# Rate Estimators for Non-stationary Point Processes

Non-stationary point processes are often used to model systems whose rates vary over time. Estimating underlying rate functions is important for input to a discrete-event simulation along with various statistical analyses. We study nonparametric estimators to the marked point process, the infinite-server queueing model, and the transitory queueing model. We conduct statistical inference for these estimators by establishing a number of asymptotic results.

For the marked point process, we consider estimating the offered load to the system over time. With direct observations of the offered load sampled at fixed intervals, we establish asymptotic consistency, rates of convergence, and asymptotic covariance through a Functional Strong Law of Large Numbers, a Functional Central Limit Theorem, and a Law of Iterated Logarithm. We also show that there exists an asymptotically optimal interval width as the sample size approaches infinity.

The infinite-server queueing model is central in many stochastic models. Specifically, the mean number of busy servers can be used as an estimator for the total load faced to a multi-server system with time-varying arrivals and in many other applications. Through an omniscient estimator based on observing both the arrival times and service requirements for n samples of an infinite-server queue, we show asymptotic consistency and rate of convergence. Then, we establish the asymptotics for a nonparametric estimator based on observations of the busy servers at fixed intervals.

The transitory queueing model is crucial when studying a transitory system, which arises when the time horizon or population is finite. We assume we observe arrival counts at fixed intervals. We first consider a natural estimator which applies an underlying nonhomogeneous Poisson process. Although the estimator is asymptotically unbiased, we see that a correction term is required to retrieve an accurate asymptotic covariance. Next, we consider a nonparametric estimator that exploits the maximum likelihood estimator of a multinomial distribution to see that this estimator converges appropriately to a Brownian Bridge.