# Estimating a Distribution Function Subject to a Stochastic Order Restriction

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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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Contributed by: Michail Bozoudis and Vasileios Papachatzis (July 2015)
Suggested by: Michail Boutsikas
Open content licensed under CC BY-NC-SA

## Details

This Demonstration (based on [1]) shows the performance of four nonparametric estimators on a distribution function , subject to a stochastic order restriction. The estimators are:

• the usual (unrestricted) empirical distribution function (EDF) estimator,

• the nonparametric maximum likelihood estimator (np-MLE), ([2])

• the pointwise maximum likelihood estimator (p-MLE), ([3])

• and the switch maximum likelihood estimator (s-MLE), ([4])

While , the order restriction may be imposed according to three shift () alternatives:

, denoted as the "difference" shift pattern

, denoted as the "power" shift pattern

for and for , denoted as the "tail" shift pattern

The first graphic displays the construction of the np-MLE least concave majorant (LCM) according to the , ordered random walk. The second graphic displays the quantile plots for and . The third graphic displays the box plots for , , and . Finally, the fourth graphic displays the estimator errors against the theoretical standard normal quantiles.

References

[1] O. Davidov and G. Iliopoulos, "Estimating a Distribution Function Subject to a Stochastic Order Restriction: A Comparative Study," Journal of Nonparametric Statistics, 24(4), 2012 pp. 923–933.

[2] H. D. Brunk, W. E. Franck, D. L. Hanson, and R. V. Hogg, "Maximum Likelihood Estimation of the Distributions of Two Stochastically Ordered Random Variables," Journal of the American Statistical Association, 61(316), 1966 pp. 1067–1080.

[3] R. V. Hogg, "On Models and Hypotheses with Restricted Alternatives," Journal of the American Statistical Association, 60(312), 1965 pp. 1153–1162.

[4] S. Lo, "Estimation of Distribution Functions under Order Restrictions," Statistics & Risk Modeling, 5(3–4), 1987 pp. 251–262.

## Permanent Citation

Michail Bozoudis and Vasileios Papachatzis

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