Stat Comput (2012) 22:1059–1067 DOI 10.1007/s11222-011-9278-4 Estimating curves and derivatives with parametric penalized spline smoothing Jiguo Cao · Jing Cai · Liangliang Wang Received: 18 December 2010 / Accepted: 25 July 2011 / Published online: 23 September 2011 © Springer Science+Business Media, LLC 2011 Abstract Accurate estimation of an underlying function and its derivatives is one of the central problems in statis- tics. Parametric forms are often proposed based on the ex- pert opinion or prior knowledge of the underlying func- tion. However, these strict parametric assumptions may re- sult in biased estimates when they are not completely accu- rate. Meanwhile, nonparametric smoothing methods, which do not impose any parametric form, are quite flexible. We propose a parametric penalized spline smoothing method, which has the same flexibility as the nonparametric smooth- ing methods. It also uses the prior knowledge of the under- lying function by defining an additional penalty term using the distance of the fitted function to the assumed parametric function. Our simulation studies show that the parametric penalized spline smoothing method can obtain more accu- rate estimates of the function and its derivatives than the pe- nalized spline smoothing method. The parametric penalized spline smoothing method is also demonstrated by estimating the human height function and its derivatives from the real data. Keywords Growth curve · Nonlinear regression · Parameter cascading J. Cao ( ) · J. Cai Department of Statistics & Actuarial Science, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 e-mail: jca76@sfu.ca L. Wang Department of Statistics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 1 Introduction Intensive research has been conducted to model human growth over time, known as the human height function. Several parametric models have been widely adopted in re- search and thought to be sensible. For example, Preece and Baines proposed PB-1 model in 1978 (Preece and Baines 1978); Bock and Thissen suggested a triple logistic model in 1980 (Bock and Thissen 1980); Kanefuji and Shohoji de- rived two models in 1990 (Kanefuji and Shohoji 1990); and Jolicoeur and his co-workers proposed three different mod- els from 1988 to 1992 (Jolicoeur et al. 1988, 1992). How- ever, these parametric models may not be completely valid for the height function, because they cannot properly model the entire growth curve or the rate of change (Jolicoeur et al. 1992). To relax the parametric assumption, it is also popu- lar to apply nonparametric smoothing methods to estimate the human growth curve and its derivatives (Gasser et al. 1984, 1985; Ramsay et al. 1994; Ramsay and Silverman 2005). These nonparametric smoothing methods estimate the growth curve completely from the data without making any parametric assumptions on the growth curve. A short- coming of these methods is that they completely ignore the expert opinion or prior knowledge of the growth curve as reflected in the parametric models. Motivated by the above dilemma, a general paramet- ric penalized spline smoothing method is proposed in or- der to combine both information from the data and the prior knowledge. The parametric penalized spline smooth- ing method still estimates the underlying function non- parametrically. Different from the traditional nonparametric smoothing methods, the nonparametric function is evaluated with three terms: the first term measures the fit of the non- parametric function to the data, the second term measures