Research Group "Stochastic Algorithms and Nonparametric Statistics"
Research Seminar "Mathematical Statistics" Summer Semester 2013
last reviewed:April 25, 2013, Christine Schneider
Andrija Mihoci (C.A.S.E., HU Berlin)
Local adaptive multiplicative error models for high-frequency forecasts
Abstract: We propose a local adaptive multiplicative error model (MEM) accommodating time-varying parameters. MEM parameters are adaptively estimated based on a sequential testing procedure. A data- driven optimal length of local windows is selected, yielding adaptive forecasts at each point in time. Analyzing one-minute cumulative trading volumes of ve large NASDAQ stocks in 2008, we show that local windows of approximately 3 or 4 hours are reasonable to capture parameter variations while balancing modeling bias and estimation (in)eciency. In forecasting, the proposed adaptive approach signicantly outperforms a MEM where local estimation windows are xed on an ad hoc basis.
Dr. Micha Pesta(Charles University, Prag/Czech Republic)
Asymptotic consistency and inconsistency of the chain ladder
Abstract: The distribution-free chain ladder reserving method belongs to the most frequently used approaches in the general insurance. It is well known, see [1], that the estimators $widehatf_j$ of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In [2], we deal with some asymptotic properties of $widehatf_j$. Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors $f_j$ are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well. References: [1] Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. em ASTIN Bulletin, bf 23, 2, 213-225. [2] Pešta, M. and Hudecová, Š. (2012). Asymptotic consistency and inconsistency of the chain ladder. em Insurance: Mathematics and Economics, bf 51, 2, 472-479.
Alexey Kulik (Taras Shevchenko University, Kiev)
Limit theorems and statistical inference in Markov models
Abstract:The ``martingale problem approach'' for proving averaging principle and diffusion approximation type theorems for functionals of an ergodic Markov process will be discussed. Applications to statistical inference will be illustrated by two examples concerning asymptotic properties of MLE studied (a) for discretely observed solutions to SDE's with jumps; (b) for autoregressive models with hidden Markov input.
Matthias Scherer (TU München)
On the construction and use of factor copula models
Abstract: Modeling the dependence structure of high-dimensional random vectors is not an easy task. Nevertheless, it is required in many applications in the financial industry. Often, one faces a tradeoff between models that are rather simple but computationally efficient on the one hand, and very flexible dependence structures that become unhandy as the dimension of the problem increases on the other hand. Several popular families of copulas, especially when based on a factor-model construction, are extendible. Even though such structures are very convenient in large dimensions (due to the factor model / conditional i.i.d. structure), the assumption of conditional i.i.d. may be over-simplistic for real situations. One possibility to overcome extendibility without giving up the general structure is to consider hierarchical (or nested) extensions of the dependence structure in concern. Heuristically speaking, the dependence structure of hierarchical copulas is induced by some global stochastic factor affecting i.i.d. components and by additional group-specific factors that only affect certain sub-vectors. We present a survey of recent developments on hierarchical models, such as hierarchical Archimedean and Marshall-Olkin type dependence structures, and unify the literature by introducing the notion of h-extendibility. This definition generalizes extendible models in a natural way to hierarchical structures. Finally, we sketch applications to credit risk and insurance portfolios.
Ismael Castillo (Université Pierre et Marie Curie, Paris)
Some results on frequentist analysis of Bayesian posterior distributions
Abstract: In this talk I will discuss recent work on Bayesian analysis of procedures in non- and semi-parametric
settings. First, I will talk about conditions guaranteeing the asymptotic normality of the marginal posterior distribution - the so-called Bernstein-von Mises theorem - in semiparametric settings and give some examples. Second, a notion of nonparametric Bernstein-von Mises theorem will be introduced and some applications discussed (this part is joint work with Richard Nickl).
