EstSimPDMP (version 1.2)

CondPdf.DC.interval: Estimation of the density associated to the jump rate for piecewise-deterministic Markov processes (discrete state space)

Description

This function computes the estimation of the density associated to the jump rate for a piecewise-deterministic Markov (PDMP) process whose state space is finite between the two times tmin and tmax. The estimator is given in the paper mentioned in References.

Usage

CondPdf.DC.interval(dat,x,tmin,tmax,nbre,h,alpha,verbose,bound)

Arguments

dat

data from which the estimator is to be computed. It corresponds to the observation of a PDMP within a long time. dat is a matrix such that the last column contains the interarrival times, while the other columns contain the states.

x

the conditional probability density function is estimated given state=x.

tmin

the conditional probability density function is estimated between tmin and tmax given state=x.

tmax

the conditional probability density function is estimated between tmin and tmax given state=x. In addition, tmax must be less than bound.

nbre

size of the grid plot.

h

bandwith

alpha

strictly positive real number. If h is NULL, the bandwith is 1/n^alpha where n is the number of data.

verbose

if TRUE, add a plot between tmin and tmax.

bound

the estimator is computed as an integral between the times 0 and bound. bound must be less than the deterministic exit time function tstar computed at state x.

References

Azais R., Dufour F., and Gegout-Petit A. Nonparametric estimation of the conditional distribution of the inter-jumping times for piecewise-deterministic Markov processes Scandinavian Journal of Statistics, 2014.

See Also

CondPdf.DC, Simu.PDMP.DC

Examples

# CondPdf.DC.interval

# Simulation of a PDMP with discrete state space
dat<-Simu.PDMP.DC(1,500,verbose=FALSE)

# Estimation of the conditional density given state=2
CondPdf.DC.interval(dat,2,0.4,5.5,70,alpha=1/4,bound=5.8)

tmin<-0.4
tmax<-5.5
N<-70
a<-(N*tmin):(N*tmax)
a<-a/N

# Conditional density given state=2
gr<-exp(-a)
# Theoretical conditional pdf
points(a,gr,"l",col="blue")