Erich Haeusler

Large deviations for martingales via Cramér's method

We develop a new approach for proving large deviation results for martingales based on a change of probability measure. It extends to the case of martingales the conjugate distribution technique due to Cramer. To demonstrate our approach, we derive formulae for probabilities of large deviations for martingales with bounded jumps and bounded norming factor. Surprisingly enough, our result shows that the relative error in the normal range is of the same order as in the case of sums of independent random variables. It also allows to extend the range beyond the normal one.

Weighted Approximations to Continuous Time Martingales with Applications

A weighted approximation to a sequence of continuous time martingales by a time transformed Wiener process is established. The basic tool of proof is the Skorohod imbedding for martingale difference sequences. As an application of the main result a useful weighted approximation to the randomly weighted uniform empirical process is derived. A number of other applications are also discussed.

Assessing confidence intervals for the tail index by Edgeworth expansions for the Hill estimator

We establish Edgeworth expansions for the distribution function of the centered and normalized Hill estimator for the reciprocal of the index of regular variation of the tail of a distribution function.The expansions are used to derive expansions for coverage probabilities of confidence intervals for the tail index based on the Hill estimator.

Asymptotic Distributions of Trimmed Wasserstein Distances Between the True and the Empirical Distribution Function

If the distribution function F has a finite mean, then the Wasserstein distance \(d({{F}_{n}},F) = \smallint _{{ - \infty }}^{\infty }|{{F}_{n}}(x) - F(x)|dx\) between F and the corresponding empirical distribution function F n , based on a sample of size n converges almost surely to zero as n →∞. In [6] del Barrio, Gine and Matran have provided an exhaustive study of the distributional limit theorems associated with this law of large numbers. Nothing can be said about d(F n , F) = ∞ almost surely for all n ≥ 1 if F has no finite mean. In the present paper we modify d(F n , F) into a finite quantity for all F by an adaptation of the notion of trimming from statistics, and study the asymptotic distributions of these trimmed Wasserstein distances for appropriate classes of distribution functions F via weighted approximation results for uniform empirical processes.

Bootstrapping Empirical Distributions under Auxiliary Information

Being the nonparametric maximum likelihood estimator, the classical empirical distribution function is the estimator of choice for a completely unknown distribution function. As shown by Qin and Lawless (1994), in the presence of some auxiliary information the nonparametric maximum likelihood estimator is a modified empirical distribution function. It puts random masses on the observations in order to take the available information into account Zhang (1997) has proved a functional central limit theorem for this modified empirical distribution function. The centered Gaussian limit process in this fclt has a complicated covariance structure so that the result is not directly applicable in statistical problems, e.g. for the construction of confidence bands. It is shown here that the bootstrap is one possible remedy.

On Empirical Distribution Functions Under Auxiliary Information

Being the nonparametric maximum likelihood estimator, the classical empirical distribution function is the estimator of choice for a completely unknown distribution function. As shown by Qin and Lawless in (Ann Statist 22:300–325, 1994), in the presence of some nonparametric auxiliary information about the underlying distribution function the nonparametric maximum likelihood estimator is a modified empirical distribution function. It puts random masses on the observations in order to take the auxiliary information into account. Under a second moment condition Zhang in (Metrika 46:221–244, 1997) has proved a functional central limit theorem for this modified empirical distribution function. The covariance function of the centered Gaussian limit process in his result is smaller than the covariance function of the Browinan brigde limit process in Donsker’s functional central limit theorem for the classical empirical distribution function. If the auxiliary information about the underlying distribution function is knowledge of the mean, then the second moment condition in Zhang’s result requires square integrable random variables. In this note we will study integrable random variables with known mean which are not square integrable and will show that Zhang’s result is no longer true.

On Sequential Empirical Distribution Functions for Random Variables with Mean Zero

The classical sequential empirical distribution function incorporates all subsamples of a sample of independent and identically distributed random variables and is therefore well suited to construct tests for detecting a distributional change occurring somewhere in the sample. If the independent and identically distributed variables are replaced by the residuals of appropriate time series models tests for a distributional change in the unobservable errors (or innovations) of these models are obtained; see Bai (Annals of Statistics, 22:2051–2061, 1994) for the discussion of ARMA models. These errors are often assumed to have mean zero, an information which is not taken into account by the classical sequential empirical distribution function. Based upon ideas from empirical likelihood, see Owen (London/Boca Raton: Chapman & Hall/CRC, 2001), we consider a modified sequential empirical distribution function for random variables with mean zero which does exploit this information.