hazard.LL {DOvalidation}

Local Linear Hazard Estimator (Natural Weighting)

Description

Local linear estimator of the unidimensional hazard (or hazard rate) with natural weighting introduced by Nielsen and Tanggaard (2001).

Usage

  hazard.LL(xi, Oi, Ei, x, b, K="epa", Ktype="symmetric" , CI=FALSE)

Arguments

xi

Vector of time points where the counts data are given.

Oi

Vector with the number of occurrences observed at each time point (xi).

Ei

Vector with the observed exposure at each time point (xi).

x

Vector (or scalar) with the (time) grid points where the hazard estimator will be evaluated.

b

A positive scalar used as the bandwidth.

K

Indicates the kernel function to be considered in the estimator. Choose between values "epa" (for the Epanechnikov kernel) or "sextic" (see details for its expression).

Ktype

Indicates the type of kernel to be used. Choose among "symmetric" for the usual kernel definition (chosen in the argument K), "left" for the left-sided version of the kernel, or "right" for the right-sided version. See details below.

CI

Logical. If TRUE then 95% pointwise confidence intervals are provided for the hazard function.

Details

The estimator is calculated assuming that the data are given as count data i.e. number of occurrences and exposures. The function allows to consider two different kernels using the argument K. These are: Epanechnikov, K(u)=.75*(1-u^2)*(abs(u)<1), and sextic K(u)=(3003/2048)*(1-(u)^2)^6)*(abs(u)<1). The argument Ktype will define the usual estimator with whole support kernel as it is defined by K or the one-sided versions using left-sided kernel, 2*K(u)*(u<0), or right-sided kernel 2*K(u)*(u>0). See more details in Gamiz et al. (2016).

Value

x

Vector (or scalar) with the (time) grid points where the hazard estimator has been evaluated.

OLL

Vector with the smoothed occurrences (using the local linear kernel).

ELL

Vector with the smoothed exposures (using the local linear kernel).

hLL

Vector (or scalar) with the resulting hazard estimates at grid points x.

OLL.norm

Vector with the normalized smoothed occurrences (the smoothing weights are defined as for O.LL but normalized to add up one.

ELL.norm

Vector with the normalized smoothed exposures (the smoothing weights are defined as for E.LL but normalized to add up one.

CI.inf

Vector with the lower limits for the 95% confidence intervals. If CI=FALSE then NA values are provided.

CI.sup

Vector with the upper limits for the 95% confidence intervals. If CI=FALSE then NA values are provided.

Author(s)

Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.

References

Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.

Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.

See Also

hazard.LL.RH,hazard.MBC

Examples

## Calculation of the local linear hazard estimator with do-validated bandwidth.
## The hazard estimator is shown and decomposed into smoothed occurrences and exposures.
## This example is described in Gamiz et al. (2016).
data(UK)
Oi<-UK$D
Ei<-UK$E
ti<-40:110  # time is age and it goes from 40 to 110 years
M<-length(ti)
country<-'UK'
bdo<-5.11
resLL.do<-hazard.LL(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=bdo,K="sextic",Ktype="symmetric",CI=TRUE)

## The local linear hazard estimate is hLL.do below
hLL.do<-resLL.do$hLL

## The smoothed occurrences and exposures are:
ELL.norm.do<-resLL.do$ELL.norm
OLL.norm.do<-resLL.do$OLL.norm

## The 95% pointwise confidence intervals based on the asymptotics are
hLL.do.inf<-resLL.do$CI.inf
hLL.do.sup<-resLL.do$CI.sup

# Now we plot the hazard estimator with confidence intervals
old.par<-par(mar=c(3,1.5,1.5,1.5),oma=c(2,0.5,0.5,0.2),
mgp=c(1.5,0.5,0),cex.axis=1,cex.main=1.5,mfrow=c(3,2))

#hazard estimate
tit<-paste(country,"Hazard estimate",sep= ' - ' )
yy<-range(c(hLL.do.inf,hLL.do.sup),na.rm=TRUE)
plot(ti,hLL.do,main=tit,xlab='age',ylab='',type='l',lwd=2,ylim=yy)
# the confidence bands
x1<-ti;x2<-ti[M:1]   
y1<-hLL.do.sup;y2<-hLL.do.inf[M:1]       
polygon(c(x1,x2,x1[1]),c(y1,y2,y1[1]),col=gray(0.7),border=FALSE)
lines(ti,hLL.do,lty=1,lwd=2,col=1)
  
## Zooming at the old mortality
ind.ages<- -c(1:60)  ## only women with ages 100 or higher
ti2<-ti[ind.ages];M2<-length(ti2)
yy2<-range(c(hLL.do.inf[ind.ages],hLL.do.sup[ind.ages]),na.rm=TRUE)
plot(ti2,hLL.do[ind.ages],main=tit,xlab='age',ylab='',type='l',
lwd=2,ylim=yy2)
# the confidence intervals
x1<-ti2;x2<-ti2[M2:1]   
y1<-hLL.do.sup[ind.ages];hLL.do.inf2<-hLL.do.inf[ind.ages]
y2<-hLL.do.inf2[M2:1]       
polygon(c(x1,x2,x1[1]),c(y1,y2,y1[1]),col=gray(0.7),border=FALSE)
lines(ti2,hLL.do[ind.ages],lty=1,lwd=2,col=1)
  
## We decompose the estimator in the smooth occurrences and exposures
#   The occurrences with a zoom at old-age mortality
yy<-range(OLL.norm.do,na.rm=TRUE)
plot(ti,OLL.norm.do,main="Smoothed occurrences",xlab='age',ylab='',type='l',
lwd=2,ylim=yy)
yy2<-range(OLL.norm.do[ind.ages],na.rm=TRUE)
plot(ti2,OLL.norm.do[ind.ages],main="Smoothed occurrences",xlab='age',
ylab='',type='l',lwd=2,ylim=yy2)
  
#   The exposures with a zoom at old-age mortality
yy<-range(ELL.norm.do,na.rm=TRUE)
plot(ti,ELL.norm.do,main="Smoothed exposures",xlab='age',ylab='',type='l',
lwd=2,ylim=yy)
yy2<-range(ELL.norm.do[ind.ages],na.rm=TRUE)
plot(ti2,ELL.norm.do[ind.ages],main="Smoothed exposures",xlab='age',ylab='',
type='l',lwd=2,ylim=yy2)

# Revert the changes made in the graphics options
par(old.par)