mos (version 0.1.3)

mo_topple: Moments of Order Statistics from the Topp-Leone Distribution

Description

This function computes the moments of order statistic from the topp-leone distribution, based on the formula presented in Genç, A. İ. (2012).

Usage

mo_topple(r, n, k = 1, a, b = 1)

Value

The \(k\)th moment of the \(r\)th order statistic from a topp-leone 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).

a

shape parameter of the topp-leone distribution (a > 0).

b

scale parameter of the topp-leone distribution (default is 1, b > 0).

Details

This function implements the exact formula for moments of order statistics from the topp-leone distribution as provided in Genç, A. İ. (2012): $$ \text{E}[X_{r:n}^k] = \frac{n!(ab^k)}{(r-1)!(n-r)!} \sum_{j=0}^{n-r} \binom{n-r}{j} (-1)^j 2^{k + 2a(r+j)} \left[ B_{1/2}(k + a(r+j), a(r+j)) - 2 B_{1/2}(k + a(r+j) + 1, a(r+j)) \right] $$ Here, \(B_x(., .)\) is the incomplete Beta function.

References

Genç, A. İ. (2012). Moments of order statistics of Topp–Leone distribution. Statistical Papers, 53, 117-131.

Examples

# Compute the first moment of the first order statistic for n=5, a=2, b=1
mo_topple(1, 5, 1, 2)

# Compute the second moment of the second order statistic for n=10, a=1.5, b=2
mo_topple(2, 10, 2, 1.5, 2)