 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().

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 .