[SciPy-dev] Statistics Review progress

Robert Kern robert.kern at gmail.com
Fri Apr 14 01:43:14 CDT 2006

Travis Oliphant wrote:
> Robert Kern wrote:

>>Hmmm. It occurs to me that "bias" or "mean squared error" is probably best
>>defined in the logarithmic space for strictly positive quantities like variance.
>>Then the standard deviation is just a scale away, and the "biasedness" ought to
>>be invariant.
>>This is off-topic, but has anyone seen any publications taking this approach?
> I have not seen any,  but I think you are spot-on.  

Interestingly, when you define the "error" of the variance estimator to be
log(vhat/real_var), the minimum mean squared error estimator in the family

  power(x - x.mean(), 2).mean() / y

gives y=n-2 rather than any of the ones proposed in this thread. Even more
interestingly, it is also unbiased!

At least according to my numerical experiments. I haven't made any rigorous
derivations. Surely this must be Jaynes' book somewhere. I'll have to check when
I get back to the office.

Robert Kern
robert.kern at gmail.com

"I have come to believe that the whole world is an enigma, a harmless enigma
 that is made terrible by our own mad attempt to interpret it as though it had
 an underlying truth."
  -- Umberto Eco
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