EstSimPDMP (version 1.2)

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

Description

This is the main function of the package EstSimPDMP. It computes the estimation of the density associated to the jump rate for a piecewise-deterministic Markov process (PDMP) whose state space is continuous. Details about the estimator are given in the paper mentioned in References.

Usage

CondPdf.CC.interval(dat,x,epsilon,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 post-jump locations of the process.

x

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

epsilon

the probability density function is estimated given the distance between state and x is less than epsilon. If epsilon is small, this is an approximation of the exact density.

tmin

the probability density function is estimated between tmin and tmax.

tmax

the probability density function is estimated between tmin and tmax. 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.interval, Simu.PDMP

Examples

# CondPdf.CC.interval

# Simulation of a PDMP with continuous state space
dat<-Simu.PDMP(2.3,500,verbose=FALSE)

# Estimation of the conditional density given state=1.8
CondPdf.CC.interval(dat,1.8,0.3,0.5,7.5,70,h=1/3,bound=7.8)

tmin<-0.5
tmax<-7.5
N<-70
a<-tmin:N*tmax
a<-a/N

x<-1.8
# Theoretical conditional pdf given state=1.8
grid<-(1/(1+x))*exp(-(1/(1+x))*a)
points(a,grid,"l",col="blue")