# Estimating a Distribution Function Subject to a Stochastic Order Restriction

Estimating a Distribution Function Subject to a Stochastic Order Restriction

This Demonstration shows the nonparametric estimation of a standard normal variable cumulative distribution function (CDF) , under a stochastic order restriction. A pseudorandom data generation process produces a standard normal variable with distribution function and data size , and a uniform variable with data size . Mathematica's built-in inverse normal distribution function utilizes to generate another normal variable with distribution function and data size , under the stochastic restriction (). This restriction may be imposed by three different shift patterns (see Details).

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While the usual (unrestricted) empirical distribution function (EDF) estimator uses information only from variable , the maximum likelihood estimators (MLE) , , and use information from variables and or . The comparative study [1] shows that outperforms all other estimators when the underlying distributions are "close" to each other. You can use the controls to experiment on a variety of settings and observe the performance of the four estimators.

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