🚀 go-pugleaf

>>>>> "Glyn" == G M Collis <psrdj@warwick.ac.uk> writes: 

Glyn> What intrigues me is that the most elementary stats texts make a big
Glyn> fuss about using n-1 for an unbiased estimate of the variance, but
Glyn> ignore the fact that this gives a biased estimate for the SD. I
Glyn> recall that n - 1.5 is nearer the target for the SD when the sample
Glyn> is from a normally distributed population. I gather that minimising
Glyn> the bias when estimating the SD is rather sensitive to the population
Glyn> distribution - I'd like to know more about this.  But my big puzzle
Glyn> remains - why is the biasedness of the usual SD estimator (with N-1)
Glyn> so rarely mentioned, in stark contrast to the case of the variance.

What surprises me is how this quaint little thread got going at all.  The
elementary books are wrong if they make a big issue of unbiasedness, period.

In this context the *two* important quantities are (a) the sum of squares,
since it is the squared length of the orthogonal projection of the
observation vector onto the residual space, and (b) the degrees of freedom,
which is the dimension of the residual space.  This latter number is
sometimes n-1, but more often n-p where p is somewhat larger than 1.  These
two quantities, *separately*, are what you need for virtually all
inferential procedures, like testing and confidence intervals.  Whether you
divide one by the other to give an estimate of the variance is up to you.
Incidently, if you do, it turns out to be unbiased, but "so what?", really.

In my opinion statistical inference is all about reliably capturing
information from data (and elsewhere if you are a Bayesian); it's not
really about coming up with a number from a data set that you can show will
be "close" to an unknown parameter value, in some special sense of "close".
The trouble with many elementary books is that they get hung up on a narrow
definition of "estimation" and elevate unbiasedness to an importance far in
excess of what is warranted, at the same time not mentioning sufficiency,
say, a far more important concept, (but harder to describe, of course).
--
___________________________________________________________________________
Bill Venables, Dept. of Statistics, | Email: venables@stats.adelaide.edu.au
Univ. of Adelaide, South Australia. | Tel: +61 8 228 5412  Fax: ...232 5670