Page 291 - Probability and Statistical Inference

268 5. Concepts of Stochastic Convergence

+
g(x) where g(x) is the pdf of at x, for each fixed x ∈ ℜ , as ν → ∞? The
2
answer follows.
Using the limiting value of the ratio of gamma functions from (1.6.24), we
again obtain

Next, we rewrite

and obtain

using (5.4.14) as well as the facts that (1 + ν /ν x) –1/2 ν = e –1/2 ν and
x
1 2 2 1
(1 + ν /ν x) –1/2 ν = 1. Refer to the Section 1.6 as needed. We conclude:
1 2 1

Let Fν ν be the upper 100α% point of the Fν ν distribution for any fixed
1, 2,α
1, 2
0 < α < 1. Unlike the percentiles tν for the t distribution, the corresponding
ν
,α
F percentiles Fν ν do not satisfy a clear-cut monotonicity property. The sce-
1, 2,α
nario here is complicated and one may consult the two articles of Ghosh (1973)
and DasGupta and Perlman (1974) cited earlier for details. The percentile Fν ν
1, 2,α
does have the Cornish-Fisher expansion in terms of the upper 100α% point
from the distribution depending upon ν ,ν ,α for large ν when ν is
1 2 2 1
kept fixed. The explicit details are due to Scheffé and Tukey (1944). Refer to
Johnson and Kotz (1970, p. 84) for specific details in this regard.