burr_plt function

Burr coefficients after power-law transformation

Coefficients of a power-law transformed Burr distribution

burr_plt(shape1 = 2, shape2 = 1, scale = 0.5, a = 1, b = 1, inv = FALSE)

Arguments

• shape1, shape2, scale: Shape1, shape2 and scale of the Burr distribution, defaults to 2, 1 and 1 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.

Returns

Returns a named list containing

• coefficients: Named vector of coefficients

Comparing probabilites of power-law transformed transformed variables pburr(3,shape1=2,shape2=3,scale=1) coeff = burr_plt(shape1=2,shape2=3,scale=1,a=5,b=7)$coefficients pburr(5*3^7,shape1=coeff[["shape1"]],shape2=coeff[["shape2"]],scale=coeff[["scale"]])

pburr(5*0.9^7,shape1=2,shape2=3,scale=1) coeff = burr_plt(shape1=2,shape2=3,scale=1,a=5,b=7, inv=TRUE)$coefficients pburr(0.9,shape1=coeff[["shape1"]],shape2=coeff[["shape2"]],scale=coeff[["scale"]])

Comparing the first moments and sample means of power-law transformed variables for large enough samples x = rburr(1e5,shape1=2,shape2=3,scale=1) coeff = burr_plt(shape1=2,shape2=3,scale=1,a=2,b=0.5)$coefficients y = rburr(1e5,shape1=coeff[["shape1"]],shape2=coeff[["shape2"]],scale=coeff[["scale"]]) mean(2*x^0.5) mean(y) mburr(r=1,shape1=coeff[["shape1"]],shape2=coeff[["shape2"]],scale=coeff[["scale"]],lower.tail=FALSE)

Details

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

is Burr distributed with shape1 $shape1$, shape2 $b*shape2$ and scale $( \frac{scale}{a})^{\frac{1}{b}}$.