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.

[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.