Bayesian approach to non-parametric monotone function estimation

The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression s...

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Bibliographic Details
Published in:Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2009, Vol.71 (1), p.159-175
Main Authors: Shively, Thomas S., Sager, Thomas W., Walker, Stephen G.
Format: Article
Language:English
Subjects:
Asymptotic properties
Bayes estimators
Bayesian analysis
Bayesian method
Demand curves
Distribution functions
Estimating techniques
Estimation
Estimation methods
Estimators
Exact sciences and technology
General topics
Linear inference, regression
Linear regression
Markov chain Monte Carlo sampling scheme
Markovian processes
Mathematical functions
Mathematical independent variables
Mathematics
Mixture prior distributions
Monotonic functions
Monte Carlo simulation
Nonparametric tests
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Parametric inference
Probability and statistics
Quantitative analysis
Regression analysis
Regression splines
Sample properties
Sampling
Sciences and techniques of general use
Small sample properties
Statistical methods
Statistical variance
Statistics
Studies
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Summary:The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.
ISSN:1369-7412
1467-9868
DOI:10.1111/j.1467-9868.2008.00677.x