`EL`

objectThe melt package provides several functions to construct an
`EL`

object or an object that inherits from
`EL`

:

`el_mean()`

for the mean.`el_sd()`

for the standard deviation.`el_lm()`

for linear models.`el_glm()`

for generalized linear models.

We illustrate the usage of `el_mean()`

with the
`faithful`

data set.

```
data("faithful")
str(faithful)
#> 'data.frame': 272 obs. of 2 variables:
#> $ eruptions: num 3.6 1.8 3.33 2.28 4.53 ...
#> $ waiting : num 79 54 74 62 85 55 88 85 51 85 ...
summary(faithful)
#> eruptions waiting
#> Min. :1.600 Min. :43.0
#> 1st Qu.:2.163 1st Qu.:58.0
#> Median :4.000 Median :76.0
#> Mean :3.488 Mean :70.9
#> 3rd Qu.:4.454 3rd Qu.:82.0
#> Max. :5.100 Max. :96.0
```

Suppose we are interested in evaluating empirical likelihood at (3.5, 70).

```
fit <- el_mean(faithful, par = c(3.5, 70))
class(fit)
#> [1] "EL"
#> attr(,"package")
#> [1] "melt"
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
```

The `faithful`

data frame is coerced to a numeric matrix.
Simple print method shows essential information on `fit`

.

```
fit
#>
#> Empirical Likelihood
#>
#> Model: mean
#>
#> Maximum EL estimates:
#> eruptions waiting
#> 3.488 70.897
#>
#> Chisq: 8.483, df: 2, Pr(>Chisq): 0.01439
#> EL evaluation: converged
```

Note that the maximum empirical likelihood estimates are the same as the sample average. The chi-square value shown corresponds to the minus twice the empirical log-likelihood ratio. It has an asymptotic chi-square distribution of 2 degrees of freedom under the null hypothesis. Hence the \(p\)-value here is not exact. The convergence status at the bottom can be used to check the convex hull constraint.

Weighted data can be handled by supplying the `weights`

argument. For non-`NULL`

`weights`

, weighted
empirical likelihood is computed. Any valid `weights`

is
re-scaled for internal computation to add up to the total number of
observations. For simplicity, we use `faithful$waiting`

as
our weight vector.

```
w <- faithful$waiting
(wfit <- el_mean(faithful, par = c(3.5, 70), weights = w))
#>
#> Weighted Empirical Likelihood
#>
#> Model: mean
#>
#> Maximum EL estimates:
#> eruptions waiting
#> 3.684 73.494
#>
#> Chisq: 25.41, df: 2, Pr(>Chisq): 3.039e-06
#> EL evaluation: converged
```

We get different results, where the estimates are now the weighted sample average. The chi-square value and the associated \(p\)-value are based on the same limit theorem, but care must be taken when interpreting the results since they are largely affected by the limiting behavior of the weights.