estimate.tau: Estimating the convergence rate

Description

This function estimates the convergence rate of the bootstrap estimator and returns it as a function of the form tau_n = n^a, where n is the input parameter.

Usage

estimate.tau(
  data,
  statistic,
  R = 1000,
  replace = FALSE,
  min.m = 3,
  beta = seq(0.2, 0.7, length.out = 5),
  method = "variance",
  ...
)

Value

A function for the square root of the convergence rate of the variance, i.e., f(n) = tau_n. This function can directly be passed to mboot.ci.

Arguments

data: The data to be bootstrapped.
statistic: The estimator of the parameter.
R: Amount of bootstrap replicates used to estimate tau.
replace: If sampling should be done with replacement.
min.m: Minimal subsampling size used to estimate tau. Should be set to the minimum size for which the statistic makes sense.
beta: The tested subsample sizes m are n^beta.
method: Method to estimate tau, can be one of c("variance", "quantile").
...: Additional parameters to be passed to the mboot function.

Details

There are two methods to choose from, variance and quantile. The provided beta values are used to select subsample sizes m by using ms = n^beta. Note that the choice of the beta values can impact the accuracy of the estimated tau (Dalitz & Lögler, 2024). For each selected subsample size m a bootstrap with R replications is performed. The method 'variance' then fits a linear function to log(variance) of the bootstrap statistics as function of log(m). The method 'quantile' averages over multiple quantile ranges Q and fits a linear function to log(Q) as a function of log(m).

References

Bertail P. et al. (1999) On subsampling estimators with unknown rate of convergence. Journal of the American Statistical Association, 94(446):568-579.

Politis D.N. et al. (1999) Subsampling, Springer, New York.

Dalitz, C, and Lögler, F. (2024) moonboot: An R Package Implementing m-out-of-n Bootstrap Methods, doi:10.48550/arXiv.2412.05032

Examples

Run this code

data <- runif(1000)
estimate.max <- function(data, indices) {return(max(data[indices]))}
estimated.tau <- estimate.tau(data, estimate.max)
boot.out <- mboot(data, estimate.max, R = 1000, m = 2*sqrt(NROW(data)), replace = FALSE)
cis <- mboot.ci(boot.out, 0.95, estimated.tau, c("all"))
ci.basic <- cis$basic
print(ci.basic)

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