Plug-in L₂-upper error bounds in deconvolution, for a mixing density estimate in R^d and for its derivatives, via the L₁-error for the mixture

ABSTRACT

In deconvolution in $R^d,d\ge 1,$ with mixing density $p(\in \mathcal{P})$ and kernel h, the mixture density $f_p(\in {\mathcal{F}}_p)$ is estimated with MDE $f_{{\hat{p}}_n},$ having upper $L_1$-error rate, $a_n,$ in probability or in risk; ${\hat{p}}_n\in \mathcal{P}$. In one application, $\mathcal{P}$ consists of $L_1$-separable densities in R with differences changing sign at most J times and $h(x-y)$ Totally Positive. When h is known and p is $\tilde{q}$-smooth, vanishing outside a compact in $R^d,$ plug-in upper bounds are provided for the $L_2$-error rate of ${\hat{p}}_n$ and its $\left[s\right]$-th mixed partial derivative ${\hat{p}}_n^{\left(s\right)},$ via $\parallel f_{{\hat{p}}_n}-f_p{\parallel }_1$, with rates $(\log a_n^{-1}{)}^{-N_1}$ and $a_n^{N_2}$, respectively, for h super-smooth and smooth; $\tilde{q}\in R^{+},\left[s\right]\le \tilde{q},d\ge 1,$ $N_1>0,N_2>0$. For $a_n\sim (\log n{)}^{\zeta }\cdot n^{-\delta },$ the former rate is optimal for any $\delta >0$ and the latter misses the optimal by the factor $(\log n{)}^{\xi }$ when $\delta =.5;\zeta >0,\xi >0$. $N_1$ and $N_2$ appear in optimal rates and lower error and risk bounds in the deconvolution literature.

KEYWORDS:

• Deconvolution
• minimum distance estimation
• plug-in upper error/risk bounds
• totally positive kernels
• Vapnik–Chervonenkis classes

AMS SUBJECT CLASSIFICATIONS:

• 62G07
• 62G20

Acknowledgments

Many thanks are due to Professor Alexander Meister, Editor, the AE and a referee for comments and suggestions that improved the presentation of this work. In particular, very many thanks are due to the second referee, for comments and suggestions that have led to the improvement of the content of the paper and whose deep knowledge in Mathematical Statistics and sharp thinking made the refereeing process fun! Special thanks are also due to Miss Eleni Yatracos for improvements in the English writing. Many thanks, as usual, are due to the Department of Statistics and Applied Probability, National University of Singapore, for the warm hospitality during my summer visits where most of these results were obtained.

Disclosure statement

No potential conflict of interest was reported by the author.