Band 21, Heft 2 - 2

It has been proposed that the arbitrage possibility in the fractional Black–Scholes model depends on the definition of the stochastic integral. More precisely, if one uses the Wick–Itô–Skorohod integral one obtains an arbitrage-free model. However, this integral does not allow economical interpretation. On the other hand it is easy to give arbitrage examples in continuous time trading with self-financing strategies, if one uses the Riemann-Stieltjes integral. In this note we discuss the connection between two different notions of self-financing portfolios in the fractional Black–Scholes model by applying the known connection between these two integrals. In particular, we give an economical interpretation of the proposed arbitrage-free model in terms of Riemann–Stieltjes integrals.

This paper deals with the construction of efficient estimators in semiparametric models without the sample splitting technique. Schick (1987) gave sufficient conditions using the leave-one-out technique for a construction without sample splitting. His conditions are stronger and more cumbersome to verify than the necessary and sufficient conditions for the existence of efficient estimators which suffice for the construction based on sample splitting. In this paper we use a conditioning argument to weaken Schick′s conditions. We shall then show that in a large class of semiparametric models and for properly chosen estimators of the score function the resulting weaker conditions reduce to the minimal conditions for the construction with sample splitting. In other words, in these models efficient estimators can be constructed without sample splitting under the same conditions as those used for the construction with sample splitting. We demonstrate our results by constructing an efficient estimator using these ideas in a semiparametric additive regression model.

A nonparametric probability density estimation problem is studied for the Bahadur-type risk under the sup-norm losses. The risk is minimax over the Hölder classes of densities. The large sample limiting performance of this risk is found, and the links to asymptotic equivalent white-noise models are discussed.

Power functions of tests for Gaussian shift experiments on infinite dimensional Hilbert spaces usually can not be calculated explicitly. Therefore one analyzes the behavior of such tests in the neighborhood of the null hypothesis. Useful measures to compare the quality of different testing procedures are the gradient of a one-sided and the curvature of a two-sided test in the null hypothesis. Janssen (1995) showed that a principal component decomposition of the curvature exists based on a Hilbert–Schmidt operator. It follows that these tests have only acceptable power for a finite number of directions. In this paper we prove an even stronger general result for Gauss shifts under just mild additional assumptions. A certain optimality property of a one-sided test implicates that for a small level α the corresponding two-sided test acts only in a single direction. The results are applied to Kolmogorov–Smirnov type tests and the signal detection problem.

We present consistency results for estimators of the box-counting dimension of the support of probability distributions. The box-counting dimension of the support E is defined via the covering number, i.e. the minimal cardinality of a cover of E consisting of cubes of fixed side-length. Accordingly the covering number of a sample allows the definition of an estimator of the box-counting dimension of E . Consistency results for arrays of probability distributions may be applied to the distributions of innovations of Itô processes and allow the construction of consistent estimators of the dimension of the factors, i.e. of the dimension of the Brownian motion driving the process.

In 1967, Eaton successfully solved the problem of Bechhofer for ranking of populations based on observations of the corresponding random variables. In the present paper, we follow Eaton and apply the Wald′s theory of statistical decision functions to a similar problem where the moduli of parameters are ranked rather than the parameters.