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Hermite series estimates of a probability density and its derivatives

Author

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  • Greblicki, W?odzimierz
  • Pawlak, Miros?aw

Abstract

The following estimate of the pth derivative of a probability density function is examined: [Sigma]k = 0Nâkhk(x), where hk is the kth Hermite function and âk = ((-1)p/n)[Sigma]i = 1n hk(p)(Xi) is calculated from a sequence X1,..., Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n-[alpha]) and O(n-[alpha] log n), respectively, where [alpha] = 2(r - p)/(2r + 1). Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n-[beta]) and O(n-[beta]/2 log n), respectively, where [beta] = (2(r - p) - 1)/(2r + 1).

Suggested Citation

  • Greblicki, W?odzimierz & Pawlak, Miros?aw, 1984. "Hermite series estimates of a probability density and its derivatives," Journal of Multivariate Analysis, Elsevier, vol. 15(2), pages 174-182, October.
    • Handle: RePEc:eee:jmvana:v:15:y:1984:i:2:p:174-182
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    Cited by:

    1. Foster, Joshua, 2022. "Semi-nonparametric estimation of secret reserve prices in auctions," Economics Letters, Elsevier, vol. 220(C).
    2. Stephanou, Michael & Varughese, Melvin, 2021. "Sequential estimation of Spearman rank correlation using Hermite series estimators," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    3. Michael Stephanou & Melvin Varughese, 2021. "On the properties of hermite series based distribution function estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(4), pages 535-559, May.

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