wtd.moments: 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)

Value

Vector with values:

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

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)?

Author

Richard Valliant, Jill A. Dever, Frauke Kreuter

Details

The $r^{t h}$ 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 $r^{t h}$ 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.

References

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

Examples

Run this code

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)

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