Computes empirical likelihood for the mean.
el_mean(x, par, weights = NULL, control = el_control())
A numeric matrix, or an object that can be coerced to a numeric matrix. Each row corresponds to an observation. The number of rows must be greater than the number of columns.
A numeric vector of parameter values to be tested. The length of
the vector must be the same as the number of columns in x
.
An optional numeric vector of weights to be used in the
fitting process. The length of the vector must be the same as the number of
rows in x
. Defaults to NULL
, corresponding to identical weights.
If non-NULL
, weighted empirical likelihood is computed.
An object of class ControlEL constructed by
el_control()
.
An object of class EL.
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 \({\textrm{E}[X_i]} =
{\theta_0} \in {\rm{I\!R}}^p\) and that \(P\) has a positive definite
covariance matrix. Given a value of \(\theta\), the (profile) empirical
likelihood ratio is defined by
$$R(\theta) =
\max_{p_i}\left\{\prod_{i = 1}^n np_i :
\sum_{i = 1}^n p_i X_i = \theta,\
p_i \geq 0,\
\sum_{i = 1}^n p_i = 1
\right\}.$$
el_mean()
computes the empirical log-likelihood ratio statistic
\(-2\log R(\theta)\), along with other values in EL.
Owen A (1990). ``Empirical Likelihood Ratio Confidence Regions.'' The Annals of Statistics, 18(1), 90--120. doi:10.1214/aos/1176347494 .
## Scalar mean
data("precip")
fit <- el_mean(precip, 30)
fit
#>
#> Empirical Likelihood
#>
#> Model: mean
#>
#> Maximum EL estimates:
#> [1] 34.89
#>
#> Chisq: 8.285, df: 1, Pr(>Chisq): 0.003998
#> EL evaluation: converged
#>
summary(fit)
#>
#> Empirical Likelihood
#>
#> Model: mean
#>
#> Number of observations: 70
#> Number of parameters: 1
#>
#> Parameter values under the null hypothesis:
#> [1] 30
#>
#> Lagrange multipliers:
#> [1] 0.02319
#>
#> Maximum EL estimates:
#> [1] 34.89
#>
#> logL: -301.5, logLR: -4.142
#> Chisq: 8.285, df: 1, Pr(>Chisq): 0.003998
#> EL evaluation: converged
#>
## Vector mean
data("faithful")
fit2 <- el_mean(faithful, par = c(3.5, 70))
summary(fit2)
#>
#> Empirical Likelihood
#>
#> Model: mean
#>
#> Number of observations: 272
#> Number of parameters: 2
#>
#> Parameter values under the null hypothesis:
#> eruptions waiting
#> 3.5 70.0
#>
#> Lagrange multipliers:
#> [1] -0.33537 0.03043
#>
#> Maximum EL estimates:
#> eruptions waiting
#> 3.488 70.897
#>
#> logL: -1529, logLR: -4.241
#> Chisq: 8.483, df: 2, Pr(>Chisq): 0.01439
#> EL evaluation: converged
#>
## Weighted data
w <- rep(c(1, 2), each = nrow(faithful) / 2)
fit3 <- el_mean(faithful, par = c(3.5, 70), weights = w)
summary(fit3)
#>
#> Weighted Empirical Likelihood
#>
#> Model: mean
#>
#> Number of observations: 272
#> Number of parameters: 2
#>
#> Parameter values under the null hypothesis:
#> eruptions waiting
#> 3.5 70.0
#>
#> Lagrange multipliers:
#> [1] -0.30891 0.02829
#>
#> Maximum EL estimates:
#> eruptions waiting
#> 3.498 70.931
#>
#> logL: -1513, logLR: -3.641
#> Chisq: 7.282, df: 2, Pr(>Chisq): 0.02622
#> EL evaluation: converged
#>