In this paper, we study the complete graph $K_{n}$ with $n$ vertices, where we attach an independent and identically distributed (i.i.d.) weight to each of the $n(n-1)/2$ edges. We focus on the weight $W_{n}$ and the number of edges $H_{n}$ of the minimal weight path between vertex $1$ and vertex $n$.

It is shown in (Ann. Appl. Probab. 22 (2012) 29–69) that when the weights on the edges are i.i.d. with distribution equal to that of $E^{s}$, where $s>0$ is some parameter, and $E$ has an exponential distribution with mean $1$, then $H_{n}$ is asymptotically normal with asymptotic mean $s\log n$ and asymptotic variance $s^{2}\log n$. In this paper, we analyze the situation when the weights have distribution $E^{-s}$, $s>0$, in which case the behavior of $H_{n}$ is markedly different as $H_{n}$ is a tight sequence of random variables. More precisely, we use the method of Stein–Chen for Poisson approximations to show that, for almost all $s>0$, the hopcount $H_{n}$ converges in probability to the nearest integer of $s+1$ greater than or equal to $2$, and identify the limiting distribution of the recentered and rescaled minimal weight. For a countable set of special $s$ values denoted by $\mathcal{S}=\{s_{j}\}_{j\geq2}$, the hopcount $H_{n}$ takes on the values $j$ and $j+1$ each with positive probability.

Power law generalized covariance functions provide a simple model for describing the local behavior of an isotropic random field. This work seeks to extend this class of covariance functions to spatial-temporal processes for which the degree of smoothness in space and in time may differ while maintaining other desirable properties for the covariance functions, including the availability of explicit convergent and asymptotic series expansions.

Statistical inference for discrete time observations of an affine stochastic delay differential equation is considered. The main focus is on maximum pseudo-likelihood estimators, which are easy to calculate in practice. A more general class of prediction-based estimating functions is investigated as well. In particular, the optimal prediction-based estimating function and the asymptotic properties of the estimators are derived. The maximum pseudo-likelihood estimator is a particular case, and an expression is found for the efficiency loss when using the maximum pseudo-likelihood estimator, rather than the computationally more involved optimal prediction-based estimator. The distribution of the pseudo-likelihood estimator is investigated in a simulation study. Two examples of affine stochastic delay equation are considered in detail.

We propose a simple continuous time model for modeling the lead-lag effect between two financial assets. A two-dimensional process $(X_{t},Y_{t})$ reproduces a lead-lag effect if, for some time shift $\vartheta\in\mathbb{R} $, the process $(X_{t},Y_{t+\vartheta})$ is a semi-martingale with respect to a certain filtration. The value of the time shift $\vartheta$ is the lead-lag parameter. Depending on the underlying filtration, the standard no-arbitrage case is obtained for $\vartheta=0$. We study the problem of estimating the unknown parameter $\vartheta\in\mathbb{R}$, given randomly sampled non-synchronous data from $(X_{t})$ and $(Y_{t})$. By applying a certain contrast optimization based on a modified version of the Hayashi–Yoshida covariation estimator, we obtain a consistent estimator of the lead-lag parameter, together with an explicit rate of convergence governed by the sparsity of the sampling design.

We explore various estimators for the parameters of a pair-copula construction (PCC), among those the stepwise semiparametric (SSP) estimator, designed for this dependence structure. We present its asymptotic properties, as well as the estimation algorithm for the two most common types of PCCs. Compared to the considered alternatives, that is, maximum likelihood, inference functions for margins and semiparametric estimation, SSP is in general asymptotically less efficient. As we show in a few examples, this loss of efficiency may however be rather low. Furthermore, SSP is semiparametrically efficient for the Gaussian copula. More importantly, it is computationally tractable even in high dimensions, as opposed to its competitors. In any case, SSP may provide start values, required by the other estimators. It is also well suited for selecting the pair-copulae of a PCC for a given data set.

Bivariate extreme-value distributions have been used in modeling extremes in environmental sciences and risk management. An important issue is estimating the dependence function, such as the Pickands dependence function. Some estimators for the Pickands dependence function have been studied by assuming that the marginals are known. Recently, Genest and Segers [Ann. Statist. 37 (2009) 2990–3022] derived the asymptotic distributions of those proposed estimators with marginal distributions replaced by the empirical distributions. In this article, we propose a class of weighted estimators including those of Genest and Segers (2009) as special cases. We propose a jackknife empirical likelihood method for constructing confidence intervals for the Pickands dependence function, which avoids estimating the complicated asymptotic variance. A simulation study demonstrates the effectiveness of our proposed jackknife empirical likelihood method.

In this article we study post-model selection estimators that apply ordinary least squares (OLS) to the model selected by first-step penalized estimators, typically Lasso. It is well known that Lasso can estimate the nonparametric regression function at nearly the oracle rate, and is thus hard to improve upon. We show that the OLS post-Lasso estimator performs at least as well as Lasso in terms of the rate of convergence, and has the advantage of a smaller bias. Remarkably, this performance occurs even if the Lasso-based model selection “fails” in the sense of missing some components of the “true” regression model. By the “true” model, we mean the best $s$-dimensional approximation to the nonparametric regression function chosen by the oracle. Furthermore, OLS post-Lasso estimator can perform strictly better than Lasso, in the sense of a strictly faster rate of convergence, if the Lasso-based model selection correctly includes all components of the “true” model as a subset and also achieves sufficient sparsity. In the extreme case, when Lasso perfectly selects the “true” model, the OLS post-Lasso estimator becomes the oracle estimator. An important ingredient in our analysis is a new sparsity bound on the dimension of the model selected by Lasso, which guarantees that this dimension is at most of the same order as the dimension of the “true” model. Our rate results are nonasymptotic and hold in both parametric and nonparametric models. Moreover, our analysis is not limited to the Lasso estimator acting as a selector in the first step, but also applies to any other estimator, for example, various forms of thresholded Lasso, with good rates and good sparsity properties. Our analysis covers both traditional thresholding and a new practical, data-driven thresholding scheme that induces additional sparsity subject to maintaining a certain goodness of fit. The latter scheme has theoretical guarantees similar to those of Lasso or OLS post-Lasso, but it dominates those procedures as well as traditional thresholding in a wide variety of experiments.

We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coefficients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet multinomial distribution, respectively, on the continuous and the $N$-discrete $d$-dimensional simplex. Their infinite-dimensional limit distributions, respectively, the Poisson–Dirichlet distribution and Ewens’s sampling formula, are considered as well. We study, in particular, the possibility of mapping canonical correlations on the $d$-dimensional continuous simplex (i) to canonical correlation sequences on the $d+1$-dimensional simplex and/or (ii) to canonical correlations on the discrete simplex, and vice versa. Driven by this motivation, the first half of the paper is devoted to providing a full characterization and probabilistic interpretation of $n$-orthogonal polynomial kernels (i.e., sums of products of orthogonal polynomials of the same degree $n$) with respect to the mentioned marginal distributions. We establish several identities and some integral representations which are multivariate extensions of important results known for the case $d=2$ since the 1970s. These results, along with a common interpretation of the mentioned kernels in terms of dependent Pólya urns, are shown to be key features leading to several non-trivial solutions to Lancaster’s problem, many of which can be extended naturally to the limit as $d\rightarrow\infty$.

In this note we give, for a spectrally negative Lévy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period $r$. The formula involves only the scale function of the spectrally negative Lévy process and the distribution of the process at time $r$.

We develop a new formulation of Stein’s method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the “discrete equilibrium” distribution from renewal theory. We illustrate the approach in four non-trivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton–Watson process conditioned on non-extinction, the in-degree of a randomly chosen node in the uniform attachment random graph model and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model.

Consider a discrete-time martingale, and let $V^{2}$ be its normalized quadratic variation. As $V^{2}$ approaches $1$, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any $p\geq 1$, (Ann. Probab. 16 (1988) 275–299) gave a bound on the rate of convergence in this central limit theorem that is the sum of two terms, say $A_{p}+B_{p}$, where up to a constant, $A_{p}={\|V^{2}-1\|}_{p}^{p/(2p+1)}$. Here we discuss the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, (Ann. Probab. 10 (1982) 672–688) sketched a strategy to prove optimality for $p=1$. Here we extend this strategy to any $p\geq 1$, thereby justifying the optimality of the term $A_{p}$. As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term $B_{p}$, generalizing another result of (Ann. Probab. 10 (1982) 672–688).

Given a finite class of functions $F$, the problem of aggregation is to construct a procedure with a risk as close as possible to the risk of the best element in the class. A classical procedure (PAC-Bayesian statistical learning theory (2004) Paris 6, Statistical Learning Theory and Stochastic Optimization (2001) Springer, Ann. Statist. 28 (2000) 75–87) is the aggregate with exponential weights (AEW), defined by

\[\tilde{f}^{\mathrm{AEW}}=\sum_{f\in F}\widehat{\theta}(f)f,\qquad\mbox{where }\widehat{\theta}(f)=\frac{\exp(-({n}/{T})R_{n}(f))}{\sum_{g\in F}\exp(-({n}/{T})R_{n}(g))},\]

where $T>0$ is called the temperature parameter and $R_{n}(\cdot)$ is an empirical risk.

In this article, we study the optimality of the AEW in the regression model with random design and in the low-temperature regime. We prove three properties of AEW. First, we show that AEW is a suboptimal aggregation procedure in expectation with respect to the quadratic risk when $T\leq c_{1}$, where $c_{1}$ is an absolute positive constant (the low-temperature regime), and that it is suboptimal in probability even for high temperatures. Second, we show that as the cardinality of the dictionary grows, the behavior of AEW might deteriorate, namely, that in the low-temperature regime it might concentrate with high probability around elements in the dictionary with risk greater than the risk of the best function in the dictionary by at least an order of $1/\sqrt{n}$. Third, we prove that if a geometric condition on the dictionary (the so-called “Bernstein condition”) is assumed, then AEW is indeed optimal both in high probability and in expectation in the low-temperature regime. Moreover, under that assumption, the complexity term is essentially the logarithm of the cardinality of the set of “almost minimizers” rather than the logarithm of the cardinality of the entire dictionary. This result holds for small values of the temperature parameter, thus complementing an analogous result for high temperatures.

We consider the task of detecting a salient cluster in a sensor network, that is, an undirected graph with a random variable attached to each node. Motivated by recent research in environmental statistics and the drive to compete with the reigning scan statistic, we explore alternatives based on the percolative properties of the network. The first method is based on the size of the largest connected component after removing the nodes in the network with a value below a given threshold. The second method is the upper level set scan test introduced by Patil and Taillie [Statist. Sci. 18 (2003) 457–465]. We establish the performance of these methods in an asymptotic decision- theoretic framework in which the network size increases. These tests have two advantages over the more conventional scan statistic: they do not require previous information about cluster shape, and they are computationally more feasible. We make abundant use of percolation theory to derive our theoretical results, and complement our theory with some numerical experiments.