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Let a simple random sampling design in a finite univers U (i.e. all possible samples of size n are equally likely to occur).

How to calculate the covariance of sampling mean $\bar{y}$ and sampling variance $s^{2}=\sum_{i=1}^{n} (y_{i} - \bar{y})^{2}/(n-1)$?

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  • $\begingroup$ It is interesting to note that for a Gaussian distribution the sample mean and variance are independent and hence the covariance is 0. How this works for other distributions I am not sure about. Have you checked this site for other posts on this topic? $\endgroup$ – Michael R. Chernick Apr 14 '17 at 22:09
  • $\begingroup$ Please explain what kind of sampling you are doing, with enough detail to enable us to know something about the probability distribution of $(\bar y, s^2)$. $\endgroup$ – whuber Apr 14 '17 at 22:50
  • $\begingroup$ Michael, Intuitively that covariance should also be zero...Thanks. $\endgroup$ – Augusto Apr 15 '17 at 13:38
  • $\begingroup$ You could try the bootstrap --- it might give some information. To say more, we really needs more details, as asked for. $\endgroup$ – kjetil b halvorsen Apr 15 '17 at 16:52
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This is known as a moment of moment problem ... it is often convenient to express such problems in power sum notation namely: $s_r = \sum_{i=1}^n X_i^r$. The sample mean expressed in power sums in simply $\frac{s_1}{n}$, while the sample variance can be expressed (see for instance) in power sums as:

$$p \; = \; \frac{1}{n-1} \sum _{i=1}^n \left(X_i-\bar{X}\right)^2 \quad = \quad \frac{n s_2 -s_1^2}{n(n-1)}$$

We seek $\text{Cov}(\frac{s_1}{n}, p)$ where the covariance $\mu_{1,1}$ operator is just the {1,1} product central moment. The solution can then be found using the mathStatica function:

enter image description here

where:

  • $\mu_3$ denotes the $3^{\text{rd}}$ central moment of random variable $X$

The solution holds for any distribution whose moments exist. Note that if the parent distribution $X$ is symmetrical (as in the Normal case - see comments above), then $\mu_3 = 0$, and so the above covariance $\frac{\mu_3}{n}$ will also be zero.

I was not quite sure what the OP meant by 'finite universe', but if the OP is sampling without replacement from finite populations, then adjustments to these results are needed: there is some discussion in Stuart and Ord on how to do this.

More detail

There is an extensive discussion of such moments of moments problems in Chapter 7 of our book:

  • Rose and Smith, "Mathematical Statistics with Mathematica", Springer, NY

A free download of the written chapter is available here:

http://www.mathstatica.com/book/bookcontents.html

which also includes references for further reading. For manual approaches (though this is not easy and prone to error (including many of the published results)), see:

  • Stuart and Ord (1994), Kendall's Advanced Theory of Statistics, volume 1, Chapters 12 and 13.
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