Computes empirical likelihood for the standard deviation.
Usage
el_sd(x, mean, sd, weights = NULL, control = el_control())Arguments
- x
A numeric vector, or an object that can be coerced to a numeric vector.
- mean
A single numeric for the (known) mean value.
- sd
A positive single numeric for the parameter value to be tested.
- 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 length of
x. Defaults toNULL, 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 SD.
Details
Let \(X_i\) be independent and identically random variable from an
unknown distribution \(P\) for \(i = 1, \dots, n\). We assume that
\({\textrm{E}[X_i]} = {\mu_0}\) is known and that \(P\) has a variance
\(\sigma_0^2\). Given a value of \(\sigma\), the
(profile) empirical likelihood ratio is defined by
$$R(\sigma) =
\max_{p_i}\left\{\prod_{i = 1}^n np_i :
\sum_{i = 1}^n p_i (X_i - \mu_0)^2 = \sigma^2,\
p_i \geq 0,\
\sum_{i = 1}^n p_i = 1
\right\}.$$
el_sd() computes the empirical log-likelihood ratio statistic
\(-2\log R(\sigma)\), along with other values in SD.
See also
EL, SD, el_mean(), elt(),
el_control()
Examples
data("women")
x <- women$height
w <- women$weight
fit <- el_sd(x, mean = 65, sd = 5, weights = w)
fit
#>
#> Weighted Empirical Likelihood
#>
#> Model: sd
#>
#> Maximum EL estimates:
#> [1] 4.34
#>
#> Chisq: 1.843, df: 1, Pr(>Chisq): 0.1746
#> EL evaluation: converged
#>
summary(fit)
#>
#> Weighted Empirical Likelihood
#>
#> Model: sd
#>
#> Number of observations: 15
#> Number of parameters: 1
#>
#> Parameter values under the null hypothesis:
#> [1] 5
#>
#> Lagrange multipliers:
#> [1] -0.01913
#>
#> Maximum EL estimates:
#> [1] 4.34
#>
#> logL: -41.45, logLR: -0.9215
#> Chisq: 1.843, df: 1, Pr(>Chisq): 0.1746
#> EL evaluation: converged
#>