The Annals of Statistics

Asymptotics for in-sample density forecasting

Young K. Lee, Enno Mammen, Jens P. Nielsen, and Byeong U. Park

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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.

Article information

Ann. Statist. Volume 43, Number 2 (2015), 620-651.

First available in Project Euclid: 3 March 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties

Density estimation kernel smoothing backfitting chain ladder


Lee, Young K.; Mammen, Enno; Nielsen, Jens P.; Park, Byeong U. Asymptotics for in-sample density forecasting. Ann. Statist. 43 (2015), no. 2, 620--651. doi:10.1214/14-AOS1288.

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