Geodesic
Estimation in a model with infinite dimensional nuisance parameter
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 8, Tome 320 (2004), pp. 160-165 
V. N. Solev; F. Haghighi. Estimation in a model with infinite dimensional nuisance parameter. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 8, Tome 320 (2004), pp. 160-165. http://geodesic.mathdoc.fr/item/ZNSL_2004_320_a12/
@article{ZNSL_2004_320_a12,
     author = {V. N. Solev and F. Haghighi},
     title = {Estimation in a~model with infinite dimensional nuisance parameter},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {160--165},
     year = {2004},
     volume = {320},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_320_a12/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $X_1$ be a random variable with density function $f(t)$, $\Psi(t)$ be an increasing absolutely continuous function, $\Phi(t)$ be the inverse function, random variable $X_2$ be defined by $X_2=\Phi(X_1)$. We consider the maximum likelihood estimator for density $\psi$ of function $\Psi$ as we observe two independent samples from the distribution of $X_1$ and $X_2$. Under appropriate conditions on the involved distributions, we prove the consistency of maximum likelihood estimator.

[1] Wing Hung Wong, N. Xiatong Shen, “Probability inequalities and convergence rates of sieve mles”, The Annals of Statistics, 2 (1995), 2555–2591