# Estimating the parameters of a nonhomogeneous Poisson process with linear rate

## Abstract

Motivated by telecommunication applications, we investigate ways to estimate the parameters of a nonhomogeneous Poisson process with linear rate over a finite interval, based on the number of counts in measurement subintervals. Such a linear arrival-rate function can serve as a component of a piecewise-linear approximation to a general arrival-rate function. We consider ordinary least squares (OLS), iterative weighted least squares (IWLS) and maximum likelihood (ML), all constrained to yield a nonnegative rate function. We prove that ML coincides with IWLS. As a reference point, we also consider the theoretically optimal weighted least squares (TWLS), which is least squares with weights inversely proportional to the variances (which would not be known with data). Overall, ML performs almost as well as TWLS. We describe computer simulations conducted to evaluate these estimation procedures. None of the procedures differ greatly when the rate function is not near 0 at either end, but when the rate function is near 0 at one end, TWLS and ML are significantly more effective than OLS. The number of measurement subintervals (with fixed total interval) makes surprisingly little difference when the rate function is not near 0 at either end. The variances are higher with only two or three subintervals, but there usually is little benefit from going above ten. In contrast, more measurement intervals help TWLS and ML when the rate function is near 0 at one end. We derive explicit formulas for the OLS variances and the asymptotic TWLS variances (as the number of measurement intervals increases), assuming the nonnegativity constraints are not violated. These formulas reveal the statistical precision of the estimators and the influence of the parameters and the method. Knowing how the variance depends on the interval length can help determine how to approximate general arrival-rate functions by piecewise-linear ones. We also develop statistical tests to determine whether the linear Poisson model is appropriate.