Certain data follows a generalized extreme value distribution, would the variance estimator used in practice be a good unbiased estimator for such data.
References would be highly appreciated.
Regards
You could start from some general book describing distributions, such as: "Continuous Univariate Distributions", Volume 2, 2nd Edition, Wiley 1995, by Norman L. Johnson, Samuel Kotz, N. Balakrishnan, ISBN: 978-0-471-58494-0
... alternatively see Wikipedia, but that s not particularly helpful to this question.
To answer more directly, the usual sample variance formula gives an estimate that is unbiased provided that the theoretical variance exists (a restriction on the shape parameter). However the theoretical variance of the sample estimate of the variance is finite only over n even more restrictive range of the shape parameter, and you may think this a problem.
The reply by Paul Chiou may be suggesting that you estimate the GEV parameters first and then derive the variance from these ... but this will not give an unbiased estimator. In fact, there may be no other exactly unbiased estimator of the variance So it depends on how important the property of unbiasedness is to you. Nowadays "unbiasedness" is generally not thought to be a strong requirement for estimators.
If you are simply interested in estimation for the GEV, then you should know about L-moments: see for example "Regional Frequency Analysis: An Approach Based on L-Moments" Cambridge University Press 1997 by J. R. M. Hosking, James R. Wallis ISBN: 9780521430456 . Or search on L-moments.
When the data follows extreme value distribution means the data is clustered at the extremes and that require use of fat tail distribution, rather than a thin tail distribution like the normal distribution. I would suggest use of Generalized Pareto Distribution (GPD), and then deduce your variance.
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