Reduction algorithm for the NPMLE for the distribution function of bivariate interval censored data

We study computational aspects of the nonparametric maximum likelihood estimator (NPMLE) for the distribution function of bivariate interval censored data. The computation of the NPMLE consists of two steps: a parameter reduction step and an optimization step. In this paper we focus on the reduction step. We introduce two new reduction algorithms: the Tree algorithm and the HeightMap algorithm. The Tree algorithm is only mentioned briefly. The HeightMap algorithm is discussed in detail and also given in pseudo code. It is a very fast and simple algorithm of time complexity O(n^2). This is an order faster than the best known algorithm thus far, the O(n^3) algorithm of Bogaerts and Lesaffre (2003). We compare our algorithms with the algorithms of Gentleman and Vandal (2001), Song (2001) and Bogaerts and Lesaffre (2003), using simulated data. We show that our algorithms, and especially the HeightMap algorithm, are significantly faster. Finally, we point out that the HeightMap algorithm can be easily generalized to d-dimensional data with d>2. Such a multivariate version of the HeightMap algorithm has time complexity O(n^d).