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")
#> Class "EL" [package "melt"]
#>
#> Slots:
#>
#> Name: optim logp logl loglr statistic
#> Class: list numeric numeric numeric numeric
#>
#> Name: df pval nobs npar weights
#> Class: integer numeric integer integer numeric
#>
#> Name: coefficients method data control
#> Class: numeric character ANY ControlEL
#>
#> Known Subclasses:
#> Class "CEL", directly
#> Class "SD", directly
#> Class "LM", by class "CEL", distance 2
#> Class "GLM", by class "CEL", distance 3
#> Class "QGLM", by class "CEL", distance 4
```