Fitting continuous bivariate distributions to data

Abstract

SUMMARY In this paper we are concerned with fitting bivariate distributions to data. Let X and Y be bivariate random variables with cumulative distribution function (CDF) F(x, y; 0), where 0 is an unknown, possibly vector-valued, parameter. To estimate 0 on the basis of an observed random sample from F(x, y; 0), we first express the predicted values of the random variables as functions of 0; then an estimate of 0 is obtained by minimizing the sum of the squares of the distances between the observed and predicted values. In the univariate case, this is straightforward because the inverse of the CDF is a single point, but in the bivariate case the inverse is a surface. We present ways for expressing the predicted values in the bivariate case as a function of 0. The idea is to use the joint and marginal CDFs as the basis for calculating the predicted values as functions of 0, thereby extending the univariate to the bivariate case. The method is illustrated by applications to several bivariate distributions such as the bivariate logistic, Pareto and exponential distributions. Simulation results indicate that the method performs well. The method is also applied to an example of real data. Finally, we briefly discuss possible extensions to the multivariate case.