The Optimal Estimator Of Distribution Function

Given a random sample x₁,x₂,… ,x_n of size n from an unknown distribution function F, this paper considers the problem of optimal estimator of the c.d.f.. Under the loss functionsome literature have given the best invariant estimator of F when r = 2. In this paper, we obtain the best invariant estimator of F when r = 1, and then we obtained that given F is discrete distribution function the best invariant estimator is admissible.Furthermore we consider the asymmetric loss (LINEX) in parametric problem, which as an important loss function in nonparametric problem was cared by many author. The loss can be expressed asunder this loss, we considers the problem of optimal estimator of continuous distribution function F. We get the best invariant estimator and then proved it's a Minimax estimator, too.