Text S4.
Derivation of additional variance due to MISR sampling.
Let the variance be estimated using the following unbiased estimator
\(S^{2}=\ \frac{1}{n-1}\sum_{i}\left(x_{i}-\frac{\sum x}{n}\right)^{2}\)(S1)
where \(x_{i}\) are sampled values. As proven by Cho 2004 (Corollary 1),
Cho and Cho 2008 (Eq. 1), and Behnhmou (2018 Equations. 10 and 11), the
variance of the variance of a sampled population with the above
estimator and a large number of samples (n) can be approximated as,
\(\text{var}\left(S^{2}\right)\approx\ \frac{1}{n}\left(\mu_{4}-\ \frac{n-3}{n-1}(\mu_{2})^{2}\right)\)(S2)
where\(\text{\ μ}_{n}\) = E[(X – E(X)k] is the
n-th order moment and E the expected value. For a Gaussian distribution
the 4th moment is given by \(\mu_{4}=3\sigma^{4}\) and the
2nd moment is the variance \(\mu_{2}=\sigma^{2}\),
and yields the results that
\(\text{var}\left(S^{2}\right)=\ \frac{2\sigma^{4}}{n-1}\). (S3)
Thus the measured sampled variance (\(S^{2}\)) will be given by true
variance of the sample population (\(\sigma^{2})\), because the
estimator is unbiased, with one standard deviation given by the square
root of the above variance,
\(S^{2}=\ \sigma^{2}\ \pm\sqrt{\frac{2\sigma^{4}}{n-1}}=\ \sigma^{2}\left(1\ \pm\sqrt{\frac{2}{n-1}}\ \right)\).
(S4)
And the measured standard deviation (S) becomes,
\(S=\ \sigma\sqrt{1\ \pm\sqrt{\frac{2}{n-1}}\ }\). (S5)
Expanding the square root in a taylor series
(\(\sqrt{1+x}\approx 1+\ \frac{1}{2}x+\ \)higher order terms),
yields
\(S\ \approx\sigma\left(1\pm\frac{1}{2}\sqrt{\frac{2}{n-1}}\right)\)=\(\sigma\pm\frac{\sigma}{\sqrt{2n-2}}\) (S6)
with the increase in the standard deviation (relatively to that of the
unsampled population) given by the second term in S6. Note this is the
one-sigma uncertainty. While equation S6 was derived assuming Gaussian
statistics, as equation S2 shows, in general the variance (of the
variance) is proportional to 1/n and consequently for any reasonable
distribution (one with finite moments) the standard deviation of the
sampled population will asymptotically approach the true standard
deviation proportional to 1/\(\sqrt{n}\). In numerical test with
Gaussian distributions, we find Eq. S6 is quite a good approximation
(error of only a few percent), even for values of n as small as 10.
Cho, E. (2004). The Variance of Sample Variance for a Finite Population.ASA Section on Survey Research and Methods , 3345-3350
Cho, E. and Cho, M.J. (2008). Variance of sample variance. Section
on Survey Research Methods–JSM , 2 , pp.1291-1293.
Benhamou, E. (2018). A few properties of sample variance. arXiv
preprint arXiv:1809.03774 .