Fast and flexible methods for monotone polynomial fitting

We investigate an isotonic parameterization for monotone polynomials previously unconsidered in the statistical literature. We show that this parameterization is more flexible than its alternatives through enabling polynomials to be constrained to be monotone over either a compact interval or a semi-compact interval of the form [a,∞] , in addition to over the whole real line. Furthermore, algorithms based on our new parameterization estimate the fitted monotone polynomials much faster than algorithms based on previous isotonic parameterizations which in turn makes the use of standard bootstrap methodology feasible. We investigate the use of the bootstrap under monotonicity constraints to obtain confidence bands for the fitted curves and show that an adjustment by using either the ‘m out of n’ bootstrap or a post hoc symmetrization of the confidence bands is necessary to achieve more uniform coverage probabilities. We illustrate our new methodology with two real world examples which demonstrate not only the need for such techniques, but how restricting the monotonicity constraints to be over either a compact or semi-compact interval allows the fitting of even degree monotone polynomials. We also describe methods for using the ‘m out of n’ bootstrap to select the degree of the fitted monotone polynomial. All algorithms discussed in this paper are available in the R package MonoPoly (version 0.3-6 or later).