Density and Distribution Evaluation for Convolution of Independent Gamma Variables

Several numerical evaluations of the density and distribution of convolution of independent gamma variables are compared in their accuracy and speed. In application to renewal processes, an efficient formula is derived for the probability mass function of the event count

Asymptotics of a Clustering Criterion for Smooth Distributions

We develop a clustering framework for observations from a population with a smooth probability distribution function and derive its asymptotic properties. A clustering criterion based on a linear combination of order statistics is proposed. The asymptotic behavior of the point at which the observations are split into two clusters is examined. The results obtained can then be utilized to construct an interval estimate of the point which splits the data and develop tests for bimodality and presence of clusters

Convexity Bias in the Pricing of Eurodollar Swaps

The traditional use of LIBOR futures prices to obtain surrogates for the Eurodollar forward rates is proved to yield a systematic bias in the pricing of Eurodollar swaps when one assumes that the yield curve is well described by the Heath-Jarrow-Morton model. The resulting theoretical inequality is consistent with the empirical observations of Burghardt and Hoskins (1995), and it provide a theoretical basis for price anomalies that are suggested by more recent empirical data

Buses, Bullies, and Bijections

The random—or orderly—seating of passengers on a bus is used to motivate several questions about cycles of permutations. These in turn motivates the investigation of bijections between special subsets of permutations. The goal, of course, is to give simple explanations of surprising facts

Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains

Methods using gambling teams and martingales are developed and applied to find formulas for the expected value and the generating function of the waiting time until one observes an element of a finite collection of patterns in a sequence which is generated by a two-state first or higher order Markov chain