Consider a random vector $(X',Y)'$, where $X$ is $d$-dimensional and $Y$ is one-dimensional. We assume that $Y$ is subject to random right censoring. The aim of this paper is twofold. First, we propose a new estimator of the joint distribution of $(X',Y)'$. This estimator overcomes the common curse-of-dimensionality problem, by using a new dimension reduction technique. Second, we assume that the relation between $X$ and $Y$ is given by a mean regression single index model, and propose a new estimator of the parameters in this model. The asymptotic properties of all proposed estimators are obtained.

We consider the problem of nonparametric quantile regression for twice censored data. Two new estimates are presented, which are constructed by applying concepts of monotone rearrangements to estimates of the conditional distribution function. The proposed methods avoid the problem of crossing quantile curves. Weak uniform consistency and weak convergence is established for both estimates and their finite sample properties are investigated by means of a simulation study. As a by-product, we obtain a new result regarding the weak convergence of the Beran estimator for right censored data on the maximal possible domain, which is of its own interest.

Fix $n\in\mathbb{N}$. Let $\mathbf{T}_{n}$ be the set of rooted trees $(T,o)$ whose vertices are labeled by elements of $\{1,\dots,n\}$. Let $\nu$ be a strongly connected multi-type Galton–Watson measure. We give necessary and sufficient conditions for the existence of a measure $\mu$ that is reversible for simple random walk on $\mathbf{T}_{n}$ and has the property that given the labels of the root and its neighbors, the descendant subtrees rooted at the neighbors of the root are independent multi-type Galton–Watson trees with conditional offspring distributions that are the same as the conditional offspring distributions of $\nu$ when the types are $\nu$ are ordered pairs of elements of $[n]$. If the types of $\nu$ are given by the labels of vertices, then we give an explicit description of such $\mu$.

This paper introduces the class of volatility modulated Lévy-driven Volterra ($\mathcal{VMLV}$) processes and their important subclass of Lévy semistationary ($\mathcal{LSS}$) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of $\mathcal{VMLV}$ processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.

Stationary ergodic processes with finite alphabets are estimated by finite memory processes from a sample, an $n$-length realization of the process, where the memory depth of the estimator process is also estimated from the sample using penalized maximum likelihood (PML). Under some assumptions on the continuity rate and the assumption of non-nullness, a rate of convergence in $\bar{d}$-distance is obtained, with explicit constants. The result requires an analysis of the divergence of PML Markov order estimators for not necessarily finite memory processes. This divergence problem is investigated in more generality for three information criteria: the Bayesian information criterion with generalized penalty term yielding the PML, and the normalized maximum likelihood and the Krichevsky–Trofimov code lengths. Lower and upper bounds on the estimated order are obtained. The notion of consistent Markov order estimation is generalized for infinite memory processes using the concept of oracle order estimates, and generalized consistency of the PML Markov order estimator is presented.

In this paper, we investigate first the asymptotics of the minima of elliptical triangular arrays. Motivated by the findings of Kabluchko (Extremes 14 (2011) 285–310), we discuss further the asymptotic behaviour of the maxima of elliptical triangular arrays with marginal distribution functions in the Gumbel or Weibull max-domain of attraction. We present an application concerning the asymptotics of the maximum and the minimum of independent spherical processes.

This paper proposes a new estimation technique for fitting parametric Gibbs point process models to a spatial point pattern dataset. The technique is a counterpart, for spatial point processes, of the variational estimators for Markov random fields developed by Almeida and Gidas. The estimator does not require the point process density to be hereditary, so it is applicable to models which do not have a conditional intensity, including models which exhibit geometric regularity or rigidity. The disadvantage is that the intensity parameter cannot be estimated: inference is effectively conditional on the observed number of points. The new procedure is faster and more stable than existing techniques, since it does not require simulation, numerical integration or optimization with respect to the parameters.

The problem of optimal stopping with finite horizon in discrete time is considered in view of maximizing the expected gain. The algorithm proposed in this paper is completely nonparametric in the sense that it uses observed data from the past of the process up to time $-n+1$, $n\in\mathbb{N}$, not relying on any specific model assumption. Kernel regression estimation of conditional expectations and prediction theory of individual sequences are used as tools. It is shown that the algorithm is universally consistent: the achieved expected gain converges to the optimal value for $n\to\infty$ whenever the underlying process is stationary and ergodic. An application to exercising American options is given, and the algorithm is illustrated by simulated data.

Motivated by applications to goodness of fit testing, the empirical likelihood approach is generalized to allow for the number of constraints to grow with the sample size and for the constraints to use estimated criteria functions. The latter is needed to deal with nuisance parameters. The proposed empirical likelihood based goodness of fit tests are asymptotically distribution free. For univariate observations, tests for a specified distribution, for a distribution of parametric form, and for a symmetric distribution are presented. For bivariate observations, tests for independence are developed.

This paper studies various distributional properties of the Rosenblatt distribution. We begin by describing a technique for computing the cumulants. We then study the expansion of the Rosenblatt distribution in terms of shifted chi-squared distributions. We derive the coefficients of this expansion and use these to obtain the Lévy–Khintchine formula and derive asymptotic properties of the Lévy measure. This allows us to compute the cumulants, moments, coefficients in the chi-square expansion and the density and cumulative distribution functions of the Rosenblatt distribution with a high degree of precision. Tables are provided and software written to implement the methods described here is freely available by request from the authors.

Let $X_{1},X_{2},\ldots$ be independent random variables with zero means and finite variances, and let $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V^{2}_{n}=\sum_{i=1}^{n}X^{2}_{i}$. A Cramér type moderate deviation for the maximum of the self-normalized sums $\max_{1\leq k\leq n}S_{k}/V_{n}$ is obtained. In particular, for identically distributed $X_{1},X_{2},\ldots,$ it is proved that $\mathsf{P}(\max_{1\leq k\leq n}S_{k}\geq xV_{n})/(1-\Phi(x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_{1}$.

For a $\beta$-Jacobi ensemble determined by parameters $a_{1}$, $a_{2}$ and $n$, under the restriction that the three parameters go to infinity with $n$ and $a_{1}$ being of small orders of $a_{2}$, we obtain some limit theorems about the eigenvalues. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the central limit theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues.

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$\Delta$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$\Delta$Es and BSDEs are governed by drivers $f^{N}(t,\omega,y,z)$ and $f(t,\omega,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in $z$. For the convergence results it is assumed that the BS$\Delta$Es are based on $d$-dimensional random walks $W^{N}$ approximating the $d$-dimensional Brownian motion $W$ underlying the BSDE and that $f^{N}$ converges to $f$. Conditions are given under which for any bounded terminal condition $\xi$ for the BSDE, there exist bounded terminal conditions $\xi^{N}$ for the sequence of BS$\Delta$Es converging to $\xi$, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when $f^{N}$ and $f$ are convex in $z.$ We show that in this situation, the solutions of the BS$\Delta$Es converge to the solution of the BSDE for every uniformly bounded sequence $\xi^{N}$ converging to $\xi$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^{N},\xi^{N})$ is close to $(W,\xi)$ in distribution, then the solution of the $N$th BS$\Delta$E is close to the solution of the BSDE in distribution too.