EL class

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