Nonparametric function estimation subject to monotonicity, convexity and other shape constraints

• Published in: Journal of econometrics Vol. 161; no. 2; pp. 166 - 181
• Main Authors: Shively, Thomas S., Walker, Stephen G., Damien, Paul
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2011; Elsevier; Elsevier Sequoia S.A
• Series: Journal of Econometrics
• Subjects: Algorithms; Applications; Bayesian analysis; Bayesian method; Econometrics; Estimating techniques; Estimation; Exact sciences and technology; Fixed-knot splines; Fixed-knot splines Free-knot splines Log-concave likelihood functions MCMC sampling algorithm Small sample properties; Free-knot splines; Insurance, economics, finance; Log-concave likelihood functions; Mathematics; MCMC sampling algorithm; Model testing; Monte Carlo simulation; Nonparametric inference; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Parametric inference; Probability and statistics; Regression analysis; Sampling; Sciences and techniques of general use; Significance tests; Simulation; Small sample properties; Statistical models; Statistics; Studies; MCMC sampling algorithm; C11—Bayesian analysis; Small sample properties; Fixed-knot splines; Log-concave likelihood functions; C14—Semiparametric and nonparametric methods; Free-knot splines; Shape function; methods; Non parametric method; Non parametric estimation; Spline approximation; Numerical approximation; Markov chain; C14-Semiparametric and nonparametric; C11-Bayesian analysis; Smooth function; Log likelihood; Constraint theory; Regression spline; Convexity; Sampling; Likelihood function; Bayes estimation; Monte Carlo method; Gauss method; Statistical estimation; Constrained estimation; Semiparametric method; Algorithm; Concave function; Statistical method; Function estimation; Numerical analysis; Simulation; Econometrics
• ISSN: 0304-4076; 1872-6895
• DOI: 10.1016/j.jeconom.2010.12.001

This paper uses free-knot and fixed-knot regression splines in a Bayesian context to develop methods for the nonparametric estimation of functions subject to shape constraints in models with log-concave likelihood functions. The shape constraints we consider include monotonicity, convexity and functions with a single minimum. A computationally efficient MCMC sampling algorithm is developed that converges faster than previous methods for non-Gaussian models. Simulation results indicate the monotonically constrained function estimates have good small sample properties relative to (i) unconstrained function estimates, and (ii) function estimates obtained from other constrained estimation methods when such methods exist. Also, asymptotic results show the methodology provides consistent estimates for a large class of smooth functions. Two detailed illustrations exemplify the ideas.