# Compute the Pilot Bandwidth for the Nonparametric Estimators of Cure Probability and Latency This function computes local pilot bandwidths for the nonparametric estimators of the probability of cure and the latency function. ```r 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. ## 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 López-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 López-Cheda, 2018): $$ g(x0) = 0.5(d_k^{+}(x0) + d_k^{-}(x0))(100/n)^{1/9}g(x0) =0.5 * (d_k^{+}(x0) + d_k^{-}(x0)) * (100/n)^{1/9} $$ 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 González-Manteiga (1993) for the uncensored case. ## Returns A numeric vector of local pilot bandwidths corresponding to each one of the values of the grid of covariate values given by `x0`. ## References Cao R., González-Manteiga W. (1993). Bootstrap methods in regression smoothing. Journal of Nonparametric Statistics, 2: 379-388. [https://doi.org/10.1080/10485259308832566](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](https://doi.org/10.1023/A:1014644700806). López-Cheda A. (2018). **Nonparametric Inference in Mixture Cure Models**. PhD dissertation, Universidade da Coruña. Spain. López-Cheda, A., Cao, R., Jácome, M. A., Van Keilegom, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis, 105: 144–165. [https://doi.org/10.1016/j.csda.2016.08.002](https://doi.org/10.1016/j.csda.2016.08.002). López-Cheda, A., Jácome, M. A., Cao, R. (2017). Nonparametric latency estimation for mixture cure models. TEST, 26: 353–376. [https://doi.org/10.1007/s11749-016-0515-1](https://doi.org/10.1007/s11749-016-0515-1). ## Author(s) Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut] ## See Also `controlpars`, `latencyhboot`, `probcurehboot` ## Examples ```r ## 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) ```

hpilot function