<div><p>Saddlepoint techniques have been used successfully in many applications, owing to the high accuracy with which they can approximate intractable densities and tail probabilities. This article concerns their use for the estimation of high-order integer-valued autoregressive, INAR(<i>p</i>), processes. Conditional least squares estimation and maximum likelihood estimation have been proposed for INAR(<i>p</i>) models, but the first is inefficient for estimating parametric models, and the second becomes difficult to implement as the order <i>p</i> increases. We propose a simple saddlepoint approximation to the log-likelihood that performs well even in the tails of the distribution and with complicated INAR models. We consider Poisson and negative binomial innovations, and show empirically that the estimator that maximises the saddlepoint approximation behaves very similarly to the maximum likelihood estimator in realistic settings. The approach is applied to data on meningococcal disease counts. Supplementary materials for this article are available online.</p></div
