The melt package provides several functions to construct an
EL object or an object that inherits from
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) #>  "EL" #> attr(,"package") #>  "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
faithful data frame is coerced to a numeric matrix. Simple print method shows essential information on
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-
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.
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.