Introduction to the Mathematical and Statistical Foundations of Econometrics
The Standard Cauchy Distribution
The ti distribution is also known as the standard Cauchy distribution. Its density is
|
|
|
||
|
|
||
where the second equality follows from (4.36), and its characteristic function is
Vh(t) = exp(-|t |).
The latter follows from the inversion formula for characteristic functions:
|
|||
|
|||
|
|
||
See Appendix 4.A. Moreover, it is easy to verify from (4.39) that the expectation of the Cauchy distribution does not exist and that the second moment is infinite.
4.6.2. The F Distribution
Let Xm ~ x2 and Yn ~ x2, where Xm and Yn are independent. Then the distribution of the random variable
F _ Xm / m Yn/n
is said to be F with m and n degrees of freedom and is denoted by Fm, n. Its distribution function is
 |
and its density is
mm/2 Y(m/2 + n/2) xm/2—1
nm/2 T(m/2)T(n/2) [1 + m ■ x/n]m/2+n/2 ’
See Appendix 4.A.
Moreover, it is shown in Appendix 4.A that
E[F] = n/(n — 2) if n > 3,
 |
 |
 |
= to if n = 1, 2,
Furthermore, the moment-generating function of the Fm, n distribution does not exist, and the computation of the characteristic function is too tedious an exercise and is therefore omitted.
