Summary
A kernel density estimator, constructed from a combination of disaggregate data subject to sampling bias and aggregate data, is described. The asymptotic performance of the estimator is explored, and details of an algorithm for its implementation are given. The issue of bandwidth selection is addressed. Use of the estimator is illustrated through two examples. The first involves simulated data while the second example concerns traffic speed data collected by automatic vehicle detectors on Interstate 5 near Seattle.
Similar content being viewed by others
References
Coifman, B. (2001), “Improved velocity estimation using single loop detectors”, Transportation Research 35A, 863–880.
Dailey, D. (1999), “A statistical algorithm for estimating speed from single loop volume and occupancy measurements”, Transportation Research 33B, 313–322.
Hall, P. (1987), “On Kullback-Leibler loss and density estimation”, Annals of Statistics 15, 1491–1519.
Hall, P. & Turlach, B. (1999), “Reducing bias in curve estimation by use of weights”, Computational Statistics & Data Aanlysis 30, 67–86.
Hazelton, M. (2003), “Estimating vehicle speed from traffic count and occupancy data”, Submitted for publication.
Ihaka, R. & Gentleman, R. (1996), “R: A language for data analysis and graphics”, Journal of Computational and Graphical Statistics 5, 299–314.
Jones, M. (1991), “Kernel density estimation for length biased data”, Biometrika 78, 511–519.
Padgett, W. & McNichols, D. (1984), “Nonparametric density estimation from censored data”, Communications in Statistics A: Theory and Methods 13, 1581–1611.
Sheather, S. & Jones, M. (1991), “A reliable data-based bandwidth selection method for kernel density estimation”, Journal of the Royal Statistical Society, Series B 53, 683–690.
Simonoff, J. (1996), Smoothing Methods in Statistics, Springer, New York.
Stefanski, L. & Carroll, R. J. (1990), “Deconvoluting kernel density estimators”, Statistics 21, 169–184.
Wand, M. & Jones, M. (1995), Kernel Smoothing, Chapman & Hall, London.
White, H. (1982), “Maximum likelihood estimation of misspecified models”, Econometrica 50, 1–25.
Acknowledgments
The helpful comments of two anonymous referees are gratefully acknowledged. The author also acknowledges useful discussions with Dr. Berwin Turlach.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hazelton, M. Density Estimation from Aggregate Data. CompStat 19, 407–423 (2004). https://doi.org/10.1007/BF03372104
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03372104