npcure (version 0.1-4)

hpilot: Compute the Pilot Bandwidth for the Nonparametric Estimators of Cure Probability and Latency

Description

This function computes local pilot bandwidths for the nonparametric estimators of the probability of cure and the latency function.

Usage

hpilot(x, x0, nnfrac = 0.25)

Arguments

x

A numeric vector of observed covariate values.

x0

A numeric vector specifying a grid of covariate values.

nnfrac

A numeric value giving the fraction of the sample size that determines the order of the nearest neighbor. This is taken as floor(length(x)*nnfrac). The default is 0.25.

Value

A numeric vector of local pilot bandwidths corresponding to each one of the values of the grid of covariate values given by x0.

Details

The function computes a data-driven local pilot bandwidth, required for the bootstrap bandwidth selector of the nonparametric estimators of the cure rate and latency functions. Simulations in L<U+00F3>pez-Cheda et al. (2017) show that the choice of pilot bandwidth has small effect on the bootstrap bandwidth. This pilot bandwidth only depends on the sample size and the distribution of the covariate x (see L<U+00F3>pez-Cheda, 2018):

$$g(x0) = 0.5(d_k^{+}(x0) + d_k^{-}(x0))(100/n)^{1/9}$$

where \(d_k^{+}(x0)\) and \(d_k^{-}(x0)\) are the distances from \(x0\) to the \(k\)-th nearest neighbor on the right and the left, respectively, and \(k\) is a suitable integer depending on the sample size \(n\). If there are not at least \(k\) neighbors on the right or on the left, we use \(d_k^{+}(x0) = d_k^{-}(x0)\). The default value of \(k\) is \(n/4\). The order \(n^{-1/9}\) satisfies the conditions in Theorem 1 of Li and Datta (2001) and coincides with the order obtained by Cao and Gonz<U+00E1>lez-Manteiga (1993) for the uncensored case.

References

Cao R., Gonz<U+00E1>lez-Manteiga W. (1993). Bootstrap methods in regression smoothing. Journal of Nonparametric Statistics, 2: 379-388. https://doi.org/10.1080/10485259308832566.

Li, G., Datta, S. (2001). A bootstrap approach to nonparametric regression for right censored data. Annals of the Institute of Statistical Mathematics, 53(4): 708-729. https://doi.org/10.1023/A:1014644700806.

L<U+00F3>pez-Cheda A. (2018). Nonparametric Inference in Mixture Cure Models. PhD dissertation, Universidade da Coru<U+00F1>a. Spain.

L<U+00F3>pez-Cheda, A., Cao, R., J<U+00E1>come, M. A., Van Keilegom, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis, 105: 144<U+2013>165. https://doi.org/10.1016/j.csda.2016.08.002.

L<U+00F3>pez-Cheda, A., J<U+00E1>come, M. A., Cao, R. (2017). Nonparametric latency estimation for mixture cure models. TEST, 26: 353<U+2013>376. https://doi.org/10.1007/s11749-016-0515-1.

See Also

controlpars, latencyhboot, probcurehboot

Examples

# NOT RUN {
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)

## Computing pilot bandwidths for covariate values -1, -0.8, ..., 1
## by taking the 5-th nearest neighbor
hpilot(data$x, x0 = seq(-1, 1, by = .2), nnfrac = .05)
# }