rmammen: Distributions suggested by Mammen for Wild Bootstrap.

In reyesem/IntroAnalysis: Functions for introductory statistics using linear models

Distributions suggested by Mammen for Wild Bootstrap.

Description

Generates random variates from one of three distributions suggested by Mammen (1993) for use in Wild Bootstrap.

Usage

rmammen(n, construct = c("normal-2", "normal-1", "two-point mass"))

Arguments

n	Number of observations. If length(n) > 1, the length is taken to be the number required.
construct	Character string; indicates which of three constructs from which to generate. One of "two-point mass", "normal-1", normal-2" (default).

Details

Generates a random variable U. If construct = "two-point mass", the random variable U is defined as

Pr(U = u) = \begin{cases} \frac{\sqrt{5} + 1}{2\sqrt{5}} & u = \frac{-(\sqrt{5}-1)}{2} \\ \frac{\sqrt{5} - 1}{2\sqrt{5}} & u = \frac{\sqrt{5}+1}{2} \end{cases}

If construct = "normal-1", the random variable U is defined as

U = \frac{V}{\sqrt{2}} + \frac{V^2 - 1}{2}

where V \sim N(0,1). If construct = "normal-2", the random variable U is defined as

U = \left(\delta_1 + V_1/\sqrt{2}\right) \left(\delta_2 + V_2/\sqrt{2}\right) - \delta_1\delta_2

where V_1,V_2 are independent N(0,1) random variables and

\delta_1 = \left(3/4 + \sqrt{17}/12\right)^{1/2}

\delta_2 = \left(3/4 - \sqrt{17}/12\right)^{1/2}

Value

A vector of length n.

References

Mammen E. "Bootstrap and Wild Bootstrap for High Dimensional Linear Models." The Annals of Statistics (1993) 21(1):255-285.

Examples

mammen_sample <- rmammen(10000, construct = "normal-2")
mean(mammen_sample)
mean(mammen_sample^2)
mean(mammen_sample^3)

plot(density(mammen_sample,
main = "Mammen's Suggested Distribution for the Wild Bootstrap"))

reyesem/IntroAnalysis documentation built on March 29, 2025, 3:29 p.m.