EL-class | R Documentation |
S4 class for empirical likelihood.
Let X_i
be independent and identically distributed
p
-dimensional random variable from an unknown distribution P
for i = 1, \dots, n
. We assume that P
has a positive definite
covariance matrix. For a parameter of interest
\theta(F) \in {\rm{I\!R}}^p
, consider a p
-dimensional smooth
estimating function g(X_i, \theta)
with a moment condition
\textrm{E}[g(X_i, \theta)] = 0.
We assume that there exists an unique \theta_0
that solves the above
equation. Given a value of \theta
, the (profile) empirical likelihood
ratio is defined by
R(\theta) =
\max_{p_i}\left\{\prod_{i = 1}^n np_i :
\sum_{i = 1}^n p_i g(X_i, \theta) = 0, p_i \geq 0, \sum_{i = 1}^n p_i = 1
\right\}.
The Lagrange multiplier \lambda \equiv \lambda(\theta)
of the dual
problem leads to
p_i = \frac{1}{n}\frac{1}{1 + \lambda^\top g(X_i, \theta)},
where \lambda
solves
\frac{1}{n}\sum_{i = 1}^n \frac{g(X_i, \theta)}
{1 + \lambda^\top g(X_i, \theta)} = 0.
Then the empirical log-likelihood ratio is given by
\log R(\theta) = -\sum_{i = 1}^n
\log(1 + \lambda^\top g(X_i, \theta)).
This problem can be efficiently solved by the Newton-Raphson method when
the zero vector is contained in the interior of the convex hull of
\{g(X_i, \theta)\}_{i = 1}^n
.
It is known that -2\log R(\theta_0)
converges in
distribution to \chi^2_p
, where \chi^2_p
has a chi-square
distribution with p
degrees of freedom. See the references below for
more details.
optim
A list of the following optimization results:
par
A numeric vector of the specified parameters.
lambda
A numeric vector of the Lagrange multipliers of the dual
problem corresponding to par
.
iterations
A single integer for the number of iterations performed.
convergence
A single logical for the convergence status.
cstr
A single logical for whether constrained EL optimization is
performed or not.
logp
A numeric vector of the log probabilities of the empirical likelihood.
logl
A single numeric of the empirical log-likelihood.
loglr
A single numeric of the empirical log-likelihood ratio.
statistic
A single numeric of minus twice the empirical log-likelihood ratio with an asymptotic chi-square distribution.
df
A single integer for the degrees of freedom of the statistic.
pval
A single numeric for the p
-value of the statistic.
nobs
A single integer for the number of observations.
npar
A single integer for the number of parameters.
weights
A numeric vector of the re-scaled weights used for the model fitting.
coefficients
A numeric vector of the maximum empirical likelihood estimates of the parameters.
method
A single character for the method dispatch in internal functions.
data
A numeric matrix of the data for the model fitting.
control
An object of class ControlEL constructed by
el_control()
.
Owen A (2001). Empirical Likelihood. Chapman & Hall/CRC. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9781420036152")}.
Qin J, Lawless J (1994). “Empirical Likelihood and General Estimating Equations.” The Annals of Statistics, 22(1), 300–325. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/aos/1176325370")}.
showClass("EL")
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