Table of contents, Bernoulli Journal, vol. 11, no. 1 (2005)

On data depth and distribution-free discriminant analysis using separating surfaces; Authors: Anil K. Ghosh, Probal Chaudhuri

On the quantiles of Brownian motion and their hitting times; Authors: Angelos Dassios

On L²-convergence of controlled branching processes with random control function; Authors: Miguel González, Manuel Molina, Inés Del Puerto

Sharp estimates in signed Poisson approximation of Poisson mixtures; Authors: José Antonio Adell, Alberto Lekuona

Adaptive estimation for affine stochastic delay differential equations; Authors: Markus Reiss

Consistent and asymptotically normal parameter estimates for hidden Markov mixtures of Markov models; Authors: Pierre Vandekerkhove

Asymptotics for L₂ functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances; Authors: Eustasio Del Barrio, Evarist Giné, Frederic Utzet

A very well-known traditional approach in discriminant analysis is to use some linear (or nonlinear) combination of measurement variables which can enhance class separability. For instance, a linear (or a quadratic) classifier finds the linear projection (or the quadratic function) of the measurement variables that will maximize the separation between the classes. These techniques are very useful in obtaining good lower-dimensional views of class separability. Fisher's discriminant analysis, which is primarily motivated by the multivariate normal distribution, uses the first- and second-order moments of the training sample to build such classifiers. These estimates, however, are highly sensitive to outliers, and they are not reliable for heavy-tailed distributions. This paper investigates two distribution-free methods for linear classification, which are based on the notions of statistical depth functions. One of these classifiers is closely related to Tukey's half-space depth, while the other is based on the concept of regression depth. Both these methods can be generalized for constructing nonlinear surfaces to discriminate among competing classes. These depth-based methods assume some finite-dimensional parametric form of the discriminating surface and use the distributional geometry of the data cloud to build the classifier. We use a few simulated and real data sets to examine the performance of these discriminant analysis tools and study their asymptotic properties under appropriate regularity conditions.

The distribution of the α-quantile of a Brownian motion on an interval [0,t] has been obtained motivated by a problem in financial mathematics. In this paper we generalize these results by calculating an explicit expression for the joint density of the α-quantile of a standard Brownian motion, its first and last hitting times and the value of the process at time t. Our results can easily be generalized to a Brownian motion with drift. It is shown that the first and last hitting times follow a transformed arcsine law.

The topic of this paper is Yanev's class of controlled branching processes with a random control function. We investigate the L²-convergence of the suitably normed process to a non-degenerate limit. Necessary and sufficient conditions for this convergence are established under certain constraints on the means and variances of the control variables.

We give sharp estimates in total variation and certain kinds of stop-loss metrics in signed Poisson approximation of Poisson mixtures. We provide closed-form solutions to the problem of best choice of the Poisson parameter in simple Poisson approximation with respect to the total variation distance. The important special case of the negative binomial distribution is also discussed. To obtain our results, we apply a differential calculus based on different Taylor formulae for the Poisson process which allows us to give simple unified proofs.

For stationary solutions of the affine stochastic delay differential equation $$dX\left(t\right)=\left({\gamma }_{0}X\right(t)+{\gamma }_{\mathrm{rX}}(t-r)+{\int }_{-r}^{0}X(t+u\left)g\right(u)du)dt+\sigma dW\left(t\right),$$ we consider the problem of nonparametric inference for the weight function g and for γ₀,γ_r from the continuous observation of one trajectory up to time T>0. For weight functions in the scale of Besov spaces B^s_p,1 and L^ρ-type loss functions, convergence rates are established for long-time asymptotics. The estimation problem is equivalent to an ill-posed inverse problem with error in the data and unknown operator. We propose a wavelet thresholding estimator that achieves the rate (T/logT)^-s/(2s+3) under certain restrictions on p and ρ. This rate is shown to be optimal in a minimax sense.

We introduce a new missing-data model, based on a mixture of K Markov processes, and consider the general problem of identifying its parameters. We point out in detail the main difficulties of statistical inference for such models: complete likelihood calculation, parametrization of the stationary distribution and identifiability. We propose a general tractable approach for estimating these models (admitting parametrization of the stationary distribution and identifiability) and check in detail that our assumptions are fully satisfied for a Markov mixture of two linear AR(1) models with Gaussian noise. Finally, a Monte Carlo method is proposed to calculate the split data likelihood of this model when no analytic expression for the invariant probability densities of the Markov processes is known.

Weighted L₂ functionals of the empirical quantile process appear as a component of many test statistics, in particular in tests of fit to location-scale families of distributions based on weighted Wasserstein distances. An essentially complete set of distributional limit theorems for the squared empirical quantile process integrated with respect to general weights is presented. The results rely on limit theorems for quadratic forms in exponential random variables, and the proofs use only simple asymptotic theory for probability distributions in Rⁿ. The limit theorems are then applied to determine the asymptotic distribution of the test statistics on which weighted Wasserstein tests are based. In particular, this paper contains an elementary derivation of the limit distribution of the Shapiro-Wilk test statistic under normality.