Order Statistic Distribution and Moments
Usage
bd_order_stat_cdf(t, i, n, a, b, c, k)
bd_order_stat_moments(r, i, n, a, b, c, k, rel.tol = 1e-10)Arguments
- t
Vector of times, for `bd_order_stat_cdf`.
- i
Index of the order statistic, from 1 to `n`.
- n
Sample size.
- a
Shape parameter (beta generator).
- b
Shape parameter governing the tail.
- c
Shape parameter (baseline).
- k
Scale parameter (baseline).
- r
Order of the moment, for `bd_order_stat_moments`.
- rel.tol
Passed to [stats::integrate()].
Value
`bd_order_stat_cdf` returns a vector the length of `t`; `bd_order_stat_moments` returns a single number.
Details
\(F_{i:n}(t) = I_{F(t)}(i, n - i + 1)\), and moments are taken against the order-statistic density on the Beta scale. The \(i\)-th order statistic of a sample of size \(n\) has a finite \(r\)-th moment when \(b(n - i + 1) > r\), so extremes need heavier restrictions than the parent distribution, and the maximum needs \(b > r\) exactly as the parent does.
Examples
bd_order_stat_cdf(c(0.5, 1, 2), i = 3, n = 5, a = 1.5, b = 3, c = 2, k = 1)
#> [1] 0.02256538 0.31306186 0.85578769
bd_order_stat_moments(1, i = 1, n = 5, a = 1.5, b = 3, c = 2, k = 1)
#> [1] 0.53635