Superpositioned Stationary Count Time Series

Abstract

This paper probabilistically explores a class of stationary count time series models built by superpositioning (or otherwise combining) independent copies of a binary stationary sequence of zeroes and ones. Superpositioning methods have proven useful in devising stationary count time series having prespecified marginal distributions. Here, basic properties of this model class are established and the idea is further developed. Specifically, stationary series with binomial, Poisson, negative binomial, discrete uniform, and multinomial marginal distributions are constructed; other marginal distributions are possible. Our primary goal is to derive the autocovariance function of the resulting series.

Multivariate integer-valued time series with flexible autocovariances and their application to major hurricane counts

“…When we fitted this negative binomial version of the model to the count data, the estimated parameters migrated toward a degenerate bivariate Gaussian with correlation between components of −1: the fit pushed up against "model class boundaries" and was numerically unstable. Other marginal distributions are possible; these are currently being probabilistically formalized in Jia and Lund (2016).…”

Latent Gaussian Count Time Series

“…Blight [5] and [9] take a different approach, constructing the desired count marginal distribution by combining IID copies of a correlated Bernoulli series {B t } in various ways. By using a binary {B t } constructed from a stationary renewal sequence [5,9,27], a variety of marginal distributions, including binomial, Poisson, and negative binomial, were produced; moreover, these models can have negative correlations. While these models do not necessarily produce the most negatively correlated count structures possible, they often come close to achieving this bound.…”