Hermite expansion and estimation of monotonic transformations of Gaussian data

Abstract

This paper describes a semiparametric method for estimating a generic probability distribution using a basis expansion in . We express the given distribution as a monotonic transformation of the Gaussian cumulative distribution function, expanded in a basis of Hermite polynomials. The coefficients in the basis expansion are functionals of the quantile function, and can be consistently estimated to give a smooth estimate of the transformation function. For situations in which the estimated function is not monotone, a projection approach is used to adjust the estimated transformation function to guarantee monotonicity. Two applications are presented which focus on the analysis of model residuals. The first is a data example which uses the residuals from the 2012 Small Area Income and Poverty Estimates model. The Hermite estimation method is applied to these residuals as a graphical method for detection of departures from normality and to construct credible intervals. The second example analyses residuals from time series models for the purpose of estimating the variance of the mean and median and comparing the results to the AR-sieve. This paper concludes with a set of numerical examples to illustrate the theoretical results.

Keywords:

• Hermite polynomials
• small area
• time series
• AR-sieve
• SAIPE
• Hilbert space

AMS Subject Classification:

• 62G05
• 62G07

Acknowledgements

This report is released to inform interested parties of ongoing research and to encourage discussion of work in progress. The views expressed are those of the authors and not necessarily those of the U.S. Census Bureau. The authors wish to thank the referees and the associate editor for their constructive comments and suggestions which greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.