Nonparametric curve estimation under monotonicity constraint

A finite sample comparison is carried out for three recent nonparametric methodologies in estimating the mono-tone regression function F and its inverse F−1. The methods are (1) the inverse kernel method DNP (Dette et al. (2005), Dette and Scheder (2010)), (2) the monotone spline (Kong and Eubank (2006)) and (3) the data adaptive method NAM (Bhattacharya and Lin (2010), (2011)), with roots in isotonic regression (Ayer et al. (1955), Bhat-tacharya and Kong (2007)). Comparative studies are carried out mainly in terms of length of the confidence interval in this article. Let there be m distinct values of the independent variable x among N observations y. The results show that if m is relatively small compared to N then generally the NAM performs best, while the DNP outperforms the other methods when m is O(N) unless there is a substantial clustering of the values of the independent variable x