Asymptotics for in-sample density forecasting

TL;DR

This paper develops a general theoretical framework for in-sample density forecasting models with structural assumptions, including seasonal effects, and provides asymptotic results supported by a practical insurance example.

Contribution

It introduces a general theory for density forecasting models with product-structured densities and seasonal effects, including asymptotic properties and practical estimation methods.

Findings

Asymptotic properties of smoothing estimators are established.

The models effectively incorporate seasonal effects.

Empirical example demonstrates practical applicability in insurance claims forecasting.

Abstract

This paper generalizes recent proposals of density forecasting models and it develops theory for this class of models. In density forecasting, the density of observations is estimated in regions where the density is not observed. Identification of the density in such regions is guaranteed by structural assumptions on the density that allows exact extrapolation. In this paper, the structural assumption is made that the density is a product of one-dimensional functions. The theory is quite general in assuming the shape of the region where the density is observed. Such models naturally arise when the time point of an observation can be written as the sum of two terms (e.g., onset and incubation period of a disease). The developed theory also allows for a multiplicative factor of seasonal effects. Seasonal effects are present in many actuarial, biostatistical, econometric and statistical studies. Smoothing estimators are proposed that are based on backfitting. Full asymptotic theory is derived for them. A practical example from the insurance business is given producing a within year budget of reported insurance claims. A small sample study supports the theoretical results.