Sampling Covariance of two sample variances (for a finite population and a small N[population size])) What is the expression for the sampling covariance of two sample variances i.e. $\operatorname{Cov}(s_1^2, s_2^2)$. Where $s_1^2=\frac {1}{n-1} \sum(x_i-\bar x )^2$ and $s_2^2=\frac {1}{n-1} \sum(y_i-\bar y )^2$?
 A: This is a problem of finding moments of moments. Given a sample of size $n$, namely $((X_1, Y_1), \dots, (X_n, Y_n))$, we seek the covariance:
$$\text{ Cov}\big(\frac{1}{n-1}\sum _{i=1}^n \left(X_i-\bar{X}\right)^2, \frac{1}{n-1}\sum _{i=1}^n \left(Y_i-\bar{Y}\right)^2\big)$$
The modus operandi for solving such problems is to work with power sum notation $s_{r,t}$, namely:
$$s_{r,t}=\sum _{i=1}^n X_i^r Y_i^t$$
Let:


*

*$m_{20} = \frac{1}{n-1}\sum _{i=1}^n \left(X_i-\bar{X}\right){}^2 \left(Y_i-\bar{Y}\right){}^0$ and

*$m_{02} = \frac{1}{n-1}\sum _{i=1}^n \left(X_i-\bar{X}\right){}^0 \left(Y_i-\bar{Y}\right){}^2$ 


... denote our two unbiased estimators of the variance of $X$ and $Y$ respectively, which can be expressed in power sum notation as:

Solution
The covariance operator is just the $\mu_{1,1}$ central moment ... so:


*

*$\text{Cov}(m_{20},m_{02})$ is given by:



where:


*

*$\mu _{r,s}$ denotes the product central moment:


$$\mu _{r,s}=E\left[(X-E[X]]^r (Y-E[Y])^s\right]$$
... so $\mu_{1,1} = \text{Cov}(X,Y)$, $\mu_{2,0}= Var(X)$ and $\mu_{0,2}= Var(Y)$


*

*CentralMomentToCentral is a function from the mathStatica package for Mathematica.


The OP further notes that he is sampling from a finite population. Presumably this means without replacement. This requires a further 'weighting' adjustment, typically something like $ \frac{N}{N-1}$ where $N$ denotes the finite population size, reducing to the above solution as $N$ becomes large. The extension to modify results to finite populations without replacement can be done by the Irwin-Kendall principle ... a discussion of which is provided in Stuart and Ord: Kendall's Advanced Theory of Statistics - volume 1 (6th edition) at Section 12.20 and 12.21. 
