Estimation and Inference for Non-Stationary Arrival Models with a Linear Trend

This paper is concerned with building statistical models for non-stationary input processes with a linear trend. Under a Poisson assumption, we investigate the use of the maximum likelihood (ML) method to estimate the model and establish limiting behavior for the ML estimator in an asymptotic regime that naturally arises in applications with high-volume inputs. We also develop likelihood ratio tests for the presence of a linear trend and discuss the asymptotic efficiency. Change-point detection procedures are discussed to identify an unknown point when the model switches from a stationary mode to non-stationarity with a linear trend. Numerical experiments on an e-commerce data set are included. Incorporating a linear trend into an input model can improve prediction accuracy and potentially enhance associated performance evaluations and decision making.