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Maximum likelihood estimation of linear functionals in the inverse problem ofdeconvolution is considered. Given observations of a random sample from adistribution $P_0\equiv P_{F_0}$ indexed by a (potentiallyinfinite-dimensional) parameter $F_0$, which is the distribution of the latentvariable in a standard additive Laplace measurement error model, one wants toestimate a linear functional of $F_0$. Asymptotically efficient maximumlikelihood estimation (MLE) of integral linear functionals of the mixingdistribution $F_0$ in a convolution model with the Laplace kernel density isinvestigated. Situations are distinguished in which the functional of interestcan be consistently estimated at $n^{-1/2}$-rate by the plug-in MLE, which isasymptotically normal and efficient, in the sense of achieving the variancelower bound, from those in which no integral linear functional can be estimatedat parametric rate, which precludes any possibility for asymptotic efficiency.The $\sqrt{n}$-convergence of the MLE, valid in the case of a degenerate mixingdistribution at a single location point, fails in general, as does asymptoticnormality. It is shown that there exists no regular estimator sequence forintegral linear functionals of the mixing distribution that, when recenteredabout the estimand and $\sqrt{n}$-rescaled, is asymptotically efficient,\emph{viz}., has Gaussian limit distribution with minimum variance. One canthus only expect estimation with some slower rate and, often, with anon-Gaussian limit distribution.

On asymptotically efficient maximum likelihood estimation of linear functionals in Laplace measurement error models