The Nontruncated Marginal of a Truncated Bivariate Normal Distribution

Abstract

Inference is considered for the marginal distribution of X , when ( X, Y ) has a truncated bivariate normal distribution. The Y variable is truncated, but only the X values are observed. The relationship of this distribution to Azzalini's “skew-normal” distribution is obtained. Method of moments and maximum likelihood estimation are compared for the three-parameter Azzalini distribution. Samples that are uniformative about the skewness of this distribution may occur, even for large n . Profile likelihood methods are employed to describe the uncertainty involved in parameter estimation. A sample of 87 Otis test scores is shown to be well-described by this model.