Volume 22, Issue 2 - 2

The local behavior of the power of weighted χ 2 -tests and Bayes tests is studied for simple null hypothesis in Gaussian shift experiments. A second order expansion of the power function is given. This expansion provides a shrinking family of ellipsoids (δ E ) 0<δ<1 so that the power of the weighted χ 2 -test is locally constant on the boundary ∂(δ E ). Approximating the weighted χ 2 -test by a sequence of Bayes tests with priors on ∂(δ E ), the weighted χ 2 -test is shown to be locally maximin in the sense of Giri and Kiefer for the family of restricted alternatives given by the complements of the (δ E ) 0<δ<1 .

In this paper we establish that the natural single point update Markov chain (also known as Glauber dynamics) for counting the number of Euler orientations of 2-dimensional Cartesian grids is rapidly mixing. This extends a result of Luby, Randall, and Sinclair (2001) who consider the case where orientations in the boundary are fixed. Similarly, we also obtain a rapid mixing result for the 3-colouring of rectangular Cartesian grids without fixing the boundaries. The proof uses path coupling and comparison to related Markov chains which allow additional transitions and which can be analysed directly.

We consider the two-sample problem with dispersion alternatives. Starting with a sufficient characterization of the alternative by means of two integrals we come up with a test which is based on the empirical counterparts of the integrals. Especially the critical region is now the inverse image of an infinite rectangle in ℝ 2 . The common limit distribution of the empirical integrals is determined under the hypothesis of randomness and under a broad class of nonparametric local alternatives. In each case it turns out to be normal. It enables the construction of an asymptotic level-α test, which is consistent on the alternative. In addition we are able to make local power investigations. In our example the new test is superior to the classical Cramér–von Mises test.

For any utility function with asymptotic elasticity equal to one, we construct a market model in countable discrete time, such that the utility maximization problem with proportional transaction costs admits no solution. This proves that the necessity of the reasonable asymptotic elasticity condition, established by Kramkov and Schachermayer in the frictionless case, remains also valid in the presence of transaction costs.