distributionsrd (version 0.0.6)

lnorm: The Lognormal distribution

Description

Raw moments for the Lognormal distribution.

Usage

mlnorm(r = 0, truncation = 0, meanlog = -0.5, sdlog = 0.5, lower.tail = TRUE)

Arguments

r

rth raw moment of the distribution, defaults to 1.

truncation

lower truncation parameter, defaults to 0.

meanlog, sdlog

mean and standard deviation of the distribution on the log scale with default values of 0 and 1 respectively.

lower.tail

logical; if TRUE (default), moments are \(E[x^r|X \le y]\), otherwise, \(E[x^r|X > y]\)

Value

Provides the y-bounded, rth raw moment of the distribution.

Details

Probability and Cumulative Distribution Function:

$$f(x) = \frac{1}{{x Var \sqrt {2\pi } }}e^{- (lnx - \mu )^2/ 2Var^2} , \qquad F_X(x) = \Phi(\frac{lnx- \mu}{Var})$$

The y-bounded r-th raw moment of the Lognormal distribution equals:

$$\mu^r_y = e^{\frac{r (rVar^2 + 2\mu)}{2}}[1-\Phi(\frac{lny - (rVar^2 + \mu)}{Var})] $$

Examples

# NOT RUN {
## The zeroth truncated moment is equivalent to the probability function
plnorm(2, meanlog = -0.5, sdlog = 0.5)
mlnorm(truncation = 2)

## The (truncated) first moment is equivalent to the mean of a (truncated) random sample,
#for large enough samples.
x <- rlnorm(1e5, meanlog = -0.5, sdlog = 0.5)
mean(x)
mlnorm(r = 1, lower.tail = FALSE)

sum(x[x > quantile(x, 0.1)]) / length(x)
mlnorm(r = 1, truncation = quantile(x, 0.1), lower.tail = FALSE)
# }