Mehrdad Shahshahani

On Nonlinear Functions of Linear Combinations

Projection pursuit algorithms approximate a function of p variables by a sum of nonlinear functions of linear combinations: \[ (1)\qquad f\left( {x_1 , \cdots ,x_p } \right) \doteq \sum_{i = 1}^n {g_i \left( {a_{i1} x_1 + \cdots + a_{ip} x_p } \right)} . \] We develop some approximation theory, give a necessary and sufficient condition for equality in (1), and discuss nonuniqueness of the representation.

The performance of trellis-coded MDPSK with multiple symbol detection

The idea of using a multiple (more than two) symbol observation interval to improve error probability performance is applied to differential detection of trellis-coded multiple phase-shift keying (MPSK) over an additive white Gaussian noise (AWGN) channels. An equivalent Euclidean distance measure per trellis branch is determined for this detection scheme. This is used to define an augmented (larger multiplicity) trellis code whose distance measure is the conventional squared Euclidean distance typical of conventional trellis-coded modulation on the AWGN. Such an augmented multiple trellis code is a convenient mathematical tool for simplifying the analysis. Results are obtained by a combination of analysis and computer simulation. It is shown that only a slight increase (e.g. one symbol) in the length of the observation interval will provide a significant improvement in bit error probability performance. >

The Subgroup Algorithm for Generating Uniform Random Variables

We suggest a simple algorithm for Monte Carlo generation of uniformly distributed variables on a compact group. Example include random permutations, Rubiks cube positions, orthogonal, unitary, and symplectic matrices, and elements of GL n over a finite field. the algorithm reduces to the “standard” fast algorithm when there is one, but many new example are included.

On the Number of Component Failures in Systems Whose Component Lives are Exchangeable

We consider a system that is composed of finitely many independent components each of which is either “on” or “off” at any time. The components are initially on and they have common on-time distributions. Once a component goes off, it remains off forever. The system is monotone in the sense that if the system is off whenever each component in a subset S called a cut set of components is off, then that is also true for every subset of components containing S. We are interested in studying the properties of N, the number of components that are off at the moment the system goes off. We compute the factorial moments of N in terms of the reliability function. We also prove that N is an increasing failure rate average random variable and present a duality result. We consider the special structure in which the minimal cut sets do not overlap and we prove a conjecture of El-Neweihi, Proschan and Sethuraman which states that N is an increasing failure rate random variable. Then we consider the special case of nonoverlapping minimal path sets, and in the final section we present an application to a shock model.

On square roots of the uniform distribution on compact groups

Let G be a compact separable topological group. When does there exist a probability P such that P P = U, where U is Haar measure and P + U? We show that such square roots exist if and only if G is not abelian, nor the product of the quaternions and a product of two element groups. In the course of proving this we classify compact groups with the property that every closed subgroup is normal.

On the Duration of the Problem of the Points

Abstract We consider an r-player version of the famous problem of the points, which was the stimulus for the correspondence between Pascal and Fermat in the 17th century. At each play of a game, exactly one of the players wins a point, player i winning with probability pi. The game ends the first time a player has accumulated his or her required number of points—this requirement being ni for player i. A reliability application would be to suppose that a system is subject to r different types of shocks and failure occurs the first time there have been ni type i shocks for any i = 1, …, r. Our main result is to show that N, the total number of plays, is an increasing failure-rate random variable. In addition, we prove some Schur convexity results regarding P{N ≤ k} as a function of p (for ni ≡ n) and as a function of n (for pi ≡ 1/r).