll–Ross process and the one-dimensional Wright–Fisher process, which have been investigated before. The dual we obtain for the Cox–Ingersoll–Ross process appears to be new in the literature.

While effective concentration inequalities for suprema of empirical processes exist under boundedness or strict tail assumptions, no comparable results have been available under considerably weaker assumptions. In this paper, we derive concentration inequalities assuming only low moments for an envelope of the empirical process. These concentration inequalities are beneficial even when the envelope is much larger than the single functions under consideration.

We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that is for the density of a fixed number of consecutive observations. Using conditions on the prior, we are then able to define a consistent Bayesian estimator of the number of hidden states. It is known that the likelihood ratio test statistic for overfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. Inference of parameters is a much more difficult task than inference of marginal densities, we still provide a precise description of the situation when the observations are i.i.d. and we allow for $2$ possible hidden states.

This paper is concerned with Crump–Mode–Jagers branching processes, describing spread of an epidemic depending on the proportion of the population that is vaccinated. Births in the branching process are aborted independently with a time-dependent probability given by the fraction of the population vaccinated. Stochastic monotonicity and continuity results for a wide class of functions (e.g., extinction time and total number of births over all time) defined on such a branching process are proved using coupling arguments, leading to optimal vaccination schemes to control corresponding functions (e.g., duration and final size) of epidemic outbreaks. The theory is illustrated by applications to the control of the duration of mumps outbreaks in Bulgaria.

Let $T$ be the Student one- or two-sample $t$-, $F$-, or Welch statistic. Now release the underlying assumptions of normality, independence and identical distribution and consider a more general case where one only assumes that the vector of data has a continuous joint density. We determine asymptotic expressions for $\mathbf{P}(T>u)$ as $u\to\infty$ for this case. The approximations are particularly accurate for small sample sizes and may be used, for example, in the analysis of High-Throughput Screening experiments, where the number of replicates can be as low as two to five and often extreme significance levels are used. We give numerous examples and complement our results by an investigation of the convergence speed – both theoretically, by deriving exact bounds for absolute and relative errors, and by means of a simulation study.

We consider spherical data $X_{i}$ noised by a random rotation $\varepsilon_{i}\in\operatorname{SO} (3)$ so that only the sample $Z_{i}=\varepsilon_{i}X_{i}$, $i=1,\dots,N$ is observed. We define a nonparametric test procedure to distinguish $H_{0}$: “the density $f$ of $X_{i}$ is the uniform density $f_{0}$ on the sphere” and $H_{1}$: “$\|f-f_{0}\|_{2}^{2}\geq\mathcal{C} \psi_{N}$ and $f$ is in a Sobolev space with smoothness $s$”. For a noise density $f_{\varepsilon}$ with smoothness index $\nu$, we show that an adaptive procedure (i.e., $s$ is not assumed to be known) cannot have a faster rate of separation than $\psi_{N}^{\mathrm{ad}}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2\nu+1)}$ and we provide a procedure which reaches this rate. We also deal with the case of super smooth noise. We illustrate the theory by implementing our test procedure for various kinds of noise on $\operatorname{SO}(3)$ and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.

A characterization of the support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable is proved. The result is a consequence of an approximation theorem, in the convergence of probability, for a sequence of evolution equations driven by a family of regularizations of the driving noise.

This paper gives a precise characterization of the fundamental limits of adaptive sensing for diverse estimation and testing problems concerning sparse signals. We consider in particular the setting introduced in (IEEE Trans. Inform. Theory 57 (2011) 6222–6235) and show necessary conditions on the minimum signal magnitude for both detection and estimation: if $\mathbf{x}\in\mathbb{R}^{n}$ is a sparse vector with $s$ non-zero components then it can be reliably detected in noise provided the magnitude of the non-zero components exceeds $\sqrt{2/s}$. Furthermore, the signal support can be exactly identified provided the minimum magnitude exceeds $\sqrt{2\log s} $. Notably there is no dependence on $n$, the extrinsic signal dimension. These results show that the adaptive sensing methodologies proposed previously in the literature are essentially optimal, and cannot be substantially improved. In addition, these results provide further insights on the limits of adaptive compressive sensing.

In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means $(\alpha,\beta)$ and of the criticality parameter $\varrho:=\alpha+\beta$ for a 2-type critical doubly symmetric positively regular Galton–Watson branching process with immigration is described.

In statistical analysis, measuring a score of predictive performance is an important task. In many scientific fields, appropriate scoring rules were tailored to tackle the problems at hand. A proper scoring rule is a popular tool to obtain statistically consistent forecasts. Furthermore, a mathematical characterization of the proper scoring rule was studied. As a result, it was revealed that the proper scoring rule corresponds to a Bregman divergence, which is an extension of the squared distance over the set of probability distributions. In the present paper, we introduce composite scoring rules as an extension of the typical scoring rules in order to obtain a wider class of probabilistic forecasting. Then, we propose a class of composite scoring rules, named Hölder scores, that induce equivariant estimators. The equivariant estimators have a favorable property, implying that the estimator is transformed in a consistent way, when the data is transformed. In particular, we deal with the affine transformation of the data. By using the equivariant estimators under the affine transformation, one can obtain estimators that do no essentially depend on the choice of the system of units in the measurement. Conversely, we prove that the Hölder score is characterized by the invariance property under the affine transformations. Furthermore, we investigate statistical properties of the estimators using Hölder scores for the statistical problems including estimation of regression functions and robust parameter estimation, and illustrate the usefulness of the newly introduced scoring rules for statistical forecasting.

Székely, Rizzo and Bakirov (Ann. Statist. 35 (2007) 2769–2794) and Székely and Rizzo (Ann. Appl. Statist. 3 (2009) 1236–1265), in two seminal papers, introduced the powerful concept of distance correlation as a measure of dependence between sets of random variables. We study in this paper an affinely invariant version of the distance correlation and an empirical version of that distance correlation, and we establish the consistency of the empirical quantity. In the case of subvectors of a multivariate normally distributed random vector, we provide exact expressions for the affinely invariant distance correlation in both finite-dimensional and asymptotic settings, and in the finite-dimensional case we find that the affinely invariant distance correlation is a function of the canonical correlation coefficients. To illustrate our results, we consider time series of wind vectors at the Stateline wind energy center in Oregon and Washington, and we derive the empirical auto and cross distance correlation functions between wind vectors at distinct meteorological stations.