Volume 23, Issue 3 - 3

In this paper, we discuss the global L 2 error of the nonlinear wavelet estimators of the density function in the Besov space B s pq , when the survival times form a stationary α-mixing sequence, and prove that the nonlinear wavelet estimators can achieve the optimal rate of convergence, which is similar to the result of Donoho et al. (1996). Also, the optimal convergence rates of the nonlinear wavelet estimators of the hazard rate function in the Besov space B s pq are considered, which had not been discussed by Donoho et al. (1996) for complete data in the i.i.d. case.

In traditional nonparametric EB (empirical Bayes) setting, the paper proposes generalization of the linear EB estimation method which takes advantage of the flexibility of the wavelet techniques. A nonparametric EB estimator is represented as a wavelet series expansion and the coefficients are estimated by minimizing the prior risk of the estimator. Although wavelet series have been used previously for EB estimation, the method suggested in the paper is completely novel since the EB estimator as a whole is represented as a wavelet series rather than its components. Moreover, the method exploits de-correlating property of wavelets which is not instrumental for the former wavelet-based EB techniques. As a result, estimation of wavelet coefficients requires solution of a well-posed sparse system of linear equations. The technique provides asymptotically optimal EB estimators posterior risks of which tend to zero at the optimal rate as the number of observations tends to infinity.

Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utility of the terminal wealth in a general incomplete market model. We also allow for very general sets of prior models. In particular, we do not assume that all prior models are equivalent to each other, which allows us to handle many economically meaningful robust utility functionals such as those defined by AVaR λ , concave distortions, or convex capacities. We also show that dropping the equivalence of prior models may lead to new effects such as the existence of arbitrage strategies under the least favorable model.

The paper aims at drawing attention to the potential that qualitative stability theory of stochastic programming bears for the study of asymptotic properties of statistical estimators. Stability theory of stochastic programming yields a unifying approach to convergence (almost surely, in probability, and in distribution) of solutions to random optimization problems and can hence be applied to many estimation problems. Non-unique solutions of the underlying optimization problems, constraints for the solutions, and discontinuous objective functions can be dealt with. Making use of stability results, it is often possible to extend existing consistency statements and assertions on the asymptotic distribution, especially to non-standard cases. In this paper we will exemplify how stability results can be employed. For this aim existing results are supplemented by assertions which convert the stabilit results in a form which is more convenient for asymptotic statistics. Furthermore, it is shown, how ε n -optimal solutions can be dealt with. Emphasis is on strong and weak consistency and asymptotic distribution in the case of non-unique solutions. M-estimation, constrained M-estimation for location and scatter, quantile estimation and the behavior of the argmin functional for càdlàg processes are considered.