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 $[0,\infty ], k(x,y) = y^{ - 1} 1(x \leq y)$.