Hermite Expansion and Estimation of Monotonic Transformations of Gaussian Data

• Activity Number: 76
• Type: Contributed
• Date/Time: Sunday, August 3, 2014 : 4:00 PM to 5:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Ryan Janicki*+ and Tucker Sprague McElroy
• Companies: U.S. Census Bureau and U.S. Census Bureau
• Keywords: Hermite polynomials ; Hilbert space ; Small area estimation ; AR Sieve ; SAIPE ; Time series

Abstract

This paper describes a semi-parametric method for estimating a generic probability distribution using a basis expansion in L2. We express the given distribution as a monotonic transformation of the Gaussian cumulative distribution function, expanded in a basis of Hermite polynomials. For situations in which the estimated function is not monotone, a projection approach is presented which adjusts the estimated transformation function and guarantees 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 (SAIPE) 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 analyzes residuals from time series models for the purpose of estimating the variance of the median and comparing the results to the AR-sieve. This paper concludes with a set of numerical examples to illustrate the theoretical results.