Volume 26, Issue 4

This paper addresses the Stein conjecture in the simultaneous estimation of a matrix mean of a multivariate normal distribution with a known covariance matrix. Stein (1973) derived an unbiased estimator of a risk function for orthogonally equivariant estimators and considered to isotonize the estimator which minimizes the main part of the unbiased risk-estimator. We call it the Stein risk-minimization estimator (RM) in this paper. Although the Stein RM estimator has been recognized as an excellent procedure with a nice risk-performance, it has a complicated form based on the isotonizing algorithm, and no analytical properties such as minimaxity have been shown. The aim of this paper is to fix this conjecture in lower dimensional cases, that is, the minimaxity of the Stein RM estimator is established for the two and three dimensions.

We consider the problem of estimating the size of a random digital search tree on the basis of the maximal node depth observed along a specific path. We show that the maximum likelihood estimator exists and we investigate its properties. A similar problem arises in the context of approximate counting. In both cases a simple pure birth process plays a central role. We also construct confidence bounds.

This paper is concerned with evaluation of American options in discrete time, also called Bermudan options. We use the dual approach to derive upper bounds on the price of such options using only a reduced number of nested Monte Carlo steps. The key idea is to apply nonparametric regression to estimate continuation values and all other required conditional expectations and to combine the resulting estimate with another estimate computed by using only a reduced number of nested Monte Carlo steps. The expectation of the resulting estimate is an upper bound on the option price. It is shown that the estimates of the option prices are universally consistent, i.e., they converge to the true price regardless of the structure of the continuation values. The finite sample behavior is validated by experiments on simulated data.

The paper deals with the problem of finding an optimal one-time rebalancing strategy assuming that in the Black–Scholes model the drift term of the stock may change its value spontaneously at some random non-observable (hidden) time. The problem is studied on a finite time interval under two criteria of optimality (logarithmic and linear). The methods of the paper are based on the results for the quickest detection of drift change for Brownian motion.

In this paper we consider the problem of optimal risk allocation or risk exchange with respect to convex risk functionals, which not necessarily are monotone or cash invariant. General existence and characterization results are given for optimal risk allocations minimizing the total risk as well as for Pareto optimal allocations. We establish a general uniqueness result for optimal allocations. As particular consequence we obtain in case of cash invariant, strictly convex risk functionals the uniqueness of Pareto optimal allocations up to additive constants. In the final part some tools are developed useful for the verification of the basic intersection condition made in the theorems which are applied in several examples.