Lahiri, Furukawa, and Lee (2007) Nonparametric Plug-In "NPPI" Rule to Select the Optimal Block-Length

This function implements the Nonparametric Plug-In (NPPI) algorithm, as proposed by Lahiri, Furukawa, and Lee (2007), to select the optimal block length for block bootstrap procedures. The NPPI method estimates the optimal block length by balancing bias and variance in block bootstrap estimators, particularly for time series and other dependent data structures. The function also leverages the Moving Block Bootstrap (MBB) method of (Kunsch, 1989) and the Moving Blocks Jackknifte (MBJ) of Liu and Singh (1992).

nppi(
  data,
  stat_function = mean,
  r = 1,
  a = 1,
  l = NULL,
  m = NULL,
  num_bootstrap = 1000,
  c_1 = 1L,
  epsilon = 1e-08,
  plots = TRUE
)

Arguments

data: A numeric vector, ts, or single-column data.frame representing the time series or dependent data.
stat_function: A function to compute the statistic of interest (*e.g.*, mean, variance). The function should accept a numeric vector as input and return a scalar value (default is mean).
r: The rate parameter for the MSE expansion (default is 1). This parameter controls the convergence rate in the bias-variance trade-off.
a: The bias exponent (default is 1). Adjust this based on the theoretical properties of the statistic being bootstrapped.
l: Optional. The initial block size for bias estimation. If not provided, it is set to max(2, round(c_1 * n^{1 / (r + 4)})), where n is the sample size.
m: Optional. The number of blocks to delete in the Jackknife-After-Bootstrap (JAB) variance estimation. If not provided, it defaults to floor(c_2 * n^{1/3} * l^{2/3}).
num_bootstrap: The number of bootstrap replications for bias estimation (default is 1000).
c_1: A tuning constant for initial block size calculation (default is 1).
epsilon: A small constant added to the variance to prevent division by zero (default is 1e-8).
plots: A logical value indicating whether to plot the JAB diagnostic

Value

A object of class nppi with the following components:

optimal_block_length: The estimated optimal block length for the block bootstrap procedure.
bias: The estimated bias of the block bootstrap estimator.
variance: The estimated variance of the block bootstrap estimator using the JAB method.
jab_point_values: The point estimates of the statistic for each deletion block in the JAB variance estimation. Used for diagnostic plots
jab_pseudo_values: The pseudo-values of each JAB point value.
l: The initial block size used for bias estimation.
m: The number of blocks to delete in the JAB variance estimation.

Details

Jackknife-After-Bootstrap (JAB) variance estimation (Lahiri, 2002).

References

Efron, B. (1992), 'Jackknife-after-bootstrap standard errors and influence functions (with discussion)', Journal of Royal Statistical Society, Series B 54, 83-111.

Kunsch, H. (1989) The Jackknife and the Bootstrap for General Stationary Observations. The Annals of Statistics, 17(3), 1217-1241. Retrieved February 16, 2021, from doi:10.1214/aos/1176347265

Lahiri, S. N., Furukawa, K., & Lee, Y.-D. (2007). A nonparametric plug-in rule for selecting optimal block lengths for Block Bootstrap Methods. Statistical Methodology, 4(3), 292-321. DOI: doi:10.1016/j.stamet.2006.08.002

Lahiri, S. N. (2003). 7.4 A Nonparametric Plug-in Method. In Resampling methods for dependent data (pp. 186-197). Springer.

Liu, R. Y. and Singh, K. (1992), Moving blocks jackknife and bootstrap capture weak dependence, in R. Lepage and L. Billard, eds, 'Exploring the Limits of the Bootstrap', Wiley, New York, pp. 225-248.

Examples

# Generate AR(1) time series
set.seed(32)
sim <- stats::arima.sim(list(order = c(1, 0, 0), ar = 0.5),
                        n = 500, innov = rnorm(500))

# Estimate the optimal block length for the sample mean
result <- nppi(data = sim, stat_function = mean, num_bootstrap = 500, m = 2)
#>  Setting l to recomended value: 3



print(result$optimal_block_length)
#> [1] 4.583894e-06

# Use S3 method to plot JAB diagnostic
plot(result)