Statistics and Its Interface

Volume 13 (2020)

Number 1

Copula Modeling for Data with Ties

Pages: 103 – 117



Yan Li (Department of Statistics, University of Connecticut, Mansfield, Ct., U.S.A.)

Yang Li (School of Statistics, Center for Applied Statistics, Renmin University of China, Beijing, China)

Yichen Qin (Department of Operations, Business Analytics and Information Systems, University of Cincinnati, Ohio, U.S.A.)

Jun Yan (Department of Statistics, University of Connecticut, Mansfield, Ct., U.S.A.)


Tied observations in copula modeling may cause serious problems to rank-based inference methods that are intended for data with no ties. Simple methods such as breaking the ties at random or using midrank could lead to bias in estimation and invalidity in naive bootstrap inferences. We propose to treat the ranks of tied observations as being interval censored and estimate the copula parameters by maximizing a pseudo-likelihood based on interval censored pseudo-observations. A parametric bootstrap procedure that preserves the tied ranks in the observed data is adapted to do interval estimation and goodness-of-fit test. The proposed approach is shown to be very competitive in comparison to the simple treatments in a large scale simulation study. The utility of the method is illustrated in real data examples.


interval censored data, multivariate distribution, pseudo-observations, rank-based method

J. Yan’s research was partially supported by an NSF grant (DMS 1521730).

Y. Li’s research was partially supported by the National Natural Science Foundation of China (No. 71771211).

Received 30 January 2019

Accepted 6 September 2019

Published 7 November 2019