This Demonstration (based on ) 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),
• the pointwise maximum likelihood estimator (p-MLE),
• and the switch maximum likelihood estimator (s-MLE),
, the order restriction
may be imposed according to three shift (
, denoted as the "difference" shift pattern
, denoted as the "power" shift pattern
, 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
. The third graphic displays the box plots for
. Finally, the fourth graphic displays the estimator errors against the theoretical standard normal quantiles.
 O. Davidov and G. Iliopoulos, "Estimating a Distribution Function Subject to a Stochastic Order Restriction: A Comparative Study," Journal of Nonparametric Statistics
(4), 2012 pp. 923–933.
 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
(316), 1966 pp. 1067–1080.
 R. V. Hogg, "On Models and Hypotheses with Restricted Alternatives," Journal of the American Statistical Association
(312), 1965 pp. 1153–1162.
 S. Lo, "Estimation of Distribution Functions under Order Restrictions," Statistics & Risk Modeling
(3–4), 1987 pp. 251–262.