EL-class {melt}R Documentation

EL class

Description

S4 class for empirical likelihood.

Details

Let XiX_i be independent and identically distributed pp-dimensional random variable from an unknown distribution PP for i=1,,ni = 1, \dots, n. We assume that PP has a positive definite covariance matrix. For a parameter of interest θ(F)I ⁣Rp\theta(F) \in {\rm{I\!R}}^p, consider a pp-dimensional smooth estimating function g(Xi,θ)g(X_i, \theta) with a moment condition

E[g(Xi,θ)]=0.\textrm{E}[g(X_i, \theta)] = 0.

We assume that there exists an unique θ0\theta_0 that solves the above equation. Given a value of θ\theta, the (profile) empirical likelihood ratio is defined by

R(θ)=maxpi{i=1nnpi:i=1npig(Xi,θ)=0,pi0,i=1npi=1}.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 λλ(θ)\lambda \equiv \lambda(\theta) of the dual problem leads to

pi=1n11+λg(Xi,θ),p_i = \frac{1}{n}\frac{1}{1 + \lambda^\top g(X_i, \theta)},

where λ\lambda solves

1ni=1ng(Xi,θ)1+λg(Xi,θ)=0.\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

logR(θ)=i=1nlog(1+λg(Xi,θ)).\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(Xi,θ)}i=1n\{g(X_i, \theta)\}_{i = 1}^n.

It is known that 2logR(θ0)-2\log R(\theta_0) converges in distribution to χp2\chi^2_p, where χp2\chi^2_p has a chi-square distribution with pp 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 pp-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")

[Package melt version 1.11.4 Index]