mo_weibull: Moments of Order Statistics from the Weibull Distribution

In mos: Simulation and Moment Computation for Order Statistics

Moments of Order Statistics from the Weibull Distribution

Description

This function computes the moments of order statistics from the weibull distribution.

Usage

mo_weibull(r, n, k = 1, shape, scale = 1)

Arguments

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
shape	shape parameter of the weibull distribution.
scale	scale parameter of the weibull distribution (default is 1).

Details

The function calculates the kth moment using the formula:

\text{E}[X_{r:n}^k] = \frac{n!}{(r-1)!(n-r)!} \Gamma\left(1 + \frac{k}{\text{shape}}\right) \sum_{j=0}^{r-1} (-1)^j \binom{r-1}{j} \frac{1}{(n-r+1+j)^{1 + \frac{k}{\text{shape}}}}

For non-standard weibull distributions (scale not equal to 1), the relationship \text{E}[Z_{r:n}^k] = \text{scale}^k \text{E}[X_{r:n}^k] is used.

Value

The kth moment of the rth order statistic from a weibull distribution.

References

Harter, H. L., & Balakrishnan, N. (1996). CRC handbook of tables for the use of order statistics in estimation. CRC press.

Examples

# Example 1: Standard weibull distribution (shape = 2, scale = 1)
mo_weibull(r = 2, n = 5, k = 1, shape = 2)

# Example 2: Non-standard weibull distribution (shape = 2, scale = 3)
mo_weibull(r = 3, n = 6, k = 2, shape = 2, scale = 3)

mos documentation built on June 16, 2025, 5:09 p.m.