| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
24 Jun 2005 12:44:46 AM |
| Object: |
Rotation vs Expansion-Contraction 5 |
From Osher Doctorow
One of the most surprising results concerns the F distribution which is
key to ANOVA (analysis of variance) used in laboratory statistical
scientific experiments in biomedical, psychological, and other
scientific scenarios, and which is also used in regression analysis
(least squares, etc.) and other fields.
We have:
1) fX(x) = kx^a/(m + nx)^b
where a = n/2 - 1, k is a constant involving positive integers n, m
called the (two) degrees of freedom of F, b is (n+m)/2. By
differentiation:
2) dfX(x)/dx = [kfX(x)/(m+nx)^b][a/x - nb/(m + nx)]
The second bracket is just the sum of two first degree polynomials
respectively in 1/x and 1/(m + nx), but the first bracket on the right
hand side has the exponent b which is a positive integer.
Should we generalize the criterion for "admissibility" of the F
distribution to PI MaxEnt because (m + nx)^b is merely a positive
integer power of the linear polynomial m + nx? Yes. The reason is
quite direct: such polynomials have an extreme simplicity compared to
arbitrary higher degree polynomials which involve powers of quadratics.
The decision turns out to be a wise one because the F distribution is
nonzero only on the positive real line so that it is asymmetric, and
also because the F distribution is so important to experiment and
regression.
It turns out that t^2 = F provided that F has degrees of freedom 1 and
v, in which case the Student's t distribution has degrees of freedom v.
However, as indicated previously, t does not satisfy the
"admissibility" criterion for PI MaxEnt. Also note that t is symmetric
on the whole real line.
Curiously enough, t is used in mainstream probability-statistics to
conduct small sample tests for means or else tests for means with
unknown population variances (textbooks and researchers differ on
which) and in a few other hypothesis tests. However, t in such
hypothesis tests requires an assumption of underlying normality and
equal variances (the latter if two populations are involved). Similar
but generalized requirements hold for F tests which are usually on 3 or
more means (although 2 are possible) and any number of "factors" (a
factor being something like hair color, occupation, etc., with
subfactors or levels being particular colors, particular occupations,
etc.).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Rotation vs Expansion-Contraction 5 |
24 Jun 2005 12:57:56 AM |
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From Osher Doctorow
I remind readers that the only irreducible polynomials over the complex
field are linear, which is to say that polynomials in variable z with
complex coefficients are products of a constant times (z - zo)(z -
z1)...(z - zn) for n a positive integer and zi complex constants, i = 1
to n. See Birkhoff and MacLane (1962), Theorem 6 (p. 109). So
polynomials over the reals are products of irreducible quadratics and
linear polynomials (with various powers). An irreducible quadratic
over the reals has real coefficients but can't be factoried into a
product of linear polynomials with real coefficients.
Osher Doctorow
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