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Keep it Tighter -- A Story on Analytical Mean Embeddings

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  • Linda Chamakh
  • Zoltan Szabo

Abstract

Kernel techniques are among the most popular and flexible approaches in data science allowing to represent probability measures without loss of information under mild conditions. The resulting mapping called mean embedding gives rise to a divergence measure referred to as maximum mean discrepancy (MMD) with existing quadratic-time estimators (w.r.t. the sample size) and known convergence properties for bounded kernels. In this paper we focus on the problem of MMD estimation when the mean embedding of one of the underlying distributions is available analytically. Particularly, we consider distributions on the real line (motivated by financial applications) and prove tighter concentration for the proposed estimator under this semi-explicit setting; we also extend the result to the case of unbounded (exponential) kernel with minimax-optimal lower bounds. We demonstrate the efficiency of our approach beyond synthetic example in three real-world examples relying on one-dimensional random variables: index replication and calibration on loss-given-default ratios and on S&P 500 data.

Suggested Citation

  • Linda Chamakh & Zoltan Szabo, 2021. "Keep it Tighter -- A Story on Analytical Mean Embeddings," Papers 2110.09516, arXiv.org, revised Nov 2024.
  • Handle: RePEc:arx:papers:2110.09516
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    References listed on IDEAS

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    1. Jorion, Philippe, 1986. "Bayes-Stein Estimation for Portfolio Analysis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 279-292, September.
    2. Chris J. Oates & Mark Girolami & Nicolas Chopin, 2017. "Control functionals for Monte Carlo integration," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 695-718, June.
    3. Niklas Pfister & Peter Bühlmann & Bernhard Schölkopf & Jonas Peters, 2018. "Kernel‐based tests for joint independence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(1), pages 5-31, January.
    4. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    5. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    6. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    7. Lotfi Boudabsa & Damir Filipović, 2019. "Machine Learning With Kernels for Portfolio Valuation and Risk Management," Swiss Finance Institute Research Paper Series 19-34, Swiss Finance Institute.
    8. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    9. Chalabi, Yohan & Wuertz, Diethelm, 2012. "Portfolio optimization based on divergence measures," MPRA Paper 43332, University Library of Munich, Germany.
    10. Lionel Martellini & Volker Ziemann, 2010. "Improved Estimates of Higher-Order Comoments and Implications for Portfolio Selection," The Review of Financial Studies, Society for Financial Studies, vol. 23(4), pages 1467-1502, April.
    11. repec:bla:jfinan:v:58:y:2003:i:4:p:1651-1684 is not listed on IDEAS
    12. Scott, Robert C & Horvath, Philip A, 1980. "On the Direction of Preference for Moments of Higher Order Than the Variance," Journal of Finance, American Finance Association, vol. 35(4), pages 915-919, September.
    13. Kim, Woo Chang & Fabozzi, Frank J. & Cheridito, Patrick & Fox, Charles, 2014. "Controlling portfolio skewness and kurtosis without directly optimizing third and fourth moments," Economics Letters, Elsevier, vol. 122(2), pages 154-158.
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