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We prove a stochastic expansion for a residual-based estimator of the error distribution function in a partly linear regression model. It implies a functional central limit theorem. As special cases we cover nonparametric, nonlinear and linear regression models.

Consider a probability model P θ,α , where θ=(θ 1 ,θ 2 ,…,θ k ) T is a parameter vector of interest, and α is some nuisance parameter. The problem of testing null hypothesis H 0 : θ 1 =θ 2 =…=θ k against selecting one of k alternative hypotheses H i :θ i =θ [ k ] > θ [1] , i =1,2,…, k , where θ [ k ] =max{θ 1 ,θ 2 ,…,θ k } and θ [1] =min{θ 1 ,θ 2 ,…,θ k }, is formulated from a Bayesian decision theoretic point of view. This problem can be viewed as selecting a component with the largest parameter value if the null hypothesis is rejected. General results are obtained for the Bayes rule under monotonic permutation invariant loss functions. Bayes rules are obtained for k one-parameter exponential families of distributions under conjugate priors. The example of normal populations is considered in more detail under the non-informative (improper) priors. It is demonstrated through this example that the classical hypothesis testing yields a poor power as compared to the Bayes rules when the alternatives are such that a small fraction of the components of θ have significantly high values while most of them have low values. Consequences of this for the high dimensional data such as microarray data are pointed out.

The present paper establishes finite sample most powerful tests for certain nonparametric null hypotheses P 0 which admit a sufficient statistic S . The underlying alternatives are of semiparametric or nonparametric nature. Optimal one-sided S -conditional test are offered for families with nonparametric isotone likelihood ratio. Similarly two-sided optimal locally unbiased S -conditional test are introduced for alternatives with nonparametric convex likelihood. If in addition S is P 0 -complete then of course we arrive at most powerful α -similar tests. Special examples are randomization tests, permutation tests for two-sample problems and symmetry tests for the null hypothesis of 0-symmetry. The results rely on a new conditional Neyman–Pearson Lemma which can be found in the appendix and which is of own interest. This Lemma is used to solve conditional optimization problems for tests.

This paper gives an asymptotic analysis of the mean-variance (Markowitz-type) portfolio selection under mild assumptions on the market behavior. Theoretical results show the rate of underperformance of the risk aware Markowitz-type portfolio strategy in growth rate compared to the log-optimal portfolio strategy, which does not have explicit risk control. Statements are given with and without full knowledge of the statistical properties of the underlying process generating the market, under the only assumption that the market is stationary and ergodic. The experiments show how the achieved wealth depends on the coefficient of risk aversion measured on past NYSE data.