EL-class {melt}

EL class

Description

S4 class for empirical likelihood.

Details

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.

Slots

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().

References

Owen A (2001). Empirical Likelihood. Chapman & Hall/CRC. doi:10.1201/9781420036152.

Qin J, Lawless J (1994). “Empirical Likelihood and General Estimating Equations.” The Annals of Statistics, 22(1), 300–325. doi:10.1214/aos/1176325370.

Examples

showClass("EL")