## Multiscale inference for multivariate deconvolution.(English)Zbl 1380.62143

The multivariate statistical model for density deconvolution is defined by: $$Y_{i}=Z_{i}+\epsilon_{i}$$, $$i=1,\dots ,n$$, where the pairs $$(Z_{i},\epsilon_{i}) \in \mathbb{R}^{d}\times \mathbb{R}^{d}$$ are independent and identically distributed random variables. Moreover, it is assumed that the noise terms $$\epsilon_{i}$$ are distributed according to a known density, say $$f_{\epsilon}$$, and that they are independent of all the $$Z_{i}$$. In this statistical frame, based on the sample $$Y_{1},\dots ,Y_{n}$$, the interest is focused on the properties of the density, say $$f$$, of $$Z_{i}$$. To be more specific, in this paper, initially, a method is developed which detects regions of monotonicity of $$f$$ at a controlled level. The statistical inference regarding the monotonicity properties of $$f$$ is performed based on simultaneous local tests of the directional derivatives of the density $$f$$ for a significant deviation from zero at arbitrary points in arbitrary directions. The multiple level of these statistics is controlled by investigating the asymptotic properties of the maximum of appropriately normalized statistics. Afterwards, as an application, a significance test for the presence of a local maximum at a pre-specified point is proposed. Finally, the finite sample properties are illustrated by means of a small simulation.

### MSC:

 62G07 Density estimation 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 62H10 Multivariate distribution of statistics
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