## Fachbereich Mathematik

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- Mixture Models (2)
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- AR-ARCH (1)
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Knowledge about the distribution of a statistical estimator is important for various purposes like, for example, the construction of confidence intervals for model parameters or the determiation of critical values of tests. A widely used method to estimate this distribution is the so-called bootstrap which is based on an imitation of the probabilistic structure of the data generating process on the basis of the information provided by a given set of random observations. In this paper we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space. We establish consistency results for bootstrap estimates of the distribution of parameter estimates.

In this paper we derive nonparametric stochastic volatility models in discrete time. These models generalize parametric autoregressive random variance models, which have been applied quite successfully to nancial time series. For the proposed models we investigate nonparametric kernel smoothers. It is seen that so-called nonparametric deconvolution estimators could be applied in this situation and that consistency results known for nonparametric errors- in-variables models carry over to the situation considered herein.

Kernel smoothing in nonparametric autoregressive schemes offers a powerful tool in modelling time series. In this paper it is shown that the bootstrap can be used for estimating the distribution of kernel smoothers. This can be done by mimicking the stochastic nature of the whole process in the bootstrap resampling or by generating a simple regression model. Consistency of these bootstrap procedures will be shown.

In the following, we discuss a procedure for interpolating a spatial-temporal stochastic process. We stick to a particular, moderately general model but the approach can be easily transered to other similar problems. The original data, which motivated this work, are measurements of gas concentrations (SO2, NO, O2) and several meteorological parameters (temperature, sun radiation, procipitation, wind speed etc.). These date have been and are still recorded twice every hour at several irregularly located places in the forests of the state Rheinland-Pfalz as part of a program monitoring the air pollution in the forests.

In this paper we deal with the problem of fitting an autoregression of order p to given data coming from a stationary autoregressive process with infinite order. The paper is mainly concerned with the selection of an appropriate order of the autoregressive model. Based on the so-called final prediction error (FPE) a bootstrap order selection can be proposed, because it turns out that one relevant expression occuring in the FPE is ready for the application of the bootstrap principle. Some asymptotic properties of the bootstrap order selection are proved. To carry through the bootstrap procedure an autoregression with increasing but non-stochastic order is fitted to the given data. The paper is concluded by some simulations.

Neural networks are now a well-established tool for solving classification and forecasting problems in financial applications (compare, e.g., Bol et al., 1996, Evans, 1997, Rehkugler and Zimmermann, 1994, Refenes 1995, and Refenes et al. 1996a) though many practioners are still suspicious against too evident success stories. One reason may be that the construction of an appropriate network which provides a reasonable solution to a complex data-analytic problem is rarely made explicit in the literature. In this paper, we try to contribute to filling this gap by discussing in detail the problem of dynamically allocating capital to various components of a currency portfolio in such a manner that the average gain will be larger than for certain benchmark portfolios. We base our solution on feedforward neural networks which are constructed employing various statistical model selection procedures described in, e.g., (Anders, 1997, or Refenes et al., 1996b). Neural networks which are used as the basis of trading strategies in finance should be assessed differently than in technical applications. The task is not to construct a network which provides good forecasts with respect to mean-square error of some quantities of interest or to provide good approximation of some given target values, but to achieve a good performance in economic terms. For portfolio allocation, the main goal is to achieve on the average a large return combined with a small risk. Therefore, we do not consider forecasts of the foreign exchange (FX-) rate time series using neural networks, but we try to get the allocation directly as the output of a network. Furthermore, we do not minimize some estimation or prediction error, but we try to maximize an economically meaningful performance measure, the risk-adjusted return, directly (compare also Heitkamp, 1996). In the subsequent chapter, we describe the details of the portfolio allocation problem. The following two chapters provide some technical information on how the networks were fitted to the available data and how the network inputs and outputs were selected. In chapter 5, finally, we discuss the promising results.