dIH: Density of Irwin-Hall distribution

In cobin: Cobin and Micobin Regression Models for Continuous Proportional Data

Density of Irwin-Hall distribution

Description

Irwin-Hall distribution is defined as a sum of m uniform (0,1) distribution. Its density is given as

f(x;m) = \frac{1}{(m-1)!}\sum_{k=0}^{m} (-1)^k {m \choose k} \max(0,x-k)^{m-1}, 0 < x < m

The density of Bates distribution, defined as an average of m uniform (0,1) distribution, can be obtained from change-of-variable (y = x/m),

h(y;m) = \frac{m}{(m-1)!}\sum_{k=0}^{m} (-1)^k {m \choose k} \max(0,my-k)^{m-1}, 0 < y < 1

Usage

dIH(x, m, log = FALSE)

Arguments

x	vector of quantities, between 0 and m
m	integer, parameter
log	logical, return log density if TRUE

Details

Due to alternating series representation, m > 80 may yield numerical issues

Value

(log) density evaluated at x

Examples

m = 8
xgrid= seq(0, m, length = 500)
hist(colSums(matrix(runif(m*1000), nrow = m, ncol = 1000)), freq = FALSE) 
lines(xgrid, dIH(xgrid, m, log = FALSE))
# Bates distribution
xgrid= seq(0, 1, length = 500)
hist(colMeans(matrix(runif(m*1000), nrow = m, ncol = 1000)), freq = FALSE) 
lines(xgrid, m*dIH(xgrid*m, m, log = FALSE))

cobin documentation built on Sept. 2, 2025, 1:08 a.m.