Probability Weighted Moments

\(\tau_{r,s,w} = E\{Z^r F(Z)^s (1 - F(Z))^w\}\).

Usage

bd_pwm(r, s = 0, w = 0, a, b, c, k, rel.tol = 1e-10)

Arguments

r

Power on \(Z\).

s

Power on \(F(Z)\).

w

Power on \(1 - F(Z)\).

a

Shape parameter (beta generator).

b

Shape parameter governing the tail.

c

Shape parameter (baseline).

k

Scale parameter (baseline).

rel.tol

Passed to [stats::integrate()].

Value

A single number.

Details

Existence follows the same rule as the raw moments in the worst case, \(b > r\), and the weights can only reduce the tail contribution.

Examples

# tau_{1,0,0} is the mean
bd_pwm(1, 0, 0, a = 1.5, b = 3, c = 2, k = 1)
#> [1] 1.84375
bd_moments(1, a = 1.5, b = 3, c = 2, k = 1)
#> [1] 1.84375

bd_pwm(1, 1, 0, a = 1.5, b = 3, c = 2, k = 1)
#> [1] 1.355136