Abstract
The present paper studies density deconvolution in the presence of small Berkson errors, in particular, when the variances of the errors tend to zero as the sample size grows. It is known that when the Berkson errors are present, in some cases, the unknown density estimator can be obtained by simple averaging without using kernels. However, this may not be the case when Berkson errors are asymptotically small. By treating the former case as a kernel estimator with the zero bandwidth, we obtain the optimal expressions for the bandwidth. We show that the density of Berkson errors acts as a regularizer, so that the kernel estimator is unnecessary when the variance of Berkson errors lies above some threshold that depends on the shapes of the densities in the model and the number of observations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Berkson, “Are There Two Regressions?”, J. Amer. Statist. Assoc. 45, 164–180 (1950).
J. Bovy, J. F. Hennawi, D. W. Hogg, A. D. Myers, J. A. Kirkpatrick, D. J. Schlegel, N. P. Ross, E. S. Sheldon, I. D. McGreer, D. P. Schneider, and B. A. Weaver, “Think outside the color box: Probabilistic target selection and the SDSS-XDQSO Quasar targeting catalog”, The Astrophysical Journal 729(2), #141 (2011).
R. J. Carroll, D. Rupport, L. A. Stefanski, and C. M. Crainiceanu, Measurement Error in Nonlinear Models, A Modern Perspective (Chapman and Hall, New York, 2006), 2nd ed..
R. J. Carroll, A. Delaigle, and P. Hall, “Nonparametric Prediction in Measurement Error Models”, J. Amer. Statist. Assoc. 104, 993–1003 (2009).
F. Comte and J. Kappus, “Density Deconvolution from Repeated Measurements without Symmetry Assumption on the Errors”, J. Multivar. Anal. 140, 31–46 (2015).
A. Delaigle, “Nonparametric Density Estimation from Data with a Mixture of Berkson and Classical Errors”, Can. J. Statist. 35, 89–104 (2007).
A. Delaigle, “An Alternative View of the Deconvolution Problem”, Statist. Sinica 18, 1025–1045 (2008).
R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic Press Inc., London-New York, 1973).
L. Du, C. Zou, and Z. Wang, “Nonparametric Regression Function Estimation for Errors-in-variables Models with Validation Data”, Statist. Sinica 21, 1093–1113 (2011).
P. Geng and H. L. Koul, “Minimum Distance Model Checking in Berkson Measurement Error Models with Validation Data”, Test 28 (3), 879–899 (2019).
A. Goldenshluger, “On Pointwise Adaptive Nonparametric Deconvolution”, Bernoulli 5, 907–925 (1999).
A. Goldenshluger and O. Lepski, “Bandwidth Selection in Kernel Density Estimation: Oracle Inequality and Adaptive Minimax Optimality”, Ann. Statist. 39, 1608–1632 (2011).
K. H. Kim, W. K. Härdle, and S.-K. Chao, “Simultaneous Inference for the Partially Linear Model with a Multivariate Unknown Function when the Covariates are Measured with Errors”, SFB 649 Discussion Papers SFB649DP2016-024 (Humboldt Univ., Berlin, Germany, 2016).
C. Lacour and F. Comte, “Data-Driven Density Estimation in the Presence of Additive Noise with Unknown Distribution”, J. Roy. Statist. Soc., Ser. B, 73, 601–627 (2011).
O. V. Lepski and V. G. Spokoiny, “Optimal Pointwise Adaptive Methods in Nonparametric Estimation”, Ann. Statist. 25, 2512–2546 (1997).
J. P. Long, N. E. Karoui, and J. A. Rice, “Kernel Density Estimation with Berkson Error”, Canad. J. Statist. 44, 142–160 (2016).
A. Meister, Deconvolution Problems in Nonparametric Statistics (Springer, New York, 2009).
E. A. Robinson, Seismic Inversion and Deconvolution: Part B: Dual-Sensor Technology (Elsevier, Oxford, 1999).
A. B. Tsybakov, Introduction to Nonparametric Estimation (Springer, New York, 2009).
L. Wang, “Estimation of Nonlinear Berkson-type Measurement Error Models”, Statist. Sinica 13, 1201–1210 (2003).
L. Wang, “Estimation of Nonlinear Models with Berkson Measurement Errors”, Ann. Statist. 32, 2559–2579 (2004).
C. B. Wason, J. L. Black, and G. A. King, “Seismic Modeling and Inversion”, Proc. IEEE. 72, 1385–1393 (1984).
R. Wong, Asymptotic Approximations of Integrals (SIAM, Philadelphia, 2001).
Acknowledgments
Marianna Pensky and Ramchandra Rimal were partially supported by National Science Foundation (NSF), grants DMS-1407475 and DMS-1712977. The authors also thank Alexander Tsybakov for the help with the proof of Lemma 3.
Author information
Authors and Affiliations
Corresponding authors
About this article
Cite this article
Rimal, R., Pensky, M. Density Deconvolution with Small Berkson Errors. Math. Meth. Stat. 28, 208–227 (2019). https://doi.org/10.3103/S1066530719030025
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530719030025