Elliptical Models Subject to Hidden Truncation or Selective Sampling
Hidden truncation (or selective sampling) models have received considerable attention in the last 20 years. The seminal paper for reigniting interest in such distributions was Azzalini (1985). He introduced the skew-normal density from which a cornucopia of variants have evolved. Earlier discussion of related distributions (e.g., Birnbaum, 1950) did not lead to a ﬂurry of developments. The time for such developments was ripe in the late 20th century, partially because of the ready availability of powerful computers for any simulation and numerical integration that might be required. We will survey the spectrum of hidden truncation models with special attention being paid to those involving elliptically contoured components in their development. The resulting models provide ﬂexible alternatives to the somewhat restrictive elliptically contoured distributions for modeling multivariate data. Speciﬁcally, in Section 6.2 we present various univariate skew-normal models and discuss estimation issues in Section 6.3. Other univariate skewed distributions are presented in Section 6.4. We describe multivariate skewed distributions in Section 6.5 and further generalizations in Section 6.6. Finally, we discuss skew-elliptical distributions in Section 6.7 and conclude in Section 6.8.