lpint {lpint}

Martingale estimating equation local polynomial estimator of counting process intensity function and its derivatives

Description

This local polynomial estimator is based on a biased martingale estimating equation.

Usage

lpint(jmptimes, jmpsizes = rep(1, length(jmptimes)),
      Y = rep(1,length(jmptimes)), bw = NULL,
      adjust = 1, Tau = max(1, jmptimes), p = nu + 1,
      nu = 0, K = function(x) 3/4 * (1 - x^2) * (x <= 1 & x >= -1),
      n = 101, bw.only=FALSE)

Arguments

jmptimes

a numeric vector giving the jump times of the counting process

jmpsizes

a numeric vector giving the jump sizes at each jump time. Need to be of the same length as jmptimes

Y

a numeric vector giving the value of the exposure process (or size of the risk set) at each jump times. Need to be of the same length as jmptimes

bw

a numeric constant specifying the bandwidth used in the estimator. If left unspecified the automatic bandwidth selector will be used to calculate one.

adjust

a positive constant giving the adjust factor to be multiplied to the default bandwith parameter or the supplied bandwith

Tau

a numric constant >0 giving the censoring time (when observation of the counting process is terminated)

p

the degree of the local polynomial used in constructing the estimator. Default to 1 plus the degree of the derivative to be estimated

nu

the degree of the derivative of the intensity function to be estimated. Default to 0 for estimation of the intensity itself.

K

the kernel function

n

the number of evenly spaced time points to evaluate the estimator at

bw.only

TRUE or FALSE according as if the rule of thumb bandwidth is the only required output or not

Value

either a list containing

x

the vector of times at which the estimator is evaluated

y

the vector giving the values of the estimator at times given in x

se

the vector giving the standard errors of the estimates given in y

bw

the bandwidth actually used in defining the estimator equal the automatically calculated or supplied multiplied by adjust

or a numeric constant equal to the rule of thumb bandwidth estimate

Author(s)

Feng Chen <[e-mail].>

References

Chen, F., Yip, P.S.F., & Lam, K.F. (2011) On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives. Scandinavian Journal of Statistics 38(4): 631 - 649. http://dx.doi.org/10.1111/j.1467-9469.2011.00733.x

See Also

lplikint

Examples

##simulate a Poisson process on [0,1] with given intensity
int <- function(x)100*(1+0.5*cos(2*pi*x))
censor <- 1
set.seed(2)
N <- rpois(1,150*censor);
jtms <- runif(N,0,censor);
jtms <- jtms[as.logical(mapply(rbinom,n=1,size=1,prob=int(jtms)/150))];

##estimate the intensity
intest <- lpint(jtms,Tau=censor)
##plot and compare
plot(intest,xlab="time",ylab="intensity",type="l",lty=1)
curve(int,add=TRUE,lty=2)

## Example estimating the hazard function from right censored data:
## First simulate the (not directly observable) life times and censoring
## times:
lt <- rweibull(500,2.5,3); ct <- rlnorm(500,1,0.5)
## Now the censored times and censorship indicators delta (the
## observables): 
ot <- pmin(lt,ct); dlt <- as.numeric(lt <= ct);
## Estimate the hazard rate based on the censored observations:
jtms <- sort(ot[dlt==1]);
Y <- sapply(jtms,function(x)sum(ot>=x));
haz.est <- lpint(jtms,Y=Y);
## plot the estimated hazard function:
matplot(haz.est$x,
        pmax(haz.est$y+outer(haz.est$se,c(-1,0,1)*qnorm(0.975)),0),
        type="l",lty=c(2,1,2),
        xlab="t",ylab="h(t)",
        col=1);
## add the truth:
haz <- function(x)dweibull(x,2.5,3)/pweibull(x,2.5,3,lower.tail=FALSE)
curve(haz, add=TRUE,col=2)