TY - JOUR
T1 - Grouped Dirichlet distribution
T2 - A new tool for incomplete categorical data analysis
AU - Ng, Kai Wang
AU - Tang, Man Lai
AU - Tan, Ming
AU - Tian, Guo Liang
N1 - Funding Information:
The authors thank the Associate Editor and two referees for their helpful comments and suggestions. The research of M.L. Tang was fully supported by a grant (CUHK4371/04M) from the Research Grant Council of the Hong Kong Special Administrative Region. The research of M. Tan and G.L. Tian was supported in part by U.S. National Cancer Institute Grants CA106767 and CA119758.
PY - 2008/3
Y1 - 2008/3
N2 - Motivated by the likelihood functions of several incomplete categorical data, this article introduces a new family of distributions, grouped Dirichlet distributions (GDD), which includes the classical Dirichlet distribution (DD) as a special case. First, we develop distribution theory for the GDD in its own right. Second, we use this expanded family as a new tool for statistical analysis of incomplete categorical data. Starting with a GDD with two partitions, we derive its stochastic representation that provides a simple procedure for simulation. Other properties such as mixed moments, mode, marginal and conditional distributions are also derived. The general GDD with more than two partitions is considered in a parallel manner. Three data sets from a case-control study, a leprosy survey, and a neurological study are used to illustrate how the GDD can be used as a new tool for analyzing incomplete categorical data. Our approach based on GDD has at least two advantages over the commonly used approach based on the DD in both frequentist and conjugate Bayesian inference: (a) in some cases, both the maximum likelihood and Bayes estimates have closed-form expressions in the new approach, but not so when they are based on the commonly-used approach; and (b) even if a closed-form solution is not available, the EM and data augmentation algorithms in the new approach converge much faster than in the commonly-used approach.
AB - Motivated by the likelihood functions of several incomplete categorical data, this article introduces a new family of distributions, grouped Dirichlet distributions (GDD), which includes the classical Dirichlet distribution (DD) as a special case. First, we develop distribution theory for the GDD in its own right. Second, we use this expanded family as a new tool for statistical analysis of incomplete categorical data. Starting with a GDD with two partitions, we derive its stochastic representation that provides a simple procedure for simulation. Other properties such as mixed moments, mode, marginal and conditional distributions are also derived. The general GDD with more than two partitions is considered in a parallel manner. Three data sets from a case-control study, a leprosy survey, and a neurological study are used to illustrate how the GDD can be used as a new tool for analyzing incomplete categorical data. Our approach based on GDD has at least two advantages over the commonly used approach based on the DD in both frequentist and conjugate Bayesian inference: (a) in some cases, both the maximum likelihood and Bayes estimates have closed-form expressions in the new approach, but not so when they are based on the commonly-used approach; and (b) even if a closed-form solution is not available, the EM and data augmentation algorithms in the new approach converge much faster than in the commonly-used approach.
KW - Beta-Liouville distribution
KW - Data augmentation
KW - Dirichlet distribution
KW - EM algorithm
KW - Grouped Dirichlet distribution
KW - Incomplete categorical data
UR - http://www.scopus.com/inward/record.url?scp=38349144584&partnerID=8YFLogxK
U2 - 10.1016/j.jmva.2007.01.010
DO - 10.1016/j.jmva.2007.01.010
M3 - Journal article
AN - SCOPUS:38349144584
SN - 0047-259X
VL - 99
SP - 490
EP - 509
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
IS - 3
ER -