mos (version 0.1.3)

mo_beta: Moments of Order Statistics from the Beta Distribution (Simulated)

Description

This function computes the moments of order statistics from the beta distribution using simulation.

Usage

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

Value

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

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

non-negative parameters of the beta distribution.

rep

number of simulations (default is 1e5).

seed

optional seed for random number generation to ensure reproducibility (default is 42).

Details

This function estimates the \(k\)th moment of the \(r\)th order statistic in a sample of size \(n\) drawn from a beta distribution with specified shape parameters. The estimation is done via Monte Carlo simulation using the formula:

$$\text{E}[X^k] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^k,$$ where \(X_i\) are the simulated order statistics from the beta distribution.

The function relies on the ros() function to generate order statistics.

See Also

ros for generating random samples of order statistics.

Examples

# Compute the first moment of the 2nd order statistic from Beta(3, 4) with sample size 5
mo_beta(r = 2, n = 5, k = 1, a = 3, b = 4)

# Compute the second moment with 10000 simulations
mo_beta(r = 2, n = 5, k = 2, a = 2, b = 2.5, rep = 1e4)