Package 'lpint' reference manual

Title:	Local Polynomial Estimators of the Intensity Function and Its Derivatives
Description:	Functions to estimate the intensity function and its derivative of a given order of a multiplicative counting process using the local polynomial method.
Authors:	Feng Chen <[email protected]>
Maintainer:	Feng Chen <[email protected]>
License:	GPL (>= 2.0)
Version:	2.1
Built:	2025-01-10 05:18:40 UTC
Source:	https://github.com/cran/lpint

Local Polynomail Estimators of the Intensity Function of a Counting Process and Its Derivatives

Description

Estimates the intensity function or its derivative of a give a given order using the local polynomial method with automatic bandwidth selection using a rule of thumb plug-in approach.

Details

Package:	lpint
Type:	Package
Version:	1.0
Date:	2012-09-21
License:	GPL (>=2.0)
LazyLoad:	yes

Author(s)

Feng Chen <[email protected]> Maintainer: Feng Chen <[email protected]>

References

Chen, F. (2011) Maximum local partial likelihood estimators for the counting process intensity function and its derivatives. Statistica Sinica 21(1): 107 -128. http://www3.stat.sinica.edu.tw/statistica/j21n1/J21N14/J21N14.html

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

Chen, F., Higgins, R.M., Yip, P.S.F. & Lam, K.F. (2008) Nonparametric estimation of multiplicative counting process intensity functions with an application to the Beijing SARS epidemic, Communications in Statistics - Theory and Methods 37: 294 - 306. http://www.tandfonline.com/doi/abs/10.1080/03610920701649035

Chen, F., Higgins, R.M., Yip, P.S.F. & Lam, K.F. (2008) Local polynomial estimation of Poisson intensities in the presence of reporting delays, Journal of the Royal Statistical Society Series C (Applied Statistics) 57(4): 447 - 459. http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9876.2008.00624.x/full

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)
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 <[email protected].>

References

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)
##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)

Partial likelihood based local polynomial estimators of the counting process intensity function and its derivatives

Description

This local polynomial estimator is based on the (localized) partial likelihood

Usage

lplikint(jmptimes, jmpsizes = rep(1, length(jmptimes)),
         Y = rep(1,length(jmptimes)),
         K = function(x) 3/4 * (1 - x^2) * (x <= 1 & x >= -1),
         bw, adjust = 1, nu = 0, p = 1, Tau = 1, n = 101,
         tseq = seq(from = 0, to = Tau, length = n), tol = 1e-05,
         maxit = 100, us = 10, gd = 5) 
lplikint(jmptimes, jmpsizes = rep(1, length(jmptimes)),
         Y = rep(1,length(jmptimes)),
         K = function(x) 3/4 * (1 - x^2) * (x <= 1 & x >= -1),
         bw, adjust = 1, nu = 0, p = 1, Tau = 1, n = 101,
         tseq = seq(from = 0, to = Tau, length = n), tol = 1e-05,
         maxit = 100, us = 10, gd = 5)

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
`K`	the kernel function
`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
`nu`	the degree of the derivative of the intensity function to be estimated. Default to 0 for estimation of the intensity itself.
`p`	the degree of the local polynomial used in constructing the estimator. Default to 1 plus the degree of the derivative to be estimated
`Tau`	a numric constant >0 giving the censoring time (when observation of the counting process is terminated)
`n`	the number of evenly spaced time points to evaluate the estimator at. Not used when `tseq` is provided.
`tseq`	the time sequence at which to evaluate the estimator
`tol`	the parameter error tolerance used to stop the iterations in optimizing the local likelihood
`maxit`	maximum number of iterations allowed in the optimization used in a single estimation point
`us`	a numeric constants used together with `gd` to grid search for a decent start value in solving the local score equation. The starting value is 0 except on its first dimension, which was chosen so that the starting value is the minimizer of the L^1 norm of the score function among the values: average intensity X `us^(-gd:gd)`
`gd`	a numeric constant used together with `us` to search for a decent start value in solving the local score equation

Details

The estimator is based on solving the local score equation using the Newton-Raphson method and extract the appropriate dimension.

Value

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`
`fun`	the intensity (or derivative) estimator as a function of the estimation point, which can be called to evaluate the estimator at points not included in `tseq`

Author(s)

Feng Chen <[email protected].>

References

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 <- lplikint(jtms,bw=0.15,Tau=censor)
#plot and compare
plot(intest,xlab="time",ylab="intensity",type="l",lty=1)
curve(int,add=TRUE,lty=2)
##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 <- lplikint(jtms,bw=0.15,Tau=censor)
#plot and compare
plot(intest,xlab="time",ylab="intensity",type="l",lty=1)
curve(int,add=TRUE,lty=2)

Package 'lpint'

Help Index

Local Polynomail Estimators of the Intensity Function of a Counting Process and Its Derivatives

Description

Details

Author(s)

References

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

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Partial likelihood based local polynomial estimators of the counting process intensity function and its derivatives

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples