Tikhonov’s Regularization to the Deconvolution Problem

We are interested in estimating a density function f of i.i.d. random variables X1, …, Xn from the model Yj = Xj + Zj, where Zj are unobserved error random variables, distributed with the density function g and independent of Xj. This problem is known as the deconvolution problem in nonparametric statistics. The most popular method of solving the problem is the kernel one in which, we assume gft(t) ≠ 0, for all t∈R$t\in \mathbb {R}$, where gft(t) is the Fourier transform of g. The more general case in which gft(t) may have real zeros has not been considered much. In this article, we will consider this case. By estimating the Lebesgue measure of the low level sets of gft and combining with the Tikhonov regularization method, we give an approximation fn to the density function f and evaluate the rate of convergence of supg∈Gs0,γ,M,Tsupf∈Fq,KEfn-fL2R2$\mathop {\sup }\limits _{g \in {\mathcal {G}_{{s_0},\gamma,M,T}}} {\mathop {\sup }\limits _{f \in {\mathcal {F}_{q,K}}} \mathbb {E}\left\Vert {{f_n} - f} \right\Vert _{{L^2}\left(\mathbb {R} \right)}^2}$. A lower bound for this quantity is also provided.