Estimators of discrete probabilities under kmonotony constraint. Estimation can be done on the set of kmonotone functions or on the set of kmonotone probabilities.
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ptild 
Empirical estimator 
t.zero 
Threshold for the precision of the directionnal derivatives. (see OUTPUT below) 
t.P 
Threshold for the precision on the stopping criterion. (see OUTPUT below) 
max.sn 
The maximum support for the evaluation of the estimator 
k 
Degree of monotony 
verbose 
if TRUE, print for each iteration on the maximum support : pi, Psi and sumP (see OUTPUT below) 
The thresholds t.P and t.zero are used for the precision in the algorithm : in Step one (See REFERENCES below) the algorithm computes the directionnal derivatives of the current estimator and stops if all the directionnal derivarives are null that is to say if they are smaller than t.zero. In Step two (See REFERENCES below) the algorithm computes a stopping criterion and stops if and only if the stopping criterion is verified that is to say if some quantities are nonnegative that is to say bigger than t.P.
cvge 
cvge = 0 if the criterion Psi decreases with the support of pi. cvge = 1 if Psi increases. cvge = 2 if maximum number of iterations reached 
Spi 
Support of the positive measure pi at the last iteration 
pi 
Values of the positive measure pi at the last iteration 
p 
Values of pHat 
Psi 
Scalar value of the criterion to be minimised 
sumP 

history 
Data frame with components 
L 
The maximum of the support of pi 
Psi 
Value of the criterion for the value L 
SumP 
Value of 
Jade Giguelay jade.giguelay@ensparissaclay.fr http://maiage.jouy.inra.fr/jgiguelay
Giguelay, J., (2016), Estimation of a discrete distribution under kmonotony constraint, in revision, (arXiv:1608.06541)
Groeneboom P., Jongbloed G. Wellner J. A. (2008) <DOI:10.1111/j.14679469.2007.00588.x> The Support Reduction Algorithm for Computing NonParametric Function Estimates in Mixture Models, Scandinavian Journal of Statistics, 35, 385–399
pEmp, BaseNorm
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