GetPseudoBlocks {boodd}R Documentation

Computing Pseudo-regenerative Blocks

Description

The function computes pseudo-regenerative blocks for general Markov chains.

Usage

GetPseudoBlocks(
  x,
  s,
  eps_opt,
  delta_opt,
  p_XiXip1,
  m = min(x),
  M = max(x),
  func = sum,
  ...
)

Arguments

x

A numeric vector representing a Markov chain.

s

A real number specifying the center of the small set.

eps_opt

A numeric value for the size of the small set.

delta_opt

A numeric value for the lower bound in the minorization condition.

p_XiXip1

A numeric value representing the estimator of the transition density.

m

A numeric value; the lower truncation threshold Default is the 5th percentile of x.

M

A numeric value; the upper truncation threshold Default is the 95th percentile of x.

func

A function to apply to each block. Default is sum.

...

Additional arguments passed to the function func.

Details

The function begins by determining which elements of x are within an interval [s-esp_opt,s+eps_opt]. Then an estimated Nummelin splitting trick is performed using the estimators p_n(X_i,X_{i+1}).

Value

Returns a list containing:

  1. A data frame with the following columns:

    • Time - the index of each observation,

    • x - values of the process,

    • Bnumber - block number assigned to each observation,

    • regen - indicator (1 or 0) of regeneration times. 1 corresponds to the regeneration time.

  2. A matrix summarizing block characteristics with the following columns:

    • Block number - the block index,

    • Block length - number of observations in the block,

    • Truncated sum - the value of func applied to truncated observations in the block,

    • Valid points - number of observations within the truncation thresholds,

    • Winsorized value - the Winsorized value of func applied to the block,

    • Start index - the starting index of the block,

    • End index - the ending index of the block.

  3. Total blocks - the total number of regeneration blocks.

References

Bertail, P. and Dudek, A. (2025). Bootstrap for Dependent Data, with an R package (by Bernard Desgraupes and Karolina Marek) - submitted.

Bertail, P. and Clémençon, S. (2006). Regenerative block bootstrap for Markov chains. Bernoulli, 12, 689-712.

See Also

findBestEpsilon, ftrunc, regenboot, smallEnsemble.

Examples


n=200# the length of the process
# Generating the AR(1) process
coeff=0.75
X = arima.sim(n=n, list(ar = c(coeff)))
# Find the small ensemble with the largest number of regeneration
sm <- findBestEpsilon(X,s=0,plotIt=FALSE)
f =sm$trans
eps = sm$epsilon
delta = sm$delta
m = sm$s
Pseudo_blocks=GetPseudoBlocks(X, m, eps_opt = eps, delta_opt = delta, p_XiXip1 = f,func=sum)


[Package boodd version 0.1 Index]