wtd.moments: Compute moments of a variable from either a population or...

In PracTools: Designing and Weighting Survey Samples

Compute moments of a variable from either a population or sample

Description

Compute the 2nd, 3rd, 4th moments, skewness, and kurtosis of a variable from either population or sample input

Usage

wtd.moments(y, w=NULL, pop.sw=TRUE)

Arguments

y	variable to be analyzed
w	vector of weights if the input is a sample
pop.sw	is the input for a population (pop.sw=TRUE) or for a sample (pop.sw=FALSE)?

Details

The r^{th} population moment is defined as m_r = (1/N) \sum_{k \in U} (y_k - \bar{y}_U)^r where U is the set of population units, N is the population size, and \bar{y}_U is the population mean. When the input is for the whole population, wtd.moments evaluates this directly for r=2, 3, 4. When the input is for a sample, the r^{th} moment is estimated as \hat{m}_r = (K/\hat{N}) \sum_{k \in s} ( w_k (y_k - \hat{\bar{y}}_U)^r ), r=2, 3, 4 where s is the set of sample units, w_k is the weight for sample unit k, \hat{N} = \sum_s w_k, and \hat{\bar{y}}_U = \sum_{k \in s} w_k y_k / \hat{N}. When r=2, K=n/(n-1) so that the estimator equals the unbiased variance estimator if the sample is a simple random sample; if r=3,4, then K=1. The function also computes or estimates the population skewness, defined as m_3/m_2^{3/2} and the population kurtosis, m_4/m_2^2.

The weights should be scaled for estimating population totals. The sample can be obtained from any complex design. The estimators produced by wtd.moments are consistent for general sample designs but not unbiased (except for r=2 and SRS).

Value

Vector with values:

m2	2nd moment
m3	3rd moment
m4	4th moment
skewness	skewness
kurtosis	kurtosis

Author(s)

Richard Valliant, Jill A. Dever, Frauke Kreuter

References

Valliant, R., Dever, J., Kreuter, F. (2018, sect. 3.4). Practical Tools for Designing and Weighting Survey Samples, 2nd edition. New York: Springer.

See Also

wtdvar

Examples

require(PracTools)
wtd.moments(y = hospital$y, w = NULL)
require(sampling)
sam <- strata(data = labor, stratanames = "h", size = c(30, 20, 10), method = c("srswor"),
              description=TRUE)
samdat <- labor[sam$ID_unit,]
wtd.moments(y = samdat$WklyWage, w = 1/sam$Prob, pop.sw=FALSE)

PracTools documentation built on Sept. 17, 2026, 1:09 a.m.