uM2M3 {Umoments}

Unbiased central moment estimates

Description

Calculate unbiased estimates of central moments and their powers and products.

Usage

uM2M3(m2, m3, m5, n)

Arguments

m2

naive biased variance estimate m_2 = 1/n \sum_{i = 1}^n ((X_i - \bar{X})^2 for a vector X.

m3

naive biased third central moment estimate m_3 = 1/n \sum_{i = 1}^n ((X_i - \bar{X})^3 for a vector X.

m5

naive biased fifth central moment estimate m_5 = \sum_{i = 1}^n ((X_i - \bar{X})^5 for a vector X.

n

sample size.

Value

Unbiased estimate of a product of second and third central moments \mu_2 \mu_3, where \mu_2 and \mu_3 are second and third central moments respectively.

See Also

Other unbiased estimates (one-sample): uM2M4, uM2pow2, uM2pow3, uM2, uM3pow2, uM3, uM4, uM5, uM6

Examples

n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:5) {
  m <- c(m, mean((smp - m[1])^j))
}
uM2M3(m[2], m[3], m[5], n)