Estimating mixing density from a system of observations

Abstract

Let Y11,…Y1m,Y21…,Yn1,…Ynm be observations such that Yij=Xi+εij, where xi are i.i.d. unobservable random variables with density f and εij are independent measurement errors, with density fε. First, this paper study the estimation of f from the system of observations {Yij}n,m i=1,j=1 when fε is known. Upper bounds on the rate of quadratic mean convergence are obtained. The upper bounds depend on the geometry of the problem and on the rate of growing of m as function of n. Then, we study two special situations inside the system of observations. The first case is when a parameter of the characteristic function of ε is unknown. The second case corresponds to εij non identically distributed. A simulation study has been conducted to show the sampling behavior of the estimators for distinct values of n and m.