We introduce a new approach to shape-preserving estimation of cumulative distribution functions and probability density functions using the wavelet methodology for multivariate dependent data. Our estimators preserve shape constraints such as monotonicity, positivity and integration to one, and allow for low spatial regularity of the underlying functions. We discuss conditional quantile estimation for financial time series data as an application. Our methodology can be implemented with B-splines. We show by means of Monte Carlo simulations that it performs well in finite samples and for a data-driven choice of the resolution level.

In this paper, we give new characterizations of exponential distributions on positive semigroups and we study the corresponding gamma distributions that govern semigroup products of independent and identically distributed exponential variables. We apply these results to the positive integers under multiplication to obtain new interpretations of results on Dirichlet distributions.

Let $X_1,X_2$,… be a sequence of independent and identically distributed random variables. Let $X$ be an independent copy of $X_1$. Define $\mathbb{T}_{n}=\sqrt{n}\bar{X}/S$, where $\bar{X}$ and $S^2$ are the sample mean and the sample variance, respectively. We refer to $\mathbb{T}_{n}$ as the central or non-central (Student’s) $t$-statistic, depending on whether $\mathrm{E}X=0$ or $\mathrm{E}X≠0$, respectively. The non-central $t$-statistic arises naturally in the calculation of powers for $t$-tests. The central $t$-statistic has been well studied, while there is a very limited literature on the non-central $t$-statistic. In this paper, we attempt to narrow this gap by studying the limiting behaviour of the non-central $t$-statistic, which turns out to be quite complicated. For instance, it is well known that, under finite second-moment conditions, the limiting distributions for the central $t$-statistic are normal while those for the non-central $t$-statistic can be non-normal and can critically depend on whether or not $\mathrm{E}X=∞$. As an application, we study the effect of non-normality on the performance of the $t$-test.

We consider a density estimation problem arising in nuclear physics. Gamma photons are impinging on a semiconductor detector, producing pulses of current. The integral of this pulse is equal to the total amount of charge created by the photon in the detector, which is linearly related to the photon energy. Because the inter-arrival times of photons can be shorter than the charge collection time, pulses corresponding to different photons may overlap leading to a phenomenon known as pile-up. The distortions on the photon energy spectrum estimate due to pile-up become worse when the photon rate increases, making pile-up correction techniques a must for high counting rate experiments. In this paper, we present a novel technique to correct pile-up, which extends a method introduced by Hall and Park for the estimation of the service time from the busy period in $M/G/∞$ models. It is based on a novel formula linking the joint distribution of the energy and duration of the cluster of pulses and the distribution of the energy of the photons. We then assess the performance of this estimator by providing an expression for its integrated square error. A Monte Carlo experiment is presented to illustrate, with practical examples, the benefits of the pile-up correction.

In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point $t−1$ with observations about the time point $t$ to yield an estimator of the parameter at time point $t$. The sampling properties of this estimator are studied in a non-stationary context – in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from Hölder classes of order between 1 and 2.

In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.

We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of catching nonlinear features in spatial data and exploring local dependence structures. Our approach is different from both Markov field methods and disjunctive kriging. The asymptotic properties of the additive estimators have been established for $α$-mixing spatial processes by extending the theory of the backfitting procedure to the spatial case. This facilitates the confidence intervals for the component functions, although the asymptotic biases have to be estimated via (wild) bootstrap. Simulation results are reported. Applications to real data illustrate that the improvement in describing the data over the auto-normal scheme is significant when nonlinearity or non-Gaussianity is pronounced.

Long-range dependence induced by heavy tails is a widely reported feature of internet traffic. Long-range dependence can be defined as the regular variation of the variance of the integrated process, and half the index of regular variation is then referred to as the Hurst index. The infinite-source Poisson process (a particular case of which is the $M/G/∞$ queue) is a simple and popular model with this property, when the tail of the service time distribution is regularly varying. The Hurst index of the infinite-source Poisson process is then related to the index of regular variation of the service times. In this paper, we present a wavelet-based estimator of the Hurst index of this process, when it is observed either continuously or discretely over an increasing time interval. Our estimator is shown to be consistent and robust to some form of non-stationarity. Its rate of convergence is investigated.

Given a sequence of observations from a discrete-time, finite-state hidden Markov model, we would like to estimate the sampling distribution of a statistic. The bootstrap method is employed to approximate the confidence regions of a multi-dimensional parameter. We propose an importance sampling formula for efficient simulation in this context. Our approach consists of constructing a locally asymptotically normal (LAN) family of probability distributions around the default resampling rule and then minimizing the asymptotic variance within the LAN family. The solution of this minimization problem characterizes the asymptotically optimal resampling scheme, which is given by a tilting formula. The implementation of the tilting formula is facilitated by solving a Poisson equation. A few numerical examples are given to demonstrate the efficiency of the proposed importance sampling scheme.

We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $Δ$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms.

In this paper we characterize all the $r$-parameter families of count distributions (satisfying mild conditions) that are closed under addition and under binomial subsampling. Surprisingly, few families satisfy both properties and the resulting models consist of the $r$th-order univariate Hermite distributions. Among these, we find the Poisson $(r=1)$ and the ordinary Hermite distributions $(r=2)$.

In this paper, we study complex Wishart processes or the so-called Laguerre processes $(X_t)_{t≥0}$. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman–Watson law as well as the law of $T_0:=\inf {t, \det (X_t)=0}$ when the size of the matrix is $2$.

Let $T$ be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry–Esseen type bounds for $T$ are obtained. The bounds are the best possible for many known statistics. Applications to U-statistics, multisample U-statistics, L-statistics, random sums and functions of nonlinear statistics are discussed.