Recent progress in the nonparametric estimation of monotone curves—With applications to bioassay and environmental risk assessment

Three recent nonparametric methodologies for estimating a monotone regression function FF and its inverse F−1F−1 are (1) the inverse kernel method DNP (Dette et al., 2005 and Dette and Scheder, 2010), (2) the monotone spline (Kong and Eubank (2006)) and (3) the data adaptive method NAM (Bhattacharya and Lin, 2010 and Bhattacharya and Lin, 2011), with roots in isotonic regression (Ayer et al., 1955 and Bhattacharya and Kong, 2007). All three have asymptotically optimal error rates. In this article their finite sample performances are compared using extensive simulation from diverse models of interest, and by analysis of real data. Let there be mm distinct values of the independent variable xx among NN observations yy. The results show that if mm is relatively small compared to NN then generally the NAM performs best, while the DNP outperforms the other methods when mm is O(N)O(N) unless there is a substantial clustering of the values of the independent variable xx.