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ESTIMATES OF DERIVATIVES OF (LOG) DENSITIES AND RELATED OBJECTS

Published online by Cambridge University Press:  27 December 2021

Joris Pinkse Open the ORCID record for Joris Pinkse [Opens in a new window]  and
Karl Schurter Open the ORCID record for Karl Schurter [Opens in a new window]
Joris Pinkse
Affiliation:
Pennsylvania State University
Karl Schurter*
Affiliation:
Pennsylvania State University
*
Address correspondence to Karl Schurter, Department of Economics, Pennsylvania State University, University Park, PA, USA; e-mail: [email protected].
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Abstract

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We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative 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 or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 2 , April 2023 , pp. 321 - 356
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

REFERENCES

Andrews, D. W. (1995) Nonparametric kernel estimation for semiparametric models. Econometric Theory 11, 560–586.CrossRefGoogle Scholar
Bell, E. T. (1927) Partition polynomials. Annals of Mathematics 29, 38–46.CrossRefGoogle Scholar
Bouezmarni, T. & Scaillet, O. (2005) Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data. Econometric Theory 21, 390–412.Google Scholar
Cattaneo, M. D., Jansson, M., & Ma, X. (2020) Simple local polynomial density estimators. Journal of the American Statistical Association 115, 1449–1455.CrossRefGoogle Scholar
Cheng, M.-Y., Fan, J., & Marron, J. S. (1997) On automatic boundary corrections. Annals of Statistics 25, 1691–1708.CrossRefGoogle Scholar
Eicker, F. (1966) A multivariate central limit theorem for random linear vector forms. Annals of Mathematical Statistics 36, 1825–1828.CrossRefGoogle Scholar
Gasser, T. & Müller, H.-G. (1979) Kernel estimation of regression functions. In Smoothing Techniques for Curve Estimation , pp. 23–68. Springer.CrossRefGoogle Scholar
Guerre, E., Perrigne, I., & Vuong, Q. (2000) Optimal nonparametric estimation of first–price auctions. Econometrica 68, 525–574.CrossRefGoogle Scholar
Hickman, B. R. & Hubbard, T. P. (2015) Replacing sample trimming with boundary correction in nonparametric estimation of first-price auctions. Journal of Applied Econometrics 30, 739–762.CrossRefGoogle Scholar
Hirano, K., Imbens, G. W., & Ridder, G. (2003) Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71, 1161–1189.CrossRefGoogle Scholar
Hjort, N. L. & Jones, M. C. (1996) Locally nonparametric density estimation. Annals of Statistics 24, 1619–1647.CrossRefGoogle Scholar
Jones, M. & Foster, P. (1996) A simple nonnegative boundary correction method for kernel density estimation. Statistica Sinica 6, 1005–1013.Google Scholar
Karunamuni, R. J. & Alberts, T. (2005) On boundary correction in kernel density estimation. Statistical Methodology 2, 191–212.CrossRefGoogle Scholar
Karunamuni, R. J. & Zhang, S. (2008) Some improvements on a boundary corrected kernel density estimator. Statistics and Probability Letters 78, 499–507.CrossRefGoogle Scholar
Klein, R. W. & Spady, R. H. (1993) An efficient semiparametric estimator for binary response models. Econometrica 61, 387–421.CrossRefGoogle Scholar
Lejeune, M. & Sarda, P. (1992) Smooth estimators of distribution and density functions. Computational Statistics & Data Analysis 14, 457–471.CrossRefGoogle Scholar
Lewbel, A. & Schennach, S. M. (2007) A simple ordered data estimator for inverse density weighted expectations. Journal of Econometrics 136, 189–211.CrossRefGoogle Scholar
Loader, C. R. (1996). Local likelihood density estimation. Annals of Statistics 24, 1602–1618.CrossRefGoogle Scholar
Pinkse, J. & Schurter, K. (2019). Estimation of Auction Models with Shape Restrictions. Working paper. Pennsylvania State University.Google Scholar
Zhang, S. & Karunamuni, R. J. (1998) On kernel density estimation near endpoints. Journal of Statistical Planning and Inference 70, 301–316.CrossRefGoogle Scholar