Efficient estimation using both direct and indirect observations

The Ibragimov—Khas'minskii model postulates observing $X_1,\ldots,X_m$ independent, identically distributed according to an unknown distribution $G$ and $Y_1,\ldots,Y_n$ independent and identically distributed according to $\int {k(\,\cdot\,,y)}\,dG(y)$, where $k$ is known, for example, $Y$ is obtained from $X$ by convolution with a Gaussian density. We exhibit sieve type estimates of $G$ which are efficient under minimal conditions which include those of Vardi and Zhang (1992) for the special case, $G$ on $[0,\infty]$, $k(x,y)=y^{-1}1(x\le y)$.