An estimating function that takes the data x and parameter value
par as its arguments, returning a numeric matrix. Each row is the return
value from the corresponding row in x.
x
A numeric matrix, or an object that can be coerced to a numeric
matrix. Each row corresponds to an observation. The number of rows must be
greater than the number of columns.
par
A numeric vector of parameter values to be tested.
mu
A numeric vector of parameters for regularization. See 'Details' for more
information.
Sigma
A numeric matrix, or an object that can be coerced to a numeric matrix,
of parameters for regularization. See 'Details' for more information.
tau
A single numeric parameter for regularization. See 'Details' for more
information.
type
A single character indicating the type of regularized exponentially tilted
empirical likelihood. It must be either "full" or "reduced". Defaults
to "full". See 'Details' for more information.
opts
An optional list with optimization options for nloptr().
Defaults to NULL.
Details
Let {Xi}i=1n denote independent dx-dimensional
observations from a complete probability space
(X,F,P) satisfying the moment condition:
EP[g(Xi,θ)]=0,
where g:Rdx×Θ↦Rp is an estimating
function with the true parameter value
θ0∈Θ⊂Rp.
For a given parameter value θ, regularized exponentially
tilted empirical likelihood solves the following optimization problem:
minλ∈Rp{dn(θ,λ)+pn(θ,λ)},
where
dn(θ,λ)=n+τn1∑i=1nexp(λ⊤g(Xi,θ))
and
pn(θ,λ)=n+τnτnexp(λ⊤μn,θ+21λ⊤Σn,θλ).
Here, τn>0, μn,θ,
Σn,θ are all tuning parameters that control
the strength of pn(θ,λ) as a penalty.
Once we have determined the solution λRET, we define
the likelihood ratio function as follows:
and
cn(θ,λRET)=dn(θ,λRET)+pn(θ,λRET). The reduced version of the likelihood ratio function is defined as:
RRET(θ)=∏i=1n(n+τn)pi(θ).
See the references below for more details on derivation, interpretation,
and properties.
Value
A single numeric value representing the log-likelihood ratio. It contains
the optimization results as the attribute optim.
References
Kim E, MacEachern SN, Peruggia M (2023).
"Regularized Exponentially Tilted Empirical Likelihood for Bayesian
Inference." doi:10.48550/arXiv.2312.17015.
Examples
# Generate data
set.seed(63456)
x <- rnorm(100)
# Define an estimating function (ex. mean)
fn <- function(x, par) {
x - par
}
# Set parameter value
par <- 0
# Set regularization parameters
mu <- 0
Sigma <- 1
tau <- 1
# Call the retel function
retel(fn, x, par, mu, Sigma, tau)