mo_compbeta: Moments of Order Statistics from the Complementary Beta...

In mos: Simulation and Moment Computation for Order Statistics

Moments of Order Statistics from the Complementary Beta Distribution

Description

This function computes the moments of order statistics from the complementary beta (CB) distribution. For small values of k and integer b, a closed-form formula is used; otherwise, Monte Carlo simulation is applied.

Usage

mo_compbeta(r, n, k = 1, a, b, rep = 1e+05, seed = 42, verbose = TRUE)

Arguments

r	rank of the desired order statistic (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, b	positive parameters of the complementary beta distribution.
rep	number of simulations (used when b is non-integer, default is 1e5).
seed	optional seed for random number generation to ensure reproducibility (used when b is non-integer, default is 42).
verbose	logical; if TRUE, prints a message when Monte Carlo simulation is used.

Details

The computation method varies depending on b and k:

• For integer b and k = 1, 2: The function calculates the moments using the closed-form expression derived in Makouei et al. (2021):

  \text{E}[X_{r:n}^s] = \frac{1}{B(r, n - r + 1)} \sum_{j=0}^{n-r} \binom{n - r}{j} (-1)^j \mathcal{M}^{(s)}(a, b, r + j - 1),

  Here

  \mathcal{M}^{(s)}(a, b, k) = \frac{1}{k + 1} \left[1 - \frac{s}{B(a, b)} \sum_{j=0}^{\infty} \binom{b-1}{j} (-1)^j \mathcal{M}^{(s-1)}(a, b, a + k + j) \right], \quad s \geq 1,

  with the starting point

  \mathcal{M}^{(1)}(a, b, k) = \frac{B(a + k + 1, b + 1)}{a B(a, b)} \cdot {_3F_2}\left(a + b, 1, a + k + 1; a + 1, a + b + k + 2; 1\right),

  where B(a, b) is the beta function, _3F_2 is the generalized hypergeometric function, and the upper limit of the summation stops at j = b - 1 if b is an integer.

• For non-integer b or k > 2: When b is non-integer or k is greater than 2 the function employs Monte Carlo simulation using the following formula:

  \text{E}[X^s] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^s,

  where X_i are the simulated order statistics obtained from the complementary beta distribution. The method relies on the ros() function to generate order statistics.

When verbose = TRUE, the function prints a message only if Monte Carlo simulation is used (i.e., when k > 2 or b is non-integer).

Value

The estimated or exact kth moment of the rth order statistic from a complementary beta distribution.

Note

The closed-form formula is only available for small values of k and integer b. Monte Carlo simulation is used otherwise, and results may vary slightly depending on rep.

References

Makouei, R., Khamnei, H. J., & Salehi, M. (2021). Moments of order statistics and k-record values arising from the complementary beta distribution with application. Journal of Computational and Applied Mathematics, 390, 113386.

See Also

ros for generating random samples of order statistics.

Examples

# Exact moment when k = 1
mo_compbeta(r = 2, n = 15, k = 1, a = 0.5, b = 2)

# Simulation when k > 2 or b is non-integer
mo_compbeta(r = 2, n = 15, k = 3, a = 2.5, b = 3.7, rep = 1e4)

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