The Burr distribution

Source: R/dist_burr.R

[Stable]

The Burr distribution (Type XII) is a flexible continuous probability distribution often used for modeling income distributions, reliability data, and failure times.

dist_burr(shape1, shape2, rate = 1, scale = 1/rate)

Arguments

shape1, shape2, scale

parameters. Must be strictly positive.

rate

an alternative way to specify the scale.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_burr.html

In the following, let \(X\) be a Burr random variable with parameters shape1 = \(\alpha\), shape2 = \(\gamma\), and rate = \(\lambda\).

Support: \(x \in (0, \infty)\)

Mean: \(\frac{\lambda^{-1/\alpha} \gamma B(\gamma - 1/\alpha, 1 + 1/\alpha)}{\gamma}\) (for \(\alpha \gamma > 1\))

Variance: \(\frac{\lambda^{-2/\alpha} \gamma B(\gamma - 2/\alpha, 1 + 2/\alpha)}{\gamma} - \mu^2\) (for \(\alpha \gamma > 2\))

Probability density function (p.d.f):

$$ f(x) = \alpha \gamma \lambda x^{\alpha - 1} (1 + \lambda x^\alpha)^{-\gamma - 1} $$

Cumulative distribution function (c.d.f):

$$ F(x) = 1 - (1 + \lambda x^\alpha)^{-\gamma} $$

Quantile function:

$$ F^{-1}(p) = \lambda^{-1/\alpha} ((1 - p)^{-1/\gamma} - 1)^{1/\alpha} $$

Moment generating function (m.g.f):

Does not exist in closed form.

See also

actuar::Burr

Examples

dist <- dist_burr(shape1 = c(1,1,1,2,3,0.5), shape2 = c(1,2,3,1,1,2))
dist
#> <distribution[6]>
#> [1] Burr12(1, 1, 1)   Burr12(1, 2, 1)   Burr12(1, 3, 1)   Burr12(2, 1, 1)  
#> [5] Burr12(3, 1, 1)   Burr12(0.5, 2, 1)

mean(dist)
#> [1]      Inf 1.570796 1.209200 1.000000 0.500000      Inf
variance(dist)
#> [1]       NaN       Inf 0.9562355       Inf 0.7500000       NaN
support(dist)
#> <support_region[6]>
#> [1] [0,Inf) (0,Inf) (0,Inf) [0,Inf) [0,Inf) (0,Inf)
generate(dist, 10)
#> [[1]]
#>  [1]  0.6953820  0.3172482  0.1960645  0.3109262  1.3965302  6.2424490
#>  [7] 11.3693546  0.5244306  0.6611189  0.5220797
#> 
#> [[2]]
#>  [1] 1.4270902 0.1447616 0.6310605 0.3820480 0.1303857 1.8908569 0.7105088
#>  [8] 1.2517875 4.5507249 0.7880165
#> 
#> [[3]]
#>  [1] 0.8754316 1.1338995 0.5485422 0.9766912 0.2765234 3.8657375 0.7856267
#>  [8] 1.1919504 0.4469956 0.7802311
#> 
#> [[4]]
#>  [1] 0.22613813 0.15007801 0.35726677 1.04427563 0.40180149 0.54808338
#>  [7] 0.17286512 0.25226635 0.58828240 0.02089587
#> 
#> [[5]]
#>  [1] 0.49603562 1.71078703 0.20174431 0.66179354 0.99545397 0.02150891
#>  [7] 0.07664562 0.09673076 0.23369863 0.22289518
#> 
#> [[6]]
#>  [1] 10.3765735  2.3728921  5.7152576  1.0468940  1.0924432  0.3416535
#>  [7]  4.9972601  0.2565234  2.3819379  1.1680588
#> 

density(dist, 2)
#> [1] 0.11111111 0.16000000 0.14814815 0.07407407 0.03703704 0.17888544
density(dist, 2, log = TRUE)
#> [1] -2.197225 -1.832581 -1.909543 -2.602690 -3.295837 -1.721010

cdf(dist, 4)
#> [1] 0.8000000 0.9411765 0.9846154 0.9600000 0.9920000 0.7574644

quantile(dist, 0.7)
#> [1] 2.3333333 1.5275252 1.3263524 0.8257419 0.4938016 3.1797973