This paper addresses the deconvolution problem of estimating a square-integrable probability density from observations contaminated with additive measurement errors having a known density. The estimator begins with a density estimate of the contaminated observations and minimizes a reconstruction error penalized by an integrated squared m-th derivative. Theory for deconvolution has mainly focused on kernel- or wavelet-based techniques, but other methods including spline-based techniques and this smoothnesspenalized estimator have been found to outperform kernel methods in simulation studies. This paper fills in some of these gaps by establishing asymptotic guarantees for the smoothness-penalized approach. Consistency is established in mean integrated squared error, and rates of convergence are derived for Gaussian, Cauchy, and Laplace error densities, attaining some lower bounds already in the literature. The assumptions are weak for most results; the estimator can be used with a broader class of error densities than the deconvoluting kernel. Our application example estimates the density of the mean cytotoxicity of certain bacterial isolates under random sampling; this mean cytotoxicity can only be measured experimentally with additive error, leading to the deconvolution problem. We also describe a method for approximating the solution by a cubic spline, which reduces to a quadratic program.
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Bivariate Kernel Deconvolution with Panel Data
We consider estimation of the density of a multivariate response, that is not observed directly but only through measurements contaminated by additive error. Our focus is on the realistic sampling case of bivariate panel data (repeated contaminated bivariate measurements on each sample unit) with an unknown error distribution. Several factors can affect the performance of kernel deconvolution density estimators, including the choice of the kernel and the estimation approach of the unknown error distribution. As the choice of the kernel function is critically important, the class of flat-top kernels can have advantages over more commonly implemented alternatives. We describe different approaches for density estimation with multivariate panel responses, and investigate their performance through simulation. We examine competing kernel functions and describe a flat-top kernel that has not been used in deconvolution problems. Moreover, we study several nonparametric options for estimating the unknown error distribution. Finally, we also provide guidelines to the numerical implementation of kernel deconvolution in higher sampling dimensions.
more » « less- NSF-PAR ID:
- 10224320
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Sankhya B
- ISSN:
- 0976-8386
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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