Fisher,s z-Distribution
المؤلف:
Kenney, J. F. and Keeping, E. S
المصدر:
Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand
الجزء والصفحة:
pp. 180-181
6-4-2021
2059
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.
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