The Annals of Statistics

Operational time and in-sample density forecasting

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

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In this paper, we consider a new structural model for in-sample density forecasting. In-sample density forecasting is to estimate a structured density on a region where data are observed and then reuse the estimated structured density on some region where data are not observed. Our structural assumption is that the density is a product of one-dimensional functions with one function sitting on the scale of a transformed space of observations. The transformation involves another unknown one-dimensional function, so that our model is formulated via a known smooth function of three underlying unknown one-dimensional functions. We present an innovative way of estimating the one-dimensional functions and show that all the estimators of the three components achieve the optimal one-dimensional rate of convergence. We illustrate how one can use our approach by analyzing a real dataset, and also verify the tractable finite sample performance of the method via a simulation study.

Article information

Ann. Statist., Volume 45, Number 3 (2017), 1312-1341.

Received: July 2015
Revised: June 2016
First available in Project Euclid: 13 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Density estimation kernel smoothing backfitting chain Ladder


Lee, Young K.; Mammen, Enno; Nielsen, Jens P.; Park, Byeong U. Operational time and in-sample density forecasting. Ann. Statist. 45 (2017), no. 3, 1312--1341. doi:10.1214/16-AOS1486.

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Supplemental materials

  • Supplement to “Operational time and in-sample density forecasting”. We provide the proofs of Theorems 3 and 6 in the supplement.