mo_exp: Moments Of Order Statistics from the Exponential Distribution

In mos: Simulation and Moment Computation for Order Statistics

Moments Of Order Statistics from the Exponential Distribution

Description

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

Usage

mo_exp(r, n, k = 1, mu = 0, sigma = 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).
mu	location parameter of the exponential distribution (default is 0).
sigma	scale parameter of the exponential distribution (default is 1).

Details

The function calculates the kth moment using the following relationship:

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

For non-standard exponential distributions with \mu and \sigma parameters, the transformation X^*=\mu + \sigma X is used.

Value

The kth moment of the rth order statistic from an exponential distribution.

References

Ahsanullah, M., Nevzorov, V. B., & Shakil, M. (2013). An introduction to order statistics (Vol. 8). Paris: Atlantis Press.

Examples

# First moment (mean) of the 2nd order statistic from a sample of size 5
mo_exp(2, 5, k = 1, mu = 0, sigma = 1)

# Second moment of the 3rd order statistic from an exponential distribution
# with mu = 2 and sigma = 3
mo_exp(3, 7, k = 2, mu = 2, sigma = 3)

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