Likelihood Based Estimation of the Parameters of a Log-Linear Nonhomogeneous Poisson Process

Non-homogeneous Poisson process (NHPP) has widely been used over decades to model random processes that are time dependent (e. g. occurrences of serious road accidents). After the times of occurrence of a particular event have been observed, the problem of estimating the intensity function arises. In this paper, we consider maximum likelihood estimation of the parameters of a NHPP with log-linear intensity function. The maximum likelihood estimates of the unknown parameters of the intensity function are obtained numerically and the confidence intervals and regions are constructed from the respective graphs of the maximized and joint relative likelihood functions. We present simulation results which demonstrate the good performance of our confidence intervals and regions as compared to those based on large sample approaches. The large sample confidence intervals may be inaccurate in the sense that they exclude plausible parameter values and include values that are very implausible. Our approach is optimal for small sample inferences on the parameters of a log-linear NHPP.