pareto_plt: Pareto coefficients after power-law transformation
In distributionsrd: Distribution Fitting and Evaluation

Description Usage Arguments Details Value Examples

Coefficients of a power-law transformed Pareto distribution

1	pareto_plt(xmin = 1, k = 2, a = 1, b = 1, inv = FALSE)

`xmin, k`	Scale and shape of the Pareto distribution, defaults to 1 and 2 respectively.
`a, b`	constant and power of power-law transformation, defaults to 1 and 1 respectively.
`inv`	logical indicating whether coefficients of the outcome variable of the power-law transformation should be returned (FALSE) or whether coefficients of the input variable being power-law transformed should be returned (TRUE). Defaults to FALSE.

If the random variable x is Pareto-distributed with scale xmin and shape k, then the power-law transformed variable

y = ax^b

is Pareto distributed with scale ( \frac{xmin}{a})^{\frac{1}{b}} and shape b*k.

Returns a named list containing

coefficients: Named vector of coefficients

## Comparing probabilites of power-law transformed transformed variables
ppareto(3, k = 2, xmin = 2)
coeff <- pareto_plt(xmin = 2, k = 2, a = 5, b = 7)$coefficients
ppareto(5 * 3^7, k = coeff[["k"]], xmin = coeff[["xmin"]])

ppareto(5 * 0.9^7, k = 2, xmin = 2)
coeff <- pareto_plt(xmin = 2, k = 2, a = 5, b = 7, inv = TRUE)$coefficients
ppareto(0.9, k = coeff[["k"]], xmin = coeff[["xmin"]])

## Comparing the first moments and sample means of power-law transformed variables for
#large enough samples
x <- rpareto(1e5, k = 2, xmin = 2)
coeff <- pareto_plt(xmin = 2, k = 2, a = 2, b = 0.5)$coefficients
y <- rpareto(1e5, k = coeff[["k"]], xmin = coeff[["xmin"]])
mean(2 * x^0.5)
mean(y)
mpareto(r = 1, k = coeff[["k"]], xmin = coeff[["xmin"]], lower.tail = FALSE)