Exponentially weighted estimands and the exponential family: Filtering, prediction and smoothing

TL;DR

This paper introduces a new filtering, prediction, and smoothing method for time series based on exponentially weighted estimands within the exponential family, resulting in simple linear recursions.

Contribution

It develops a novel approach using discounted convex combinations of likelihoods for exponential family models, providing exact linear recursive filters, predictors, and smoothers.

Findings

Exact filters, predictors, and smoothers with linear recursions are derived.

The method is theoretically developed and empirically validated on simulated and real data.

Abstract

We propose using a discounted version of a convex combination of the log-likelihood with the corresponding expected log-likelihood such that when they are maximized they yield a filter, predictor and smoother for time series. This paper then focuses on working out the implications of this in the case of the canonical exponential family. The results are simple exact filters, predictors and smoothers with linear recursions. A theory for these models is developed and the models are illustrated on simulated and real data.