Likelihood estimation for the INAR(<i>p</i>) model by saddlepoint approximation

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 paper 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. This article has supplementary materials online.

鞍点技术（saddlepoint techniques）凭借可高精度近似难解密度与尾部概率的特性，已在众多应用场景中获得成功应用。本文聚焦其在高阶整数值自回归（Integer-valued AutoRegressive，INAR(<i>p</i>)）过程估计中的应用。此前已有研究针对INAR(<i>p</i>)模型提出条件最小二乘估计与极大似然估计方案，但前者在参数模型估计中效率偏低，后者则随着阶数<i>p</i>的提升愈发难以实现。本文针对对数似然函数提出一种简洁的鞍点近似方法，该方法即便在分布尾部场景与复杂INAR模型下仍表现卓越。本文考虑泊松（Poisson）与负二项（negative binomial）新息分布，并通过实证研究证明，在现实应用场景中，最大化该鞍点近似的估计量与极大似然估计量的表现高度相近。本文将所提方法应用于脑膜炎球菌病计数数据集。本文附带在线补充材料。