mo_norm: Moments of Order Statistics from the Normal Distribution...
In mos: Simulation and Moment Computation for Order Statistics

mo_norm

R Documentation

Moments of Order Statistics from the Normal Distribution (Simulated)

Description

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

Usage

mo_norm(r, n, k = 1, mean = 0, sd = 1, rep = 1e+05, seed = 42)

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`).
`mean`	mean of the normal distribution (default is `0`).
`sd`	standard deviation of the normal distribution (default is `1`).
`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 kth moment of the rth order statistic in a sample of size n drawn from a normal distribution with the specified mean and standard deviation. 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 obtained from the normal distribution.

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

Value

The estimated kth moment of the rth order statistic from a normal distribution.

Note

The accuracy of the estimated moment depends on the number of simulations (rep). The default value rep = 1e5 provides a reasonable trade-off between speed and accuracy for most practical cases. For higher order moments or when greater precision is required, users are encouraged to increase rep (e.g. 1e6).

Examples

# Compute the first moment (mean) of the 3rd order statistic from a sample of size 10
mo_norm(r = 3, n = 10, k = 1, mean = 0, sd = 1)

# Compute the second moment of the 2nd order statistic with 1 million simulations
mo_norm(r = 2, n = 10, k = 2, rep = 1e6)

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