Let $F$ be a finite model of cardinality $M$ and denote by $\operatorname{conv}(F)$ its convex hull. The problem of convex aggregation is to construct a procedure having a risk as close as possible to the minimal risk over $\operatorname{conv} (F)$. Consider the bounded regression model with respect to the squared risk denoted by $R(\cdot)$. If ${ \widehat{f}}_{n}^{\mathit{ERM}\mbox{-}C}$ denotes the empirical risk minimization procedure over $\operatorname{conv}(F)$, then we prove that for any $x>0$, with probability greater than $1-4\exp(-x)$,

\[R(\widehat{f}_{n}^{\mathit{ERM}\mbox{-}C})\leq\min_{f\in\operatorname{conv}(F)}R(f)+c_{0}\max\biggl(\psi_{n}^{(C)}(M),\frac{x}{n}\biggr),\]

where $c_{0}>0$ is an absolute constant and $\psi_{n}^{(C)}(M)$ is the optimal rate of convex aggregation defined in (In Computational Learning Theory and Kernel Machines (COLT-2003) (2003) 303–313 Springer) by $\psi _{n}^{(C)}(M)=M/n$ when $M\leq\sqrt{n}$ and $\psi _{n}^{(C)}(M)=\sqrt{\log (\mathrm{e}M/\sqrt{n})/n}$ when $M>\sqrt{n}$.

We consider the recurrence and transience problem for a time-homogeneous Markov chain on the real line with transition kernel $p(x,\mathrm{d}y)=f_{x}(y-x)\,\mathrm{d}y$, where the density functions $f_{x}(y)$, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, where $\alpha(x)\in(0,2)$. In this paper, under a uniformity condition on the density functions $f_{x}(y)$ and an additional mild drift condition, we prove that when $\lim\,\inf_{|x|\longrightarrow\infty}\alpha(x)>1$, the chain is recurrent. Similarly, under the same uniformity condition on the density functions $f_{x}(y)$ and some mild technical conditions, we prove that when $\lim\,\sup_{|x|\longrightarrow\infty}\alpha(x)<1$, the chain is transient. As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric $\alpha$-stable random walk on $\mathbb{R} $ with the index of stability $\alpha\in(0,1)\cup(1,2)$.

This paper studies decision theoretic properties of benchmarked estimators which are of some importance in small area estimation problems. Benchmarking is intended to improve certain aggregate properties (such as study-wide averages) when model based estimates have been applied to individual small areas. We study decision-theoretic properties of such estimators by reducing the problem to one of studying these problems in a related derived problem. For certain such problems, we show that unconstrained solutions in the original (unbenchmarked) problem give unconstrained Bayes and improved estimators which automatically satisfy the benchmark constraint. Also, dominance properties of constrained empirical Bayes estimators are shown in the Fay–Herriot model, a frequently used model in small area estimation.

The approximation of fixed-interval smoothing distributions is a key issue in inference for general state-space hidden Markov models (HMM). This contribution establishes non-asymptotic bounds for the Forward Filtering Backward Smoothing (FFBS) and the Forward Filtering Backward Simulation (FFBSi) estimators of fixed-interval smoothing functionals. We show that the rate of convergence of the $\mathrm{L}_{q}$-mean errors of both methods depends on the number of observations $T$ and the number of particles $N$ only through the ratio $T/N$ for additive functionals. In the case of the FFBS, this improves recent results providing bounds depending on $T/\sqrt{N}$.

We consider a nonparametric regression model $Y=r(X)+\varepsilon$ with a random covariate $X$ that is independent of the error $\varepsilon$. Then the density of the response $Y$ is a convolution of the densities of $\varepsilon$ and $r(X)$. It can therefore be estimated by a convolution of kernel estimators for these two densities, or more generally by a local von Mises statistic. If the regression function has a nowhere vanishing derivative, then the convolution estimator converges at a parametric rate. We show that the convergence holds uniformly, and that the corresponding process obeys a functional central limit theorem in the space $C_{0}(\mathbb{R})$ of continuous functions vanishing at infinity, endowed with the sup-norm. The estimator is not efficient. We construct an additive correction that makes it efficient.

A number of recent work studied the effectiveness of feature selection using Lasso. It is known that under the restricted isometry properties (RIP), Lasso does not generally lead to the exact recovery of the set of nonzero coefficients, due to the looseness of convex relaxation. This paper considers the feature selection property of nonconvex regularization, where the solution is given by a multi-stage convex relaxation scheme. The nonconvex regularizer requires two tuning parameters (compared to one tuning parameter for Lasso). Although the method is more complex than Lasso, we show that under appropriate conditions including the dependence of a tuning parameter on the support set size, the local solution obtained by this procedure recovers the set of nonzero coefficients without suffering from the bias of Lasso relaxation, which complements parameter estimation results of this procedure in (J. Mach. Learn. Res. 11 (2011) 1087–1107).

We consider the Fisher–Snedecor diffusion; that is, the Kolmogorov–Pearson diffusion with the Fisher–Snedecor invariant distribution. In the nonstationary setting, we give explicit quantitative rates for the convergence rate of respective finite-dimensional distributions to that of the stationary Fisher–Snedecor diffusion, and for the $\beta$-mixing coefficient of this diffusion. As an application, we prove the law of large numbers and the central limit theorem for additive functionals of the Fisher–Snedecor diffusion and construct $P$-consistent and asymptotically normal estimators for the parameters of this diffusion given its nonstationary observation.

In this paper, we study classes of graphs with three types of edges that capture the modified independence structure of a directed acyclic graph (DAG) after marginalisation over unobserved variables and conditioning on selection variables using the $m$-separation criterion. These include MC, summary, and ancestral graphs. As a modification of MC graphs, we define the class of ribbonless graphs (RGs) that permits the use of the $m$-separation criterion. RGs contain summary and ancestral graphs as subclasses, and each RG can be generated by a DAG after marginalisation and conditioning. We derive simple algorithms to generate RGs, from given DAGs or RGs, and also to generate summary and ancestral graphs in a simple way by further extension of the RG-generating algorithm. This enables us to develop a parallel theory on these three classes and to study the relationships between them as well as the use of each class.

This paper considers testing a covariance matrix $\Sigma$ in the high dimensional setting where the dimension $p$ can be comparable or much larger than the sample size $n$. The problem of testing the hypothesis $H_{0}:\Sigma=\Sigma_{0}$ for a given covariance matrix $\Sigma_{0}$ is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension $p$ over the sample size $n$ is bounded. A test based on a $U$-statistic is introduced and is shown to be rate optimal over this asymptotic regime. Furthermore, it is shown that the power of this test uniformly dominates that of the corrected likelihood ratio test (CLRT) over the entire asymptotic regime under which the CLRT is applicable. The power of the $U$-statistic based test is also analyzed when $p/n$ is unbounded.

We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions of quadratic forms involving integrals of weighted periodograms. Conditions for asymptotic normality of these weighted sums are simple, minimal, and resemble Lindeberg–Feller condition for weighted sums of independent and identically distributed random variables. Our results are applicable to a large class of short, long or negative memory processes. The proof is based on sharp bounds derived for Bartlett type approximation of these sums by the corresponding sums of weighted periodograms of independent and identically distributed random variables.

We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an Itô formula for the process $u(X)$, when $u$ is locally in the domain of $\mathcal{E}$.

We investigate distributions of hyperbolic Bessel processes. We find links between the hyperbolic cosine of hyperbolic Bessel processes and functionals of geometric Brownian motion. We present an explicit formula for the Laplace transform of the hyperbolic cosine of a hyperbolic Bessel process and some other interesting probabilistic representations of this Laplace transform. We express the one-dimensional distribution of a hyperbolic Bessel process in terms of other, known and independent processes. We present some applications including a new proof of Bougerol’s identity and its generalization. We characterize the distribution of the process which is the hyperbolic sine of hyperbolic Bessel process.

We construct diffusions with values in the nonnegative orthant, normal reflection along each of the axes, and two pairs of local drift/variance characteristics assigned according to rank; one of the variances is allowed to vanish, but not both. The construction involves solving a system of coupled Skorokhod reflection equations, then “unfolding” the Skorokhod reflection of a suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. 79 (2009) 534–536). Questions of pathwise uniqueness and strength are also addressed, for systems of stochastic differential equations with reflection that realize these diffusions. When the variance of the laggard is at least as large as that of the leader, it is shown that the corner of the quadrant is never visited.

In this paper, we obtain a Lamperti type representation for real-valued self-similar Markov processes, killed at their hitting time of zero. Namely, we represent real-valued self-similar Markov processes as time changed multiplicative invariant processes. Doing so, we complete Kiu’s work [Stochastic Process. Appl. 10 (1980) 183–191], following some ideas in Chybiryakov [Stochastic Process. Appl. 116 (2006) 857–872] in order to characterize the underlying processes in this representation. We provide some examples where the characteristics of the underlying processes can be computed explicitly.

This paper provides optimal testing procedures for the $m$-sample null hypothesis of Common Principal Components (CPC) under possibly non-Gaussian and heterogeneous elliptical densities. We first establish, under very mild assumptions that do not require finite moments of order four, the local asymptotic normality (LAN) of the model. Based on that result, we show that the pseudo-Gaussian test proposed in Hallin et al. (J. Nonparametr. Stat. 22 (2010) 879–895) is locally and asymptotically optimal under Gaussian densities, and show how to compute its local powers. A numerical evaluation of those powers, however, reveals that, while remaining valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rank-based procedures that remain valid under any possibly heterogeneous $m$-tuple of elliptical densities, irrespective of the existence of any moments. In elliptical families, indeed, principal components naturally can be based on the scatter matrices characterizing the density contours, hence do not require finite variances. Those rank-based tests, as usual, involve score functions, which may or may not be associated with a reference density at which they achieve optimality. A major advantage of our rank tests is that they are not only validity-robust, in the sense of surviving arbitrary elliptical population densities: unlike their pseudo-Gaussian counterparts, they also are efficiency-robust, in the sense that their local powers do not deteriorate away from the reference density at which they are optimal. We show, in particular, that in the homokurtic case, their normal-score version uniformly dominates, in the Pitman sense, the aforementioned pseudo-Gaussian generalization of Flury’s test. Theoretical results are obtained via a nonstandard application of Le Cam’s methodology in the context of curved LAN experiments. The finite-sample properties of the proposed tests are investigated via simulations.

Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory far enough into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Specifically, we elucidate their limit distributions when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their (intermediate or extreme) order with the oscillations of the quantile function and a von-Mises property of the conditional distribution. A simulation experiment and an illustration on real data were presented. The real data are the American electric data where the estimation of conditional extremes is found to be of genuine interest.

This article constructs a class of random probability measures based on exponentially and polynomially tilting operated on the laws of completely random measures. The class is proved to be conjugate in that it covers both prior and posterior random probability measures in the Bayesian sense. Moreover, the class includes some common and widely used random probability measures, the normalized completely random measures (James (Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics (2002) Preprint), Regazzini, Lijoi and Prünster (Ann. Statist. 31 (2003) 560–585), Lijoi, Mena and Prünster (J. Amer. Statist. Assoc. 100 (2005) 1278–1291)) and the Poisson–Dirichlet process (Pitman and Yor (Ann. Probab. 25 (1997) 855–900), Ishwaran and James (J. Amer. Statist. Assoc. 96 (2001) 161–173), Pitman (In Science and Statistics: A Festschrift for Terry Speed (2003) 1–34 IMS)), in a single construction. We describe an augmented version of the Blackwell–MacQueen Pólya urn sampling scheme (Blackwell and MacQueen (Ann. Statist. 1 (1973) 353–355)) that simplifies implementation and provide a simulation study for approximating the probabilities of partition sizes.

We study prediction in the functional linear model with functional outputs, $Y=SX+\varepsilon$, where the covariates $X$ and $Y$ belong to some functional space and $S$ is a linear operator. We provide the asymptotic mean square prediction error for a random input with exact constants for our estimator which is based on the functional PCA of $X$. As a consequence we derive the optimal choice of the dimension $k_{n}$ of the projection space. The rates we obtain are optimal in minimax sense and generalize those found when the output is real. Our main results hold for class of inputs $X(\cdot )$ that may be either very irregular or very smooth. We also prove a central limit theorem for the predictor. We show that, due to the underlying inverse problem, the bare estimate cannot converge in distribution for the norm of the function space.

We observe a $N\times M$ matrix $Y_{ij}=s_{ij}+\xi_{ij}$ with $\xi_{ij}\sim{ \mathcal {N}}(0,1)$ i.i.d. in $i,j$, and $s_{ij}\in\mathbb{R}$. We test the null hypothesis $s_{ij}=0$ for all $i,j$ against the alternative that there exists some submatrix of size $n\times m$ with significant elements in the sense that $s_{ij}\ge a>0$. We propose a test procedure and compute the asymptotical detection boundary $a$ so that the maximal testing risk tends to $0$ as $M\to\infty$, $N\to\infty$, $p=n/N\to0$, $q=m/M\to0$. We prove that this boundary is asymptotically sharp minimax under some additional constraints. Relations with other testing problems are discussed. We propose a testing procedure which adapts to unknown $(n,m)$ within some given set and compute the adaptive sharp rates. The implementation of our test procedure on synthetic data shows excellent behavior for sparse, not necessarily squared matrices. We extend our sharp minimax results in different directions: first, to Gaussian matrices with unknown variance, next, to matrices of random variables having a distribution from an exponential family (non-Gaussian) and, finally, to a two-sided alternative for matrices with Gaussian elements.

Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the effect of allowing the components to grow at different rates, and characterize the link between these marginal growth rates and the multivariate tail probability decay rate. Our approach leads to a whole class of univariate regular variation conditions, in place of the single but multivariate regular variation conditions that underpin the current theories. These conditions are indexed by a homogeneous function and an angular dependence function, which, for asymptotically independent random vectors, mirror the role played by the exponent measure and Pickands’ dependence function in classical multivariate extremes. We additionally offer an inferential approach to joint survivor probability estimation. The key feature of our methodology is that extreme set probabilities can be estimated by extrapolating upon rays emanating from the origin when the margins of the variables are exponential. This offers an appreciable improvement over existing techniques where extrapolation in exponential margins is upon lines parallel to the diagonal.

In this paper we investigate the problem of testing the assumption of stationarity in locally stationary processes. The test is based on an estimate of a Kolmogorov–Smirnov type distance between the true time varying spectral density and its best approximation through a stationary spectral density. Convergence of a time varying empirical spectral process indexed by a class of certain functions is proved, and furthermore the consistency of a bootstrap procedure is shown which is used to approximate the limiting distribution of the test statistic. Compared to other methods proposed in the literature for the problem of testing for stationarity the new approach has at least two advantages: On one hand, the test can detect local alternatives converging to the null hypothesis at any rate $g_{T}\to0$ such that $g_{T}T^{1/2}\to\infty$, where $T$ denotes the sample size. On the other hand, the estimator is based on only one regularization parameter while most alternative procedures require two. Finite sample properties of the method are investigated by means of a simulation study, and a comparison with several other tests is provided which have been proposed in the literature.

We consider a class of probability measures $\mu_{s,r}^{\alpha}$ which have explicit Cauchy–Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify $\mu_{s,2}^{\alpha}$ as a free compound Poisson law with Lévy measure a monotone $\alpha$-stable law. This implies the free infinite divisibility of $\mu_{s,2}^{\alpha}$. Moreover, when symmetric or positive, $\mu_{s,2}^{\alpha}$ has a representation as the free multiplication of a free Poisson law and a monotone $\alpha$-stable law. We also investigate the free infinite divisibility of $\mu_{s,r}^{\alpha}$ for $r\neq2$. Special cases include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are freely infinitely divisible if and only if $1\leq r\leq2$.

We point out a technical fact which is necessary for the proof of Theorem 8 in our original paper.