Testing Problem for Increasing Function in a Model with Infinite Dimensional Nuisance Parameter

We consider the next statistical problem arising, for example, in accelerated life testing. Let X_l be a random variable with density function f (t), Ψ (t) be an increasing absolutely continuous function, Φ(t)=Φ−1(t) be its inverse function, random variable X₂ be defined as X₂= Φ(X₁). In order to test whether function Ψ(t) belongs to a given parametric family when function f is completely unknown, we take two independent nonparametric estimators $${{\hat{f}}_{n}}$$of density function f and $${{\hat{g}}_{n}}$$ of density function g of X₂ and compare the function $${{\hat{g}}_{n}}(t)$$ with the function $${{\hat{f}}_{n}}(\Psi ({{\hat{\theta }}_{n}};t))\psi ({{\hat{\theta }}_{n}};t)$$ for a minimum distance estimator $${{\hat{\theta }}_{n}}$$. But at the begining we have to investigate the asymptotic behavior of the estimator $${{\hat{\theta }}_{n}}$$. We consider a parametric minimum distance estimator for ill, when we observe (with a mechanism of independent censoring) two independent samples from the distributions of X₁and X₂ respectively.