Data-driven bandwidth vector

Description

The function ComputeBW computes the bandwidth that minimizes the pointwise Mean Squared Error using the subsampling principle in combination with undersmoothing.

Usage

ComputeBW(data, x)

Arguments

Value

bw data-driven bandwidth vector of size length(x) containing the bandwidth value for each point in x.

References

Groeneboom, P. and Hendrickx, K. (2017). The nonparametric bootstrap for the current status model. Electronic Journal of Statistics 11(2):3446-3848.

See Also

vignette("curstatCI")

Examples

library(Rcpp)
library(curstatCI)

# sample size
n <- 1000

# truncated exponential distribution on (0,2)
set.seed(100)
t <- rep(NA, n)
delta <- rep(NA, n)
for(i in (1:n) ){
  x<-runif(1)
  y<--log(1-(1-exp(-2))*x)
  t[i]<-2*runif(1);
  if(y<=t[i]){ delta[i]<-1}
  else{delta[i]<-0}}

A<-cbind(t[order(t)], delta[order(t)], rep(1,n))

# x vector
grid<-seq(0.1,1.9 ,by = 0.1)

# data-driven bandwidth vector
bw <- ComputeBW(data =A, x = grid)
plot(grid, bw)

Pointwise Confidence Intervals under Current Status data

Description

The function ComputeConfIntervals computes pointwise confidence intervals for the distribution function under current status data. The confidence intervals are based on the Smoothed Maximum likelihood Estimator and constructed using the nonparametric bootstrap.

Usage

ComputeConfIntervals(data, x, alpha, bw)

Arguments

Details

In the current status model, the variable of interest X with distribution function F is not observed directly. A censoring variable T is observed instead together with the indicator \Delta = (X \le T). ComputeConfIntervals computes the pointwise 1-alpha bootstrap confidence intervals around the SMLE of F based on a sample of size n <- sum(data$freq2).

The bandwidth parameter vector that minimizes the pointwise Mean Squared Error using the subsampling principle in combination with undersmoothing is returned by the function ComputeBW.

The default method for constructing the confidence intervals in [Groeneboom & Hendrickx (2017)] is based on estimating the asymptotic variance of the SMLE. When the bandwidth is small for some point in x, the variance estimate of the SMLE at this point might not exist. If this happens the Non-Studentized confidence interval is returned for this particular point in x.

Value

List with 5 variables:

MLE

Maximum Likelihood Estimator. This is a matrix of dimension (m+1)x2 where m is the number of jump points of the MLE. The first column consists of the point zero and the jump locations of the MLE. The second column contains the value zero and the values of the MLE at the jump points.

SMLE

Smoothed Maximum Likelihood Estimator. This is a vector of size length(x) containing the values of the SMLE for each point in the vector x.

CI

pointwise confidence interval. This is a matrix of dimension length(x)x2. The first resp. second column contains the lower resp. upper values of the confidence intervals for each point in x.

Studentized

points in x for which Studentized nonparametric bootstrap confidence intervals are computed.

NonStudentized

points in x for which classical nonparametric bootstrap confidence intervals are computed.

References

Groeneboom, P. and Hendrickx, K. (2017). The nonparametric bootstrap for the current status model. Electronic Journal of Statistics 11(2):3446-3848.

See Also

vignette("curstatCI")

Examples

library(Rcpp)
library(curstatCI)

# sample size
n <- 1000

# Uniform data  U(0,2)
set.seed(2)
y <- runif(n,0,2)
t <- runif(n,0,2)
delta <- as.numeric(y <= t)

A<-cbind(t[order(t)], delta[order(t)], rep(1,n))

# x vector
grid<-seq(0.1,1.9 ,by = 0.1)

# data-driven bandwidth vector
bw <- ComputeBW(data =A, x = grid)

# pointwise confidence intervals at grid points:
out<-ComputeConfIntervals(data = A,x =grid,alpha = 0.05, bw = bw)

left <- out$CI[,1]
right <- out$CI[,2]

plot(grid, out$SMLE,type ='l', ylim=c(0,1), main= "",ylab="",xlab="",las=1)
points(grid, left, col = 4)
points(grid, right, col = 4)
segments(grid,left, grid, right)

Maximum Likelihood Estimator

Description

The function ComputeMLE computes the Maximum Likelihood Estimator of the distribution function under current status data.

Usage

ComputeMLE(data)

Arguments

Details

In the current status model, the variable of interest X with distribution function F is not observed directly. A censoring variable T is observed instead together with the indicator \Delta = (X \le T). ComputeMLE computes the MLE of F based on a sample of size n <- sum(data$freq2).

Value

Dataframe with two variables :

x

jump locations of the MLE

mle

MLE evaluated at the jump locations

References

Groeneboom, P. and Hendrickx, K. (2017). The nonparametric bootstrap for the current status model. Electronic Journal of Statistics 11(2):3446-3848.

See Also

ComputeConfIntervals

Examples

library(Rcpp)
library(curstatCI)

# sample size
n <- 1000

# Uniform data  U(0,2)
set.seed(2)
y <- runif(n,0,2)
t <- runif(n,0,2)
delta <- as.numeric(y <= t)

A<-cbind(t[order(t)], delta[order(t)], rep(1,n))
mle <-ComputeMLE(A)
plot(mle$x, mle$mle,type ='s', ylim=c(0,1), main= "",ylab="",xlab="",las=1)

Smoothed Maximum Likelihood Estimator

Description

The function ComputeSMLE computes the Smoothed Maximum Likelihood Estimator of the distribution function under current status data.

Usage

ComputeSMLE(data, x, bw)

Arguments

Details

In the current status model, the variable of interest X with distribution function F is not observed directly. A censoring variable T is observed instead together with the indicator \Delta = (X \le T). ComputeSMLE computes the SMLE of F based on a sample of size n <- sum(data$freq2). The bandwidth parameter vector that minimizes the pointwise Mean Squared Error using the subsampling principle in combination with undersmoothing is returned by the function ComputeBW.

Value

SMLE(x) Smoothed Maximum Likelihood Estimator. This is a vector of size length(x) containing the values of the SMLE for each point in the vector x.

References

Groeneboom, P. and Hendrickx, K. (2017). The nonparametric bootstrap for the current status model. Electronic Journal of Statistics 11(2):3446-3848.

See Also

ComputeConfIntervals

Examples

library(Rcpp)
library(curstatCI)

# sample size
n <- 1000

# Uniform data  U(0,2)
set.seed(2)
y <- runif(n,0,2)
t <- runif(n,0,2)
delta <- as.numeric(y <= t)

A<-cbind(t[order(t)], delta[order(t)], rep(1,n))
grid <-seq(0,2 ,by = 0.01)

# bandwidth vector
h<-rep(2*n^-0.2,length(grid))

smle <-ComputeSMLE(A,grid,h)
plot(grid, smle,type ='l', ylim=c(0,1), main= "",ylab="",xlab="",las=1)

Hepatitis A data

Description

A dataset on the prevalence of hepatitis A in individuals from Bulgaria with age ranging from 1 to 86 years. The data consists of a cross-sectional survey conducted in 1964.

Usage

hepatitisA

Format

A data frame with 83 rows and three variables:

t

Age of the individual

freq1

Number of individuals of age t that are seropositive for Hepatitis A

freq2

Total number of individuals of age t

References

Keiding, N. (1991). Age-specic incidence and prevalence: a statistical perspective. J. Roy. Statist. Soc. Ser. A,154(3):371-412.

Rubella data

Description

A dataset on the prevalence of rubella in 230 Austrian males older than three months for whom the exact date of birth was known. Each individual was tested at the Institute of Virology, Vienna during the period 1–25 March 1988 for immunization against Rubella.

Usage

rubella

Format

A data frame with 225 rows and three variables:

t

Age of the individual at the time of testing for immunization

freq1

Number of individuals of age t that are immune for Rubella

freq2

Total number of individuals of age t

References

Keiding, N., Begtrup, K., Scheike, T., and Hasibeder, G. (1996). Estimation from current status data in continuous time. Lifetime Data Anal., 2:119-129.