Fisher's z-Distribution

Fischer's -distribution is the general distribution defined by

(1)

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

Let  and  be independent unbiased estimators of the variance of a normally distributed variate. Define

(2)

Then let

(3)

so that  is a ratio of chi-squared variates

(4)

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

(5)

giving

(6)

The mean is

(7)

and the mode is

(8)

REFERENCES:

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