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Smallest covering regions and highest density regions for discrete distributions

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  • Ben O’Neill

    (Australian National University)

Abstract

This paper examines the problem of computing a canonical smallest covering region for an arbitrary discrete probability distribution. This optimisation problem is similar to the classical 0–1 knapsack problem, but it involves optimisation over a set that may be countably infinite, raising a computational challenge that makes the problem non-trivial. To solve the problem we present theorems giving useful conditions for an optimising region and we develop an iterative one-at-a-time computational method to compute a canonical smallest covering region. We show how this can be programmed in pseudo-code and we examine the performance of our method. We compare this algorithm with other algorithms available in statistical computation packages to compute HDRs. We find that our method is the only one that accurately computes HDRs for arbitrary discrete distributions.

Suggested Citation

  • Ben O’Neill, 2022. "Smallest covering regions and highest density regions for discrete distributions," Computational Statistics, Springer, vol. 37(3), pages 1229-1254, July.
    • Handle: RePEc:spr:compst:v:37:y:2022:i:3:d:10.1007_s00180-021-01172-6
    DOI: 10.1007/s00180-021-01172-6
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    References listed on IDEAS

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    1. Lu Tian & Rui Wang & Tianxi Cai & Lee-Jen Wei, 2011. "The Highest Confidence Density Region and Its Usage for Joint Inferences about Constrained Parameters," Biometrics, The International Biometric Society, vol. 67(2), pages 604-610, June.
    2. Kim, Jae H. & Fraser, Iain & Hyndman, Rob J., 2011. "Improved interval estimation of long run response from a dynamic linear model: A highest density region approach," Computational Statistics & Data Analysis, Elsevier, vol. 55(8), pages 2477-2489, August.
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    4. Jing Lei & James Robins & Larry Wasserman, 2013. "Distribution-Free Prediction Sets," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 278-287, March.
    5. Silvano Martello & David Pisinger & Paolo Toth, 1999. "Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 45(3), pages 414-424, March.
    6. Martello, Silvano & Pisinger, David & Toth, Paolo, 2000. "New trends in exact algorithms for the 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 123(2), pages 325-332, June.
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