Volume 26, Issue 3

In this paper, we study the behavior of a kernel estimator for the regression function in a random right-censoring model. We establish pointwise and uniform strong consistency over a compact set and give a rate of convergence for the estimate.The asymptotic normality of the estimate is also proved. Simulations are drawn for different cases to illustrate both, convergence and asymptotic normality.

We consider the expected return and the variance of the expected quadratic utility portfolio and the tangency portfolio. The expected returns on the individual assets and their covariance matrix are estimated by the sample mean and the sample covariance matrix. Replacing the unknown parameters by these estimators in the portfolio characteristics estimators of the expected portfolio return and the portfolio variance are obtained. In this paper we calculate the densities of these estimators assuming independent and multivariate normally distributed returns. Because the densities can be computed by using standard mathematical software packages these representations are very useful. These results can be applied to construct tests and confidence intervals for the parameters of the efficient frontier.

Let X, U 1 , …, U n -1 be n random vectors in ℝ p with joint density of the form f((X - θ) ´ ∑ -1 (X - θ) + ∑ n -1 j = 1 U ´ j ∑ -1 U j ) where both θ ∈ℝ p and ∑ are unknown, the scale matrix ∑ being supposed structured as a diagonal matrix, that is, ∑= diag(∑ 1 , …,∑ b ) where, for 1 ≤ i ≤ b , ∑ i is a p i × p i matrix and ∑ i = 1 b p i = p . We consider the problem of the estimation of θ with the invariant loss (δ - θ)´ ∑ -1 (δ - θ) and propose estimators which dominate the usual estimator δ 0 (X) = X . These domination results hold simultaneously for the entire class of such distributions. The proof uses a generalization of integration by parts formulae by Stein and Haff. We also consider estimating ∑ under L S (∑^,∑) = tr (∑^∑ -1 ) - log |∑^∑ -1 | - p and propose estimators that dominate the unbiased estimator ∑^ UB = diag( S 1 , …, S b )/( n - 1), where S i = ∑ j = 1 n - 1 U ij U ´ ij and dim U ji = p i , for 1 ≤ i ≤ b and 1 ≤ j ≤ n - 1. The subsequent development of expressions is analogous to the unbiased estimators of risk technique and, in fact, reduces to an unbiased estimator of risk in the normal case.

In this paper we consider an explicit dynamic risk measure for discrete-time payment processes which have a Markovian structure. The risk measure is essentially a sum of conditional Average Value-at-Risks. Analogous to the static Average Value-at-Risk, this risk measures can be reformulated in terms of the value functions of a dynamic optimization problem, namely a so-called Markov decision problem. This observation gives a nice recursive computation formula. Afterwards, the definition of the dynamic risk measure is generalized to a setting with incomplete information about the risk distribution which can be seen as model ambiguity. We choose a parametric approach here. The dynamic risk measure is again defined as the sum of conditional Average Value-at-Risks or equivalently is the solution of a Bayesian decision problem. Finally, it is possible to discuss the effect of model ambiguity on the risk measure: Surprisingly, it may be the case that the risk decreases when additional “risk” due to parameter uncertainty shows up. All investigations are illustrated by a simple but useful coin tossing game proposed by Artzner and by the classical Cox–Ross–Rubinstein model.