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Monotonic Estimation for Probability Distribution and Multivariate Risk Scales by Constrained Minimum Generalized Cross-Entropy

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  • Yang, Bill Huajian

Abstract

Minimum cross-entropy estimation is an extension to the maximum likelihood estimation for multinomial probabilities. Given a probability distribution {r_i }_(i=1)^k, we show in this paper that the monotonic estimates {p_i }_(i=1)^k for the probability distribution by minimum cross-entropy are each given by the simple average of the given distribution values over some consecutive indexes. Results extend to the monotonic estimation for multivariate outcomes by generalized cross-entropy. These estimates are the exact solution for the corresponding constrained optimization and coincide with the monotonic estimates by least squares. A non-parametric algorithm for the exact solution is proposed. The algorithm is compared to the “pool adjacent violators” algorithm in least squares case for the isotonic regression problem. Applications to monotonic estimation of migration matrices and risk scales for multivariate outcomes are discussed.

Suggested Citation

  • Yang, Bill Huajian, 2019. "Monotonic Estimation for Probability Distribution and Multivariate Risk Scales by Constrained Minimum Generalized Cross-Entropy," MPRA Paper 93400, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:93400
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    References listed on IDEAS

    as
    1. Potharst, R. & Feelders, A.J., 2002. "Classification Trees for Problems with Monotonicity Constraints," ERIM Report Series Research in Management ERS-2002-45-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    2. Nelson, Charles R & Siegel, Andrew F, 1987. "Parsimonious Modeling of Yield Curves," The Journal of Business, University of Chicago Press, vol. 60(4), pages 473-489, October.
    3. de Leeuw, Jan & Hornik, Kurt & Mair, Patrick, 2009. "Isotone Optimization in R: Pool-Adjacent-Violators Algorithm (PAVA) and Active Set Methods," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 32(i05).
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    maximum likelihood; cross-entropy; least squares; isotonic regression; constrained optimization; multivariate risk scales;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • C54 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Quantitative Policy Modeling
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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