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 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 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
#>
```