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I came across variance stabilizing transformation while reading Kaggle Essay Eval method. They use a variance stabilization transformation to transform kappa values before taking their mean and then transform them back. Even after reading the wiki on variance stabilizing transforms I can't understand, why do we actually stabilize variances? What benefit do we gain by this?

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    $\begingroup$ Usually the intent is to make the (asymptotic) variance independent of the parameter of interest. This is particularly important in inference where we need to know the reference distribution to calculate related quantities of interest. $\endgroup$ – cardinal Dec 18 '12 at 16:19
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Here's one answer: usually, the most efficient way to conduct statistical inference is when your data are i.i.d. If they are not, you are getting different amounts of information from different observations, and that's less efficient. Another way to view that is to say that if you can add extra information to your inference (i.e., the functional form of the variance, via the variance-stabilizing transformation), you will generally improve the accuracy of your estimates, at least asymptotically. In very small samples, bothering with modeling of variance may increase your small sample bias. This is a sort of econometric GMM-type argument: if you add additional moments, your asymptotic variance cannot go up; and your finite sample bias increases with the overidentified degrees of freedom.

Another answer was given by cardinal: if you have an unknown variance hanging around in your asymptotic variance expression, the convergence onto the asymptotic distribution will be slower, and you would have to estimate that variance somehow. Pre-pivoting your data or your statistics usually helps improve the accuracy of asymptotic approximations.

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  • $\begingroup$ I think I understand the first sentence in your answer and it appeals to me intuitively. Is there a name for this observation that I could google? I would like to find some thought experiments or examples that show what happens when you have different amount of information in different observations and how that is inefficient $\endgroup$ – Pushpendre Dec 19 '12 at 6:43
  • $\begingroup$ Korn & Graubard (1999) text on survey statistics discusses that. $\endgroup$ – StasK Dec 19 '12 at 15:29
  • $\begingroup$ But here the transformation is used to to compute a mean by $f^{-1}\left( {1\over n} \sum_i f(\kappa_i) \right)$. I really don’t see the point. For me, this would be the way to go for confidence interval estimation, but for point estimation it just introduces a bias. $\endgroup$ – Elvis Dec 19 '12 at 15:31
  • $\begingroup$ @PushpendreRastogi you may want to read the wikipedia article on this very transformation. It was introduced by Fisher to stabilize the variance of an empirical correlation coefficient (between normal variables). In that case, the transformed variable will be approximately normal, with variance depending only on the sample size, and not on the unknown correlation coefficient (this is why this “stabilizes” the variance). $\endgroup$ – Elvis Dec 19 '12 at 15:32
  • $\begingroup$ @Elvis, I gave the correlation example in the wikipedia article on pivotal statistic (en.wikipedia.org/wiki/Pivotal_statistic). [How on earth did you provide the nice link in the comment? I tried a href, it looked ugly.] $\endgroup$ – StasK Dec 19 '12 at 15:35

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