mo_tri: Moments of Order Statistics from the Symmetric Triangular...

In mos: Simulation and Moment Computation for Order Statistics

Moments of Order Statistics from the Symmetric Triangular Distribution

Description

This function computes the moments of order statistics from the symmetric triangular distribution.

Usage

mo_tri(r, n, k = 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).

Details

The function implements the following relationship from Nagaraja (2013) for the symmetric triangular distribution:

\text{E}[X_{r:n}^k] = \frac{n!}{(r-1)!(n-r)!} \left\{ (\frac{1}{2})^{k/2} B\left(\frac{1}{2}; \frac{k}{2} + r, n - r + 1\right) + \sum_{j=0}^k (-1)^j \binom{k}{j} (\frac{1}{2})^{j/2} B\left(\frac{1}{2}; \frac{j}{2} + n - r + 1, r\right) \right\}.

Here, B(x; a, b) is the incomplete Beta function.

Value

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

References

Nagaraja, H. N. (2013). Moments of order statistics and L-moments for the symmetric triangular distribution. Statistics & Probability Letters, 83(10), 2357-2363.

Examples

# Compute the 2nd moment of the 3rd order statistic for n=5
mo_tri(3, 5, 2)

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