Let X₁, X₂, …, X_n be a sequence of independent or locally dependent random variables taking values in ℤ₊. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum ∑_i=1ⁿX_i and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This “smoothness factor” is of order O(σ⁻²), according to a heuristic argument, where σ² denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.

Copulas have been popular to model dependence for multivariate distributions, but have not been used much in modelling temporal dependence of univariate time series. This paper demonstrates some difficulties with using copulas even for Markov processes: some tractable copulas such as mixtures between copulas of complete co- and countermonotonicity and independence (Fréchet copulas) are shown to imply quite a restricted type of Markov process and Archimedean copulas are shown to be incompatible with Markov chains. We also investigate Markov chains that are spreadable or, equivalently, conditionally i.i.d.

We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Hölder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some Lévy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin differentiable.

An interesting line of research is the investigation of the laws of random variables known as Dirichlet means. However, there is not much information on interrelationships between different Dirichlet means. Here, we introduce two distributional operations, one of which consists of multiplying a mean functional by an independent beta random variable, the other being an operation involving an exponential change of measure. These operations identify relationships between different means and their densities. This allows one to use the often considerable analytic work on obtaining results for one Dirichlet mean to obtain results for an entire family of otherwise seemingly unrelated Dirichlet means. Additionally, it allows one to obtain explicit densities for the related class of random variables that have generalized gamma convolution distributions and the finite-dimensional distribution of their associated Lévy processes. The importance of this latter statement is that Lévy processes now commonly appear in a variety of applications in probability and statistics, but there are relatively few cases where the relevant densities have been described explicitly. We demonstrate how the technique allows one to obtain the finite-dimensional distribution of several interesting subordinators which have recently appeared in the literature.

We approximate the solution of some linear systems of SDEs driven by a fractional Brownian motion B^H with Hurst parameter H∈(½, 1) in the Wick–Itô sense, including a geometric fractional Brownian motion. To this end, we apply a Donsker-type approximation of the fractional Brownian motion by disturbed binary random walks due to Sottinen. Moreover, we replace the rather complicated Wick products by their discrete counterpart, acting on the binary variables, in the corresponding systems of Wick difference equations. As the solutions of the SDEs admit series representations in terms of Wick powers, a key to the proof of our Euler scheme is an approximation of the Hermite recursion formula for the Wick powers of B^H.

Particularly in genomics, but also in other fields, it has become commonplace to undertake highly multiple Student’s t-tests based on relatively small sample sizes. The literature on this topic is continually expanding, but the main approaches used to control the family-wise error rate and false discovery rate are still based on the assumption that the tests are independent. The independence condition is known to be false at the level of the joint distributions of the test statistics, but that does not necessarily mean, for the small significance levels involved in highly multiple hypothesis testing, that the assumption leads to major errors. In this paper, we give conditions under which the assumption of independence is valid. Specifically, we derive a strong approximation that closely links the level exceedences of a dependent “studentized process” to those of a process of independent random variables. Via this connection, it can be seen that in high-dimensional, low sample-size cases, provided the sample size diverges faster than the logarithm of the number of tests, the assumption of independent t-tests is often justified.

We consider Bayesian hierarchical models for survival analysis, where the survival times are modeled through an underlying diffusion process which determines the hazard rate. We show how these models can be efficiently treated by means of Markov chain Monte Carlo techniques.

The relative log-concavity ordering ≤_lc between probability mass functions (pmf’s) on non-negative integers is studied. Given three pmf’s f, g, h that satisfy f ≤_lc g ≤_lc h, we present a pair of (reverse) triangle inequalities: if ∑_iif_i=∑_iig_i<∞, then

D(f|h)≥D(f|g)+D(g|h)

and if ∑_iig_i=∑_iih_i<∞, then

D(h|f)≥D(h|g)+D(g|f),

where D(⋅|⋅) denotes the Kullback–Leibler divergence. These inequalities, interesting in themselves, are also applied to several problems, including maximum entropy characterizations of Poisson and binomial distributions and the best binomial approximation in relative entropy. We also present parallel results for continuous distributions and discuss the behavior of ≤_lc under convolution.

Given a random word of size n whose letters are drawn independently from an ordered alphabet of size m, the fluctuations of the shape of the random RSK Young tableaux are investigated, when n and m converge together to infinity. If m does not grow too fast and if the draws are uniform, then the limiting shape is the same as the limiting spectrum of the GUE. In the non-uniform case, a control of both highest probabilities will ensure the convergence of the first row of the tableau toward the Tracy–Widom distribution.

Sequential estimation of a probability p by means of inverse binomial sampling is considered. For μ₁, μ₂>1 given, the accuracy of an estimator ̂p is measured by the confidence level P[p/μ₂≤̂p≤pμ₁]. The confidence levels c₀ that can be guaranteed for p unknown, that is, such that P[p/μ₂≤̂p≤pμ₁]≥c₀ for all p∈(0, 1), are investigated. It is shown that within the general class of randomized or non-randomized estimators based on inverse binomial sampling, there is a maximum c₀ that can be guaranteed for arbitrary p. A non-randomized estimator is given that achieves this maximum guaranteed confidence under mild conditions on μ₁, μ₂.

Under a single-index regression assumption, we introduce a new semiparametric procedure to estimate a conditional density of a censored response. The regression model can be seen as a generalization of the Cox regression model and also as a profitable tool for performing dimension reduction under censoring. This technique extends the results of Delecroix et al. [J. Multivariate Anal. 86 (2003) 213–226]. We derive consistency and asymptotic normality of our estimator of the index parameter by proving its asymptotic equivalence with the (uncomputable) maximum likelihood estimator, using martingales results for counting processes and arguments from empirical processes theory. Furthermore, we provide a new adaptive procedure which allows us both to choose the smoothing parameter involved in our approach and to circumvent the weak performances of the Kaplan–Meier estimator [Amer. Statist. Assoc. 53 (1958) 457–481] in the right-tail of the distribution. By means of a simulation study, we study the behavior of our estimator for small samples.

We consider the problem of estimating the predictive density of future observations from a non-parametric regression model. The density estimators are evaluated under Kullback–Leibler divergence and our focus is on establishing the exact asymptotics of minimax risk in the case of Gaussian errors. We derive the convergence rate and constant for minimax risk among Bayesian predictive densities under Gaussian priors and we show that this minimax risk is asymptotically equivalent to that among all density estimators.

In this work, we deal with approximations for distribution functions of non-negative random variables. More specifically, we construct continuous approximants using an acceleration technique over a well-know inversion formula for Laplace transforms. We give uniform error bounds using a representation of these approximations in terms of gamma-type operators. We apply our results to certain mixtures of Erlang distributions which contain the class of continuous phase-type distributions.

We construct a semiparametric estimator in case-control studies where the gene and the environment are assumed to be independent. A discrete or continuous parametric distribution of the genes is assumed in the model. A discrete distribution of the genes can be used to model the mutation or presence of certain group of genes. A continuous distribution allows the distribution of the gene effects to be in a finite-dimensional parametric family and can hence be used to model the gene expression levels. We leave the distribution of the environment totally unspecified. The estimator is derived through calculating the efficiency score function in a hypothetical setting where a close approximation to the samples is random. The resulting estimator is proved to be efficient in the hypothetical situation. The efficiency of the estimator is further demonstrated to hold in the case-control setting as well.