mos (version 0.1.3)

mo_gamma: Moments of Order Statistics from the Gamma Distribution (Simulated)

Description

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

Usage

mo_gamma(r, n, k = 1, shape, rate, rep = 1e+05, seed = 42)

Value

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

shape

shape parameter of the gamma distribution.

rate

rate parameter of the gamma 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 gamma distribution with specified shape and rate 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 gamma distribution.

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

See Also

ros

Examples

# Compute the first moment (mean) of the 3rd order statistic from a sample of size 10
mo_gamma(r = 3, n = 10, shape = 2, rate = 1, k = 1)

# Compute the second moment with 10000 simulations
mo_gamma(r = 2, n = 10, shape = 2, rate = 0.5, k = 2, rep = 1e4)