findBestEpsilon {boodd} | R Documentation |
Optimal Size of Small Sets
Description
Determines the optimal radius \epsilon
of small sets, which maximizes the
expected number of regeneration blocks.
Useful for identifying the parameters for splitting a Markov chain into almost
independent blocks.
Usage
findBestEpsilon(x, s = median(x), plotIt = FALSE)
Arguments
x |
A numeric vector representing a Markov chain. |
s |
A real number specifying the center of the small set.
Default is the median of |
plotIt |
Logical. If |
Details
The findBestEpsilon
calculates the optimal radius \epsilon
and the
corresponding lower bound \delta
of the transition density for choosing the
optimal small set of a Markov chain given the center s
of a small set.
It uses the fastNadaraya
function to estimate
the transition density p_n(X_i,X_{i+1})
taken at points (X_i,X_{i+1})
.
Value
Returns an object of class smallEnsemble
, which is a list containing:
-
s
: the chosen center of the small set, -
epsilon
: estimated optimal radius of the small set, -
delta
: estimated lower bound of the transition density over the optimal small set, -
trans
: estimated values ofp_n(X_i,X_{i+1})
.
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
regenboot
, fastNadaraya
, naradamar
,
GetBlocks
, GetPseudoBlocks
, smallEnsemble
.
Examples
n=200 # the length of the process
# Generating the AR(1) process
sigma=1
coeff=0.75
X=arima.sim(n=n,list(ar=coeff, sd=sigma))
# Find the small ensemble with the largest number of regeneration
sm <- findBestEpsilon(X,s=0,plotIt=FALSE)