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 |
M |
A numeric value; the upper truncation threshold
Default is the 95th percentile of |
func |
A function to apply to each block. Default is |
... |
Additional arguments passed to the function |
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:
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.
-
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 offunc
applied to truncated observations in the block, -
Valid points
- number of observations within the truncation thresholds, -
Winsorized value
- the Winsorized value offunc
applied to the block, -
Start index
- the starting index of the block, -
End index
- the ending index of the block.
-
-
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)