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摘要翻译：
我们考虑了用从$F_y=F_x\星f_{\epsilon}$中得到的样本$Y_1,...,Y_n$估计密度$F_x$的问题，其中$f_{\epsilon}$是一个未知的密度。我们假设观察到来自$F_{\epsilon}$的额外示例$\epsilon_1，...，\epsilon_m$。利用$F_y和$F_{ε}的非参数估计，并在Fourier域上应用谱截止，构造了$F_x_及其导数的估计。在误差密度为f_x$满足多项式、对数或一般源条件的情况下，我们导出了估计量的收敛速度。证明了在误差密度已知或未知的模型中，当密度$F_x$属于Sobolev空间$H_{Mathbh p}$且$F_{ε_$是普通光滑或超光滑时，所提出的估计量在极大极小意义下是渐近最优的。
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英文标题：
《Deconvolution with unknown error distribution》
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作者：
Jan Johannes
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最新提交年份：
2009
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分类信息：
一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
We consider the problem of estimating a density $f_X$ using a sample $Y_1,...,Y_n$ from $f_Y=f_X\star f_{\epsilon}$, where $f_{\epsilon}$ is an unknown density. We assume that an additional sample $\epsilon_1,...,\epsilon_m$ from $f_{\epsilon}$ is observed. Estimators of $f_X$ and its derivatives are constructed by using nonparametric estimators of $f_Y$ and $f_{\epsilon}$ and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density $f_{\epsilon}$, where it is assumed that $f_X$ satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density $f_X$ belongs to a Sobolev space $H_{\mathbh p}$ and $f_{\epsilon}$ is ordinary smooth or supersmooth.
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PDF链接：
https://arxiv.org/pdf/705.3482
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