Boris Buchmann

DECOMPOUNDING POISSON RANDOM SUMS: RECURSIVELY TRUNCATED ESTIMATES IN THE DISCRETE CASE

Given a sample from a discrete compound Poisson distribution, we consider variants of plug-in and likelihood estimators for the corresponding base distribution. These proceed recursively with an intermediate truncation step. We discuss the asymptotic behaviour of the estimators and give some numerical examples. Both procedures compare favourably with the straightforward and the naively projected plug-in estimator that we introduced in Buchmann and Grübel (2003,The Annals of Statistics,31, 1054–1074).

Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence

This paper considers the effect of least squares procedures for nearly unstable linear time series with strongly dependent innovations. Under a general framework and appropriate scaling, it is shown that ordinary least squares procedures converge to functionals of fractional Ornstein-Uhlenbeck processes. We use fractional integrated noise as an example to illustrate the important ideas. In this case, the functionals bear only formal analogy to those in the classical framework with uncorrelated innovations, with Wiener processes being replaced by fractional Brownian motions. It is also shown that limit theorems for the functionals involve nonstandard scaling and nonstandard limiting distributions. Results of this paper shed light on the asymptotic behavior of nearly unstable long-memory processes.

Limit experiments of GARCH

GARCH is one of the most prominent nonlinear time series models, both widely applied and thoroughly studied. Recently, it has been shown that the COGARCH model, which has been introduced a few years ago by Kluppelberg, Lindner and Maller, and Nelsons diffusion limit are the only functional continuous-time limits of GARCH in distribution. In contrast to Nelsons diffusion limit, COGARCH reproduces most of the stylized facts of financial time series. Since it has been proved, that Nelsons diffusion is not asymptotically equivalent to GARCH in deficiency, we investigate in the present paper the relation between GARCH and COGARCH in Le Cams framework of statistical equivalence. We show that GARCH converges generically to COGARCH, even in deficiency, provided that the volatility processes are observed. Hence, from a theoretical point of view, COGARCH can indeed be considered as a continuous-time equivalent to GARCH. Otherwise, when the observations are incomplete, GARCH still has a limiting experiment which we call MCOGARCH, and which is not equivalent, but nevertheless quite similar to COGARCH. In the COGARCH model, the jump times can be more random, as for the MCOGARCH, a fact practitioners may see as an advantage of COGARCH.

Maxima of stochastic processes driven by fractional Brownian motion

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Eulers Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Bermans condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

Integrated functionals of normal and fractional processes

t (u) = R tu 0 f(Ns) ds, t > 0, u 2 [0,1], where N = (Nt)t∈R is a normal process and f is a measurable real-valued function satisfying Ef(N0) 2 3/4, respectively, whereas our result covers H = 3/4.

Weighted empirical processes in the nonparametric inference for Lévy processes

Given observations of a Lévy process, we provide nonparametric estimators of its Lévy tail and study the asymptotic properties of the corresponding weighted empirical processes. Within a special class of weight functions, we give necessary and sufficient conditions that ensure strong consistency and asymptotic normality of the weighted empirical processes, provided that complete information on the jumps is available. To cope with infinite activity processes, we depart from this assumption and analyze the weighted empirical processes of a sampling scheme where small jumps are neglected. We establish a bootstrap principle and provide a simulation study for some prominent Lévy processes.

A Continuous Time Approximation of an Evolutionary Stock Market Model

We derive a continuous time approximation of the evolutionary market selection model of Blume and Easley (1992). Conditions on the payoff structure of the assets are identified that guarantee convergence. We show that the continuous time approximation equals the solution of an integral equation in a random environment. For constant asset returns, the integral equation reduces to an autonomous ordinary differential equation. We analyze its long-run asymptotic behavior using techniques related to Lyapunov functions, and compare our results to the benchmark of profit-maximizing investors.

Processes of r t h largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an ℝ∞

Decompounding: An estimation problem for Poisson random sums

\mathbb {R}^{\infty }

Fractional integral equations and state space transforms

-valued stochastic process in r which we denote as M(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M(r) and M(r) itself, after norming and centering. In continuous time, an analogous process Y(r) based on a two-dimensional Poisson process on ℝ+×ℝ