pkmon-package {pkmon}

Least-squares estimator under k-monotony constraint for discrete functions

Description

Description: This package implements two least-squares estimators under k-monotony constraint using a method based on the Support Reduction Algorithm from Groeneboom et al (2008). The first one is a projection estimator on the set of k-monotone discrete functions. The second one is a projection on the set of k-monotone discrete probabilities. This package provides functions to generate samples from the spline basis from Lefevre and Loisel (2013), and from mixtures of splines.

Author(s)

Jade Giguelay

References

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

Lefevre C., Loisel S. (2013) <DOI:10.1239/jap/1378401239> On multiply monotone distributions, continuous or discrete, with applications, Journal of Applied Probability, 50, 827–847.

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

pMonotone, fMonotone

Examples

####################
# Example 1
# one triangular function T_j=Q_j^2, for j=supp=20 and for k=2 and k=3
n=30;
k1=2;
k2=3;
l=2;
supp=20;
p=dSpline(supp, k=l);
X=rSpline(n=n, supp, k=l);
ptilde=pEmp(X);
phat1=pMonotone(ptilde$freq, k=k1);
phat2=pMonotone(ptilde$freq, k=k2);

x.limits=c(0, max(supp+1, phat1$Spi+1, phat2$Spi+1));
y.limits=range(p, ptilde$freq, phat1$p, phat2$p);

plot(NULL, xlim=x.limits, ylim=y.limits, xlab="Counts", ylab="Frequencies");
points(0:supp, p, pch=16, col=1, lwd=2); 
points(ptilde$supp, ptilde$freq, pch=4, col=2, lwd=2);
points(0:max(phat1$Spi), phat1$p, pch=8, col=3, lwd=2);
points(0:max(phat2$Spi), phat2$p, pch=2, col=4, lwd=2);
legend("topright", pch=c(16, 4, 8, 2), col=c(1, 2, 3, 4),
	legend=c("p", expression(tilde("p")), expression(hat("p")*" - k = 2"), 
	expression(hat("p")*" - k = 3")));

####################
# Example 2
# mixture of 3 splines Q_j^3 and for k=4 and k=3
n=30;
k1=4;
k2=3;
l=3;
supp=c(5, 10, 20);
prob=c(0.5, 0.3, 0.2);
p=dmixSpline(supp, k=l, prob=prob);
X=rmixSpline(n=n, supp, k=l, prob=prob);
ptilde=pEmp(X);
phat1=pMonotone(ptilde$freq, k=k1);
phat2=pMonotone(ptilde$freq, k=k2);

x.limits=c(0, max(supp+1, phat1$Spi+1, phat2$Spi+1));
y.limits=range(p, ptilde$freq, phat1$p, phat2$p);

plot(NULL, xlim=x.limits, ylim=y.limits, xlab="Counts", ylab="Frequencies");
points(0:max(supp), p, pch=16, col=1, lwd=2); 
points(ptilde$supp, ptilde$freq, pch=4, col=2, lwd=2);
points(0:max(phat1$Spi), phat1$p, pch=8, col=3, lwd=2);
points(0:max(phat2$Spi), phat2$p, pch=2, col=4, lwd=2);
legend("topright", pch=c(16, 4, 8, 2), col=c(1, 2, 3, 4),
	legend=c("p", expression(tilde("p")), expression(hat(p)* " - k = 4"), 
	expression(hat(p)* " - k = 3")));

####################
# Example 3
# Poisson density
n=30;
k1=2;
k2=3;
supp=10;
p=dpois(0:supp, lambda=1);
X=rpois(n, lambda=1);
ptilde=pEmp(X);
phat1=pMonotone(ptilde$freq, k=k1);
phat2=pMonotone(ptilde$freq, k=k2);

x.limits=c(0, max(supp, phat1$Spi+1, phat2$Spi+1));
y.limits=range(p, ptilde$freq, phat1$p, phat2$p);

plot(NULL, xlim=x.limits, ylim=y.limits, xlab="Counts", ylab="Frequencies");
points(0:max(supp), p, pch=16, col=1, lwd=2);
points(ptilde$supp, ptilde$freq, pch=4, col=2, lwd=2);
points(0:max(phat1$Spi), phat1$p, pch=8, col=3, lwd=2);
points(0:max(phat2$Spi), phat2$p, pch=2, col=4, lwd=2);
legend("topright", pch=c(16, 4, 8, 2), col=c(1, 2, 3, 4),
       legend=c("p", expression(tilde("p")), expression(hat(p)* " - k = 2"),
                expression(hat(p)* " - k = 3")));

## Not run: 
####################
# Simulation for comparing ptilde and pHat (p is 3-monotone, k=3)
#
#OUTPUT
#
# cvge : percentage of non-convergence of the algorithm
# r.emp : L2-risk for the empirical estimator
# r.Hat : L2-risk for the estimator under k-monotony constraint


nSim=500;

n=30;
k=3;
l=3;
supp=20;
p=dSpline(supp, k=l);

result <- matrix(nrow=nSim,ncol=3);
dimnames(result)[[2]] <- c("cvge","r.emp","r.Hat");
for (i in 1:nSim) {
  X=rSpline(n=n, supp, k=l);
  ptilde=pEmp(X);
  phat=pMonotone(ptilde$freq, k=k);
  m <- max(supp+1,length(ptilde$freq)+1,phat$Spi+1)
  pV=c(p,rep(0,m-length(p)))
  pHat=c(phat$p,rep(0,m-length(phat$p)))
  ptilde=c(ptilde$freq,rep(0,m-length(ptilde$freq)))
  result[i,] <- c(phat$cvge,sum((pV-ptilde)**2),
                  sum((pV-pHat)**2))
}

apply(result,2,mean)

#Example with set.seed(0)
#cvge       r.emp       r.Hat 
#0.000000000 0.030682552 0.004984899

## End(Not run)