Raw Moment of a Lognormal Distribution — moment

Computes a raw moment of a lognormal distribution using its closed-form expression. The lognormal distribution is specified by its log-mean ($\mu$) and log-standard deviation ($\sigma$).

Usage

moment_lognormal(mu, sigma, n)

Arguments

mu: Numeric vector of means of the log-transformed variable, corresponding to $\mu$.
sigma: Non-negative numeric vector of standard deviations of the log-transformed variable, corresponding to $\sigma$.
n: Numeric vector specifying the order of the raw moment.

Value

A numeric vector containing the raw moments of the specified lognormal distribution.

Details

Let $X \sim N(\mu, \sigma^2)$ and $Y = \exp(X)$. Then $Y$ follows a lognormal distribution, and its raw moment of order $n$ is: $$ E[Y^n] = \exp\left(n\mu + \frac{n^2\sigma^2}{2}\right). $$

In particular, the mean is: $$ E[Y] = \exp\left(\mu + \frac{\sigma^2}{2}\right), $$ and the variance can be calculated as: $$ \mathrm{Var}(Y) = E[Y^2] - E[Y]^2. $$

This function can be used to adjust predictions from generalized linear mixed models with normally distributed random parameters and a log link.

Examples

mu <- 0
sigma <- 1

# Mean of the lognormal distribution
moment_lognormal(mu, sigma, n = 1)
#> [1] 1.648721

# Variance of the lognormal distribution
moment_1 <- moment_lognormal(mu, sigma, n = 1)
moment_2 <- moment_lognormal(mu, sigma, n = 2)
moment_2 - moment_1^2
#> [1] 4.670774