S4 class for constrained empirical likelihood. It inherits from EL class. Note that the optim slot has constrained optimization results with respect to the parameters, not the Lagrange multiplier.

## Details

Let $$l(\theta)$$ denote minus twice the empirical log-likelihood ratio function. We consider a linear hypothesis of the form $$L\theta = r,$$ where the left-hand-side $$L$$ is a $$q$$ by $$p$$ matrix and the right-hand-side $$r$$ is a $$q$$-dimensional vector. Under some regularity conditions, $$l(\theta)$$ converges in distribution to $$\chi^2_q$$ under the constraint of hypothesis, i.e., $$\min_{\theta: L\theta = r} l(\theta) \to_d \chi^2_q .$$

Minimization of $$l(\theta)$$ with respect to $$\theta$$ is computationally expensive since it implicitly involves the evaluation step as described in EL. Further, depending on the form of $$g(X_i, \theta)$$ and the constraint, the optimization problem can be nonconvex and have multiple local minima. For this reason, the package melt only considers linear hypotheses and performs local minimization of $$l(\theta)$$ using projected gradient descent method. With the orthogonal projection matrix $$P$$ and a step size $$\gamma$$, the algorithm updates $$\theta$$ as $$\theta^{(k + 1)} \leftarrow \theta^{(k)} - \gamma P \nabla l(\theta^{(k)}),$$ where $$\nabla l(\theta^{(k)})$$ denotes the gradient of $$l$$ at $$\theta^{(k)}$$. The first order optimality condition is $$P \nabla l(\theta) = 0$$, which is used as the stopping criterion.

## Slots

optim

A list of the following optimization results:

• par A numeric vector of the solution to the constrained optimization problem.

• 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 constrained empirical likelihood.

logl

A single numeric of the constrained empirical log-likelihood.

loglr

A single numeric of the constrained empirical log-likelihood ratio.

statistic

A single numeric of minus twice the constrained 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

Adimari G, Guolo A (2010). “A Note on the Asymptotic Behaviour of Empirical Likelihood Statistics.” Statistical Methods & Applications, 19(4), 463–476. doi:10.1007/s10260-010-0137-9 .

Qin J, Lawless J (1995). “Estimating Equations, Empirical Likelihood and Constraints on Parameters.” Canadian Journal of Statistics, 23(2), 145–159. doi:10.2307/3315441 .

## Examples

showClass("CEL")
#> Class "CEL" [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
#>
#> Extends: "EL"
#>
#> Known Subclasses:
#> Class "LM", directly
#> Class "GLM", by class "LM", distance 2
#> Class "QGLM", by class "GLM", distance 3