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Our problem here starts when the seller of an American option H has an initial capital c and aims to control shortfall risk by optimally trading in the market. We consider the case where the initial capital c is smaller than the superhedging cost of H . Thus, it is meaningful to consider efficient strategies of partial hedging . We investigate in the American case the approach of Föllmer and Leukert of efficiently hedging European options. At the same time, we consider situations where the model R is ambiguous. This implies that the seller´s preference is going to be represented by a robust loss functional, involving a whole class of absolutely continuous probability measures Q . We apply a construction due to Föllmer and Kramkov on the existence of Fatou-convergent sequences of convex combinations of supermartingales, to show that optimal strategies minimizing such a functional exist. We then show that the robust efficient hedging problem can be reduced into a non-robust problem of efficient hedging with respect to a worst case probability measure Q * ∈ Q , if Q satisfies a compactness condition and H is essentially bounded.

This paper deals with the problem of estimating the mean vector of an elliptically contoured distribution with unknown scale matrix, where the mean vector is restricted to an unbounded and closed convex set with a smooth or a piecewise smooth boundary. A shrinkage-type estimator is shown to be better than the maximum likelihood estimator subject to the restriction relative to a quadratic loss.

The minimum risk equivariant estimator (MRE) of the regression parameter vector β in the linear regression model enjoys the finite-sample optimality property, but its calculation is difficult, with an exception of few special cases. We study some possible approximations of MRE, with distribution of the errors being known or unknown: A finite-sample approximation uses the Hájek–Hoeffding projection or the Hoeffding–van Zwet decomposition of an initial equivariant estimator of β , a large-sample approximation is based on the asymptotic representation of the same. A nonparametric approximation uses the expected value with respect to the conditional empirical distribution function, developed by Stute (1986). The only possible approximation avoiding a difficult calculation of conditional expectations is the asymptotic approximation, based on the score function of the underlying distribution of the errors.

It has been shown recently that, under an appropriate integrability condition, densities of functions of independent and identically distributed random variables can be estimated at the parametric rate by a local U-statistic, and a functional central limit theorem holds. For the sum of two squared random variables, the integrability condition typically fails. We show that then the estimator behaves differently for different arguments. At points in the support of the squared random variable, the rate of the estimator slows down by a logarithmic factor and is independent of the bandwidth, but the asymptotic variance depends on the rate of the bandwidth, and otherwise only on the density of the squared random variable at this point and at zero. A functional central limit theorem cannot hold. Of course, for bounded random variables, the sum of squares is more spread out than a single square. At points outside the support of the squared random variable, the estimator behaves classically. Now the rate is again parametric, the asymptotic variance has a different form and does not depend on the bandwidth, and a functional central limit theorem holds.

We consider a class of testing problems where the null space is the union of k -1 subgraphs of the form h j ( θ j )≤ θ k , with j =1,…, k -1, ( θ 1 ,…, θ k ) the unknown parameter, and h j given increasing functions. The data consist of k independent samples, assumed to be drawn from a distribution with parameter θ j , j =1,…, k , respectively. An important class of examples covered by this setting is that of non-inferiority hypotheses, which have recently become important in the evaluation of drugs or therapies. When the true parameter approaches the boundary at a 1/ √n rate, we give the explicit form of the asymptotic distribution of the log-likelihood ratio statistic. This extends previous work on the distribution of likelihood ratio statistics to local alternatives. We consider the prominent example of binomial data and illustrate the theory for k =2 and 3 samples. We explain how this can be used for planning a non-inferiority trial. To this end we calculate the optimal sample ratios yielding the maximal power in a binomial non-inferiority trial.