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Estimating the Probability that a Function Observed with Noise Is Convex

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  • Nanjing Jian

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14850)

  • Shane G. Henderson

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14850)

Abstract

Consider a real-valued function that can be only observed with stochastic noise at a finite set of design points within a Euclidean space. We wish to determine whether there exists a convex function that goes through the true function values at the design points. We develop an asymptotically consistent Bayesian sequential sampling procedure that estimates the posterior probability of this being true. In each iteration, the posterior probability is estimated using Monte Carlo simulation. We offer three variance reduction methods: change of measure, acceptance-rejection, and conditional Monte Carlo. Numerical experiments suggest that the conditional Monte Carlo method is preferred.

Suggested Citation

  • Nanjing Jian & Shane G. Henderson, 2020. "Estimating the Probability that a Function Observed with Noise Is Convex," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 376-389, April.
    • Handle: RePEc:inm:orijoc:v:32:y:2020:i:2:p:376-389
    DOI: 10.1287/ijoc.2018.0847
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    References listed on IDEAS

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    1. Diack, C.A.T. & Thomas-Agnan, C., 1996. "A Nonparametric Test of The Non-Convexity of Regression," Papers 976.427, Toulouse - GREMAQ.
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    4. Eunji Lim & Peter W. Glynn, 2012. "Consistency of Multidimensional Convex Regression," Operations Research, INFORMS, vol. 60(1), pages 196-208, February.
    5. Jason Abrevaya & Wei Jiang, 2005. "A Nonparametric Approach to Measuring and Testing Curvature," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 1-19, January.
    6. Gad Allon & Michael Beenstock & Steven Hackman & Ury Passy & Alexander Shapiro, 2007. "Nonparametric estimation of concave production technologies by entropic methods," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(4), pages 795-816.
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    Cited by:

    1. David J. Eckman & Shane G. Henderson, 2022. "Posterior-Based Stopping Rules for Bayesian Ranking-and-Selection Procedures," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1711-1728, May.

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