Fiducial empirical distribution vs. the empirical distribution function?
Do you refer to the "fiducial e.d.f." as defined by Xingzhong Xu, Xiaobo Ding, Shuran Zhao, New goodness-of-fit tests based on fiducial empirical distribution function, Computational Statistics & Data Analysis, Volume 53, Issue 4, 15 February 2009, Pages 1132-1141, ISSN 0167-9473, http://dx.doi.org/10.1016/j.csda.2008.10.003.
(http://www.sciencedirect.com/science/article/pii/S0167947308004829)?
It is a function both of the data x1,...,xn and of an auxiliary (independent?) sample u1,...un from the uniform(0,1) distribution. Thus given the data x1,...,xn, we don't have a single estimate of the d.f., but a probability distribution of many possible "estimates" (as one varies the realized values of u1,...,un).
Dear Samah,
I would like to advise you to go through the Glivenko-Cantelli Theorem on Convergence of Empirical Probability Distribution Functions. This theorem would be available in chapters on Order Statistics in standard text books. When you would understand this classical theorem, you will get the answer to your question automatically.
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