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Estimates of derivatives of (log) densities and related objects

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  • Joris Pinkse
  • Karl Schurter

Abstract

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density $f$. The estimator is guaranteed to be nonnegative and achieves the same optimal rate of convergence in the interior as well as the boundary of the support of $f$. The estimator is therefore well-suited to applications in which nonnegative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly in finite samples to these alternative methods when they are used with optimal inputs, i.e. an Epanechnikov kernel and optimally chosen bandwidth sequence. Further simulation evidence demonstrates that, if the researcher modifies the inputs and chooses a larger bandwidth, our approach can even improve upon these optimized alternatives, asymptotically. We provide code in several languages.

Suggested Citation

  • Joris Pinkse & Karl Schurter, 2020. "Estimates of derivatives of (log) densities and related objects," Papers 2006.01328, arXiv.org.
  • Handle: RePEc:arx:papers:2006.01328
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    References listed on IDEAS

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    1. Lejeune, Michel & Sarda, Pascal, 1992. "Smooth estimators of distribution and density functions," Computational Statistics & Data Analysis, Elsevier, vol. 14(4), pages 457-471, November.
    2. Karunamuni, R.J. & Zhang, S., 2008. "Some improvements on a boundary corrected kernel density estimator," Statistics & Probability Letters, Elsevier, vol. 78(5), pages 499-507, April.
    3. Matias D. Cattaneo & Michael Jansson & Xinwei Ma, 2020. "Simple Local Polynomial Density Estimators," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(531), pages 1449-1455, July.
    4. Lewbel, Arthur & Schennach, Susanne M., 2007. "A simple ordered data estimator for inverse density weighted expectations," Journal of Econometrics, Elsevier, vol. 136(1), pages 189-211, January.
    5. Keisuke Hirano & Guido W. Imbens & Geert Ridder, 2003. "Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score," Econometrica, Econometric Society, vol. 71(4), pages 1161-1189, July.
    6. Joris Pinkse & Karl Schurter, 2019. "Estimation of Auction Models with Shape Restrictions," Papers 1912.07466, arXiv.org.
    7. Brent R. Hickman & Timothy P. Hubbard, 2015. "Replacing Sample Trimming with Boundary Correction in Nonparametric Estimation of First‐Price Auctions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 30(5), pages 739-762, August.
    8. Klein, Roger W & Spady, Richard H, 1993. "An Efficient Semiparametric Estimator for Binary Response Models," Econometrica, Econometric Society, vol. 61(2), pages 387-421, March.
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