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Fisher's z-Distribution

Fischer's z-distribution is the general distribution defined by

g(z)=(2n_1^(n_1/2)n_2^(n_2/2))/(B((n_1)/2,(n_2)/2))(e^(n_1z))/((n_1e^(2z)+n_2)^((n_1+n_2)/2))
(1)

(Kenney and Keeping 1951) which includes the chi-squared distribution and Student's t-distribution as special cases.

Let u^2 and v^2 be independent unbiased estimators of the variance of a normally distributed variate. Define

z=ln(u/v)=1/2ln((u^2)/(v^2)).
(2)

Then let

F=(u^2)/(v^2)=((Ns_1^2)/(n_1))/((Ns_2^2)/(n_2))
(3)

so that n_1F/n_2 is a ratio of chi-squared variates

(n_1F)/(n_2)=(chi^2(n_1))/(chi^2(n_2)),
(4)

which makes it a ratio of gamma distribution variates, which is itself a beta prime distribution variate,

(gamma((n_1)/2))/(gamma((n_2)/2))=beta^'((n_1)/2,(n_2)/2),
(5)

giving

f(F)=(((n_1F)/(n_2))^(n_1/2-1)(1+(n_1F)/(n_2))^(-(n_1+n_2)/2)(n_1)/(n_2))/(B((n_1)/2,(n_2)/2)).
(6)

The mean is

<F>=(n_2)/(n_2-2),
(7)

and the mode is

(n_2)/(n_2+2)(n_1-2)/(n_1).
(8)

See also

Beta Distribution, Beta Prime Distribution, Chi-Squared Distribution, Gamma Distribution, Normal Distribution, Student's t-Distribution

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References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 180-181, 1951.

Referenced on Wolfram|Alpha

Fisher's z-Distribution

Cite this as:

Weisstein, Eric W. "Fisher's z-Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fishersz-Distribution.html

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