Linear Conditional Expectation for Discretized Distributions

Abstract

Many statistical methods for continuous distributions assume a linear conditional expectation. Components of multivariate distributions are often measured on a discrete ordinal scale based on a discretization of an underlying continuous latent variable. The results in this paper show that common examples of discretized bivariate and trivariate distributions will have a linear conditional expectation. Examples and simulations are provided to illustrate the results.

Keywords:

• Conditional expectation
• biserial/polyserial correlations
• elliptical distributions
• polychoric correlations
• tetrachoric correlations