mos (version 0.1.3)

mo_weibull: 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)

Value

The \(k\)th moment of the \(r\)th order statistic from a weibull distribution.

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 \(k\)th 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.

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)