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Estimation of convolution in the model with noise

机译:带有噪声的模型中卷积的估计

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We investigate the estimation of the l-fold convolution of the density of an unobserved variable X from n i.i.d. observations of the convolution model Y = X + epsilon. We first assume that the density of the noise e is known and define non-adaptive estimators, for which we provide bounds for the mean integrated squared error. In particular, under some smoothness assumptions on the densities of X and e, we prove that the parametric rate of convergence 1 can be attained. Then, we construct an adaptive estimator using a penalisation approach having similar performances to the non-adaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.
机译:我们调查了从n i.d.未观察到的变量X的密度的1倍卷积的估计。卷积模型Y的观测值= X + epsilon。我们首先假设噪声的密度e是已知的,并定义了非自适应估计量,为此我们为均值平方平方误差提供了界限。特别地,在关于X和e的密度的一些平滑性假设下,我们证明可以达到参数收敛速度1 / n。然后,我们使用具有与非自适应性能相似的性能的惩罚方法构造自适应估计器。其适应性的价格是对数项。在有独立噪声样本的条件下,结果扩展到未知噪声密度的情况。最后,我们报告了一项仿真研究,以支持我们的理论发现。

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