We present a method to add confidence limits to level curves, drawn on a map of a Gaussian random surface, measured with or without error on a regular or irregular grid. The method is based on the conditional level crossing intensity measure for non-stationary Gaussian fields.

For normal models we consider the problem of testing a null hypothesis against an order-restricted alternative. The alternative always consists of a cone minus the null space. We offer sufficient conditions for a class of tests to be complete and for unbiasedness of tests. Both sets of sufficient conditions are expressed in terms of the notion of cone order monotonicity. A method of constructing tests that are unbiased and in the complete class is given. The method yields new tests of value to many problems. Detailed applications and a simulation study are offered for testing homogeneity of means against the simple order alternative and for testing homogeneity against the matrix order alternative.

We derive an identity for the maximum likelihood estimator in nonparametric missing data models and biased sampling models, which almost says that this estimator is efficient. Application of empirical process theory to the identity provides us with a straightforward consistency and efficiency proof. The identity is illustrated with the random truncation model.

We study the dynamics of hybrid zones in the absence of selection. In dimensions d>1 the width of the hybrid zone grows as \sqrt{t} but in one dimension the width converges to a non-degenerate limit. We believe that tight interfaces are common in one-dimensional particle systems.

We derive an upper bound for the disconnection exponent γ of two-dimensional Brownian motion. More precisely, we show that γ≤½-(log2)²/(2π²)<0.47566. This implies in particular that γ is not equal to its trivial upper bound (i.e. ½). We also derive similar estimates of disconnection exponents for several planar Brownian motions and intersection exponents.

A continued fraction in R^d+1 is the composition of an infinite number of projectivities of R^d+1 which preserve (0,+∞)×R^d. We consider a right random walk on the semigroup of such projectivities governed by a special distribution, and we prove that the corresponding random continued fraction has a generalized inverse Gaussian distributionon R^d+1. This leads to a characterization of these distributions.

Acknowledgement of refereeing work BJ vol. 01

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