Estimation in a model with infinite-dimensional nuisance parameter

Let X₁ be a random variable with density function f(t), Ψ(t) be an increasing absolutely continuous function, Φ(t) be the inverse function to Ψ(t), and X₂ be the random variable X₂ = Φ(X₁). We consider the maximum likelihood estimator for the density ψ of the function Ψ in the case when we observe two independent samples from the distributions of X₁ and X₂. Under appropriate conditions on the involved distributions, we prove the consistency of the maximum likelihood estimator. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 160–165. An erratum to this article is available at .