el_sd: Empirical likelihood for the standard deviation

Description

Computes empirical likelihood for the standard deviation.

Usage

el_sd(x, mean, sd, weights = NULL, control = el_control())

Value

An object of class SD.

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().

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.

Examples

Run this code

data("women")
x <- women$height
w <- women$weight
fit <- el_sd(x, mean = 65, sd = 5, weights = w)
fit
summary(fit)