Penalized hyperbolic-polynomial splines

With the aim of generalizing P-splines, we here define a special type of penalized splines, called HP-splines, where polynomial splines are replaced by the richer class of hyperbolic-polynomial splines and a suitably tailored discrete penalty term is used. Hyperbolic-polynomial splines, important in several applications, are a natural generalization of polynomial splines consisting of piecewise-defined functions with segments spanned by ‘atoms'of type xreαx where r=0,…,ℓ and α∈R. HP-splines, that reduce to P-splines for α=0, are more suitable to data with an exponential trend which is frequent in applications.