Estimation of a k-monotone density , part 2 : algorithms for computation and numerical results

The iterative (2k − 1)−spline algorithm is an extension of the iterative cubic spline algorithm developed and used by Groeneboom, Jongbloed, and Wellner (2001b) to compute the Least Squares Estimator (LSE) of a nonincreasing and convex density on (0,∞), and to find an approximation of the “invelope” of the integrated two-sided Brownian motion+t4 that is involved in the limiting distribution of both the Maximum Likelihood Estimator (MLE) and the LSE (Groeneboom, Jongbloed, and Wellner (2001a)). The iterative (2k − 1)− spline algorithm was developed to compute the LSE of a k-monotone density on (0,∞) for any integer k > 2, and also to calculate an approximation of the envelopes (“ invelopes”) of the (k − 1)-fold integral of two-sided Brownian motion + (k!/(2k)!) t2k when k is odd (even) on a finite interval [−c, c] for some fixed c > 0. Existence and uniqueness of the latter processes are the subject of Balabdaoui and Wellner (2004c). To compute the MLE of a k-monotone density, another variation of the algorithm involving quadratic approximation is described. This algorithm involves the computation of a spline of degree k− 1 instead of a spline of degree 2k− 1. The principles of both algorithms are explained in detail. We also give several applications to real and artificial data. 1 Research supported in part by National Science Foundation grant DMS-0203320 2 Research supported in part by National Science Foundation grants DMS-0203320, and NIAID grant 2R01 AI291968-04 AMS 2000 subject classifications. Primary: 62G07; secondary 90C99.

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