Making Multiple Decisions Adaptively

The asymptotic behavior of multiple decision procedures is studied when the underlying distributions depend on an unknown nuisance parameter. An adaptive procedure must be asymptotically optimal for each value of this nuisance parameter, and it should not depend on its value. A necessary and sufficient condition for the existence of such a procedure is derived. Several examples are investigated in detail, and possible lack of adaptation of the traditional overall maximum likelihood rule is discussed.

Large Distinct Part Sizes in a Random Integer Partition

A partition of a positive integer n is a way of writing it as the sum of positive integers without regard to order; the summands are called parts. The number of partitions of n, usually denoted by p(n), is determined asymptotically by the famous partition formula of Hardy and Ramanujan [5]. We shall introduce the uniform probability measure P on the set of all partitions of n assuming that the probability 1/p(n) is assigned to each n-partition. The symbols E and V ar will be further used to denote the expectation and variance with respect to the measure P . Thus, each conceivable numerical characteristic of the parts in a partition can be regarded as a random variable.

Densities with Spherical Level Sets in the Gauss-exponential Domain

2000 Mathematics Subject Classification: 60F05, 60B10.

Sensitivity Analysis of Some Applied Probability Models

2000 Mathematics Subject Classi cation: Primary 90C31. Secondary 62C12, 62P05, 93C41.

Controlled Multitype Branching Models: Geometric Growth

2000 Mathematics Subject Classi cation: 60J80, 60F25.

Recent Results for Supercritical Controlled Branching Processes with Control Random Functions

2000 Mathematics Subject Classification: 60J80, 60F05

Resolvent and Scattering Matrix at the Maximum of the Potential

2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38.

On the Recursive Estimation of the Location and of the Size of the Mode of a Probability Density

2000 Mathematics Subject Classification: 62G07, 62L20.

On the Asymptotic Distribution of Closed Orbits for a Class of Open Billiards

2000 Mathematics Subject Classification: 37D40.

Wave Operators for Defocusing Matrix Zakharov-Shabat Systems with Pnonvanishing at Infinity

2000 Mathematics Subject Classification: Primary: 34L25; secondary: 47A40, 81Q10.