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Optimal Rates of Convergence for Nonparametric Deconvolution Problems

The optimal rates of convergence for nonparametric deconvolution problems depend on the smoothness of error distributions. According to the research, there are two types of optimal rates of convergence:

* Ordinary Smooth Error Distributions: For these cases, the optimal rate of convergence is achieved by deconvolution kernel density estimators.
* Supersmooth Error Distributions: In this case, the optimal rate of convergence can also be achieved by deconvolution kernel density estimators.

The difficulty of deconvolution problems increases with the smoothness of error distributions. The research paper states that "the smoother, the harder" (Source: On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems).

Reference:

* Fan, Jianqing (1991). "On the optimal rates of convergence for nonparametric deconvolution problems". The Annals of Statistics. 19 (3): 1257–1272.

Last Updated: February 22, 2026
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icon https://doi.org/10.1214%2Faos%2F1176348248

On the Optimal Rates of Convergence for Nonparametric Deconvoluti...

Deconvolution problems arise in a variety of situations in statistics. An interesting problem is to estimate the density $f$ of a random variable $X$ based on $n$ i.i.d. observations from $Y = X + \va...
icon https://en.wikipedia.org/wiki/Smoothness_%28probability_theory%29

Smoothness (probability theory) - Wikipedia

normal. Fan, Jianqing (1991). "On the optimal rates of convergence for nonparametric deconvolution problems". The Annals of Statistics. 19 (3): 1257–1272