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We consider the James–Stein problem for non-normal data for estimating a p -vector θ . It is shown how the risk may be expanded in powers of p -1 . The factor 1-2/ p that distinguishes the James–Stein estimate from the Stein estimate is shown to have only O ( p -2 ) effect on the risk. The case, where the variance must be estimated is studied for the one-way unbalanced ANOVA problem.

We propose a new approach to testing whether one random variable is stochastically non-dominated by another one. The tests compare mean-risk differences of two unknown distributions using independent samples. The test can be used for comparison of the coherent risk measures of the distributions, as well as to reject stochastic dominance relation of first, second, or higher order between the two distributions. We consider several law-invariant coherent measures of risk which are consistent with the stochastic dominance relation of first and higher order. Numerical comparisons with the Mann–Whitney test and with the F-test for comparison of variance are provided. The numerical study indicates that most of the mean-risk tests are more powerful than the Mann–Whitney test.

Consider a diffusion process ( X t ) t  ≥ 0 , with unknown drift b ( x ) and diffusion coefficient σ ( x ), which is strictly stationary, ergodic and β -mixing. At discrete times t k  =  k δ for k from 1 to N , we have at disposal noisy data of the sample path, Y k δ  =  X k δ + ε k . The random variables ( ε k ) are i.i.d., centred and independent of ( X t ). In order to reduce the noise effect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that Δ =  p δ is small whereas N δ is large. Then, the drift function b is estimated in a compact set A in a non-parametric way using a penalized least squares approach. We obtain a bound for the risk of the resulting adaptive estimator. Examples of diffusions satisfying our assumptions are given and numerical simulation results illustrate the theoretical properties of our estimators.

We derive comparison results for Markov processes with respect to stochastic orderings induced by function classes. Our main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators implies ordering of the processes. Unlike in previous work no boundedness assumptions on the function classes are needed anymore. We also present an integral version of the comparison result which does not need the local comparability assumption of the generators. The method of proof is also used to derive comparison results for time-discrete Markov processes.

This paper introduces a method of moment estimator for the time-changed Lévy processes proposed by Carr, Geman, Madan and Yor (2003). By establishing that the returns sequence is strongly mixing with exponentially decreasing rate, we prove consistency and asymptotic normality of the resulting estimators. In addition, we fit parametrized versions of the model to real data and examine the quality of our estimators by performing a simulation study. Finally, we also show how to estimate the current level of volatility.