Empirical likelihood for the mean

Computes empirical likelihood for the mean.

Usage

el_mean(x, par, weights = NULL, control = el_control())

Arguments

x

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.

par

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.

weights

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.

control

An object of class ControlEL constructed by el_control().

Value

An object of class EL.

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 \({\textrm{E}[X_i]} = {\theta_0} \in {\rm{I\!R}}^p\) 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.

References

Owen A (1990). ``Empirical Likelihood Ratio Confidence Regions.'' The Annals of Statistics, 18(1), 90--120. doi:10.1214/aos/1176347494 .

See also

EL, elt(), el_eval(), el_control()

Examples

## 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 
#>