Estimators of discrete probabilities under k-monotony constraint

Description

Estimators of discrete probabilities under k-monotony constraint. Estimation can be done on the set of k-monotone functions or on the set of k-monotone probabilities.

Usage

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pMonotone(ptild, t.zero = 1e-10, t.P = 1e-08, max.sn = NULL, k, verbose = FALSE)
fMonotone(ptild, t.zero = 1e-10, t.P = 1e-08, max.sn = NULL, k, verbose = FALSE)

Arguments

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)

Details

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 non-negative that is to say bigger than -t.P.

Value

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

sum(pHat) at convergence

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 sum(pHat)

Author(s)

Jade Giguelay jade.giguelay@ens-paris-saclay.fr http://maiage.jouy.inra.fr/jgiguelay

References

Giguelay, J., (2016), Estimation of a discrete distribution under k-monotony constraint, in revision, (arXiv:1608.06541)

Groeneboom P., Jongbloed G. Wellner J. A. (2008) <DOI:10.1111/j.1467-9469.2007.00588.x> The Support Reduction Algorithm for Computing Non-Parametric Function Estimates in Mixture Models, Scandinavian Journal of Statistics, 35, 385–399

See Also

pEmp, BaseNorm

Examples

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x=rSpline(n=50, 20, k=4)
ptild=pEmp(x);
res=pMonotone(ptild$freq, k=4)