calculate properly statistical moments with stratified sample data I have problems of how to calculate properly the third and fourth order moments: my data is a stratified sampling with three strata.
The goal for me is to make a descriptive analysis: mean, variance, coefficient of variation, the problems become when I try calculate the skewness and kurtosis.
Here the notation and math that I follow:
$\bar X _h$ = sample mean in stratum h
$x_h$ = total sample in stratum h.
$N_h$ = population size of stratum h.
$n_h$ = sample size of stratum h.
$fe_h$ = elevation factor in stratum h.
To calculate:
Total population:
$\hat X_{st} = \sum ^{L} _{h=1} \hat x _h = \sum ^L _{h=1} N_h \bar x_h = \sum ^L _{h=1} \frac{N_h}{n_h} x_h = \sum^L_{h=1} fe_hx_h$
Mean population:
$\hat{\bar{X}}_{st} = \bar x_{st} = \sum^L_{h=1} W_h \bar x_h = \sum ^L_{h=1} \frac{N_h}{N} \frac{1}{n_h}x_h = \frac{1}{N} \sum^L_{h=1} \frac{N_h}{n_h}x_h = \frac{1}{N} \sum^L_{h=1} fe_h x_h$
$\hat V (\hat X_{st}) = \sum ^L_{h=1} N^2_h(1-f_h) \frac{\hat S_h^2}{n_h}$
$\hat V (\hat{\bar{X}}_{st}) = \sum ^L_{h=1} W^2_h(1-f_h) \frac{\hat S_h^2}{n_h}$
where $\hat S ^2_h$ is sample quasivariance of stratum h.
So, how should I proceed to calculate the moments of third and fourth order (unbiased?) with this type of data.
I appreciate any help, guidance or feedback.
 A: You may be confusing 


*

*the population vs. the sample moments, on one hand, and

*the population variance and the variance of the estimate of the mean (which involves the strata variances... unfortunately... as well as a lot of other stuff like finite population corrections $1-f_h$).


The population moment of order $k$ is, obviously (with some abuse of notation switching between the flat population and the stratified population):
$$
M_k = \frac{1}{N} \sum_{i=1}^N x_i^k = \sum_{h=1}^L \sum_{i=1}^{N_h} x_{hi}^k \Bigl/ \sum_{h=1}^L N_h
$$
The population variance is
$$
V = M_2 - M_1^2
$$
The population skewness (I made up the letter) is
$$
\Gamma = \frac{M_3 - 3 M_2 M_1 + 2 M_1^3}{(M_2 - M_1^2)^{3/2}}
$$
The population kurtosis is
$$
K = \frac{M_4 - 2 M_2 M_1^2 + M_1^4}{(M_2 - M_1^2)^2}
$$
The raw moments are estimable with their plug-in analogues (with abuse of notation, again, with the index $i$ running over the sample now):
$$
\hat M_k = \sum_{h=1}^L \sum_{i=1}^{n_h} w_{hi} x_{hi}^k \Bigl/ \sum_{h=1}^L \sum_{i=1}^{n_h} w_{hi}
$$
($w_{hi}$ are the analysis weights that you referred to as elevation factors $fe_h$ -- I honestly wonder what discipline you come from to call them these way; please, please comment below and let me know; in the simplest case of a simple random sample with no nonresponse, $w_{hi}=N_h=/n_h$).
Unless your sample sizes were fixed by the sampling design, the estimated moments $\hat M_K$, being ratios of random variables, are biased estimators of their target quantities. The biases are of the order $O(1/n)$, vs. the sampling error of the order $O(1/\sqrt{n})$, and hence disappear asymptotically. The plug-in estimates of skewness and kurtosis are also biased, and their biases also disappear asymptotically. The standard errors around $\hat M_k$, $\hat \Gamma$, $\hat K$ can be obtained by the delta method / Taylor series linearization. If I needed them (I am a lucky Stata user), I would just 
svyset [pw=fe_h], strata(strata)
forvalues k=2/4 {
   generate x_`k' = x^`k'
}
svy: mean x x_2 x_3 x_4
nlcom (skew: (_b[x_3]-3*_b[x_2]*_b[x]+2*_b[x]*_b[x]*_b[x])/( (_b[x_2]-_b[x]*_b[x])^(3/2) ) )

and Stata would produce the required standard errors that take stratification into account. Note that the standard errors for the moments of order $k$ (implicitly) rely on the population moments of order $2k$, so they won't be very stable/accurate.
Depending on what's available to you, some references may be helpful as introductory reading on analysis of survey data:


*

*Korn and Graubard (1995 JRSS-A)

*Korn and Graubard (1999 Wiley book)

*Heeringa, West and Berglund (2010 Chapman and Hall)

*Kolenikov and Pitblado (2014 chapter in Wiley handbook) -- that would be me; a copy can be found on ResearchGate.

