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5. Concepts of Stochastic Convergence 267

One should refer to Ghosh (1973) and DasGupta and Perlman (1974) for
many related details.
For large ν, one can expand t in terms of z and ν. E.A.Cornish and R.
α
ν,α
A. Fisher systematically developed techniques for deriving expansions of the
percentile of a distribution in terms of that of its limiting form. These expan-
sions are customarily referred to as the Cornish-Fisher expansions. The Cor-
nish-Fisher expansion of tν in terms of z and ν is given below: For large
α
,α
values of ν,

The reader may refer to Johnson and Kotz (1970, p. 102) for related details.
At this point, the reader may ask the following question: For some fixed 0
< α < 1, how good is the expression in the rhs of (5.4.11) as an approxima-
tion for t when ν is small or moderate? Let us write
ν,α

The Table 5.4.1 provides the values of z , correct up to four decimal places,
α
and those of tν , correct up to three decimal places. We have computed
,α
t ν,α,approx , correct up to five decimal places, for α = .05, .10 and ν = 10, 20, 30.
The values of z and tν are respectively obtained from Lindley and Scott
,α
α
(1995) and the Student’s t Table 14.3.3. The entries in the Table 5.4.1 show
clearly that the approximation (5.4.12) works well.
Table 5.4.1. Comparing tν with t
,α ν,α,approx
α = .10 z = 1.2816 α = .05 z = 1.6449
α
α
ν tν ,α tν ,α,approx tν ,α tν ,α,approx
10 1.3722 1.37197 1.8125 1.81149
20 1.3253 1.32536 1.7247 1.72465
30 1.3104 1.31046 1.6973 1.69727

Next, recall from (1.7.34) that for the F random variable with ν , ν de-
grees of freedom, written as Fν ,ν , the pdf is given by, 1 2
1 2

indexed by ν , ν , with b = b(ν , ν ) = (ν /ν ) ν Γ((ν + ν )/2){Γ(ν /2) ×
1/2
1 2 1 2 1 2 1 1 2 1
Γ(ν /2)} , ν , ν = 1, 2, 3, ... . We had seen earlier in (5.4.4) that
–1
2 1 2
when ν is held fixed, but ν ? ∞. But, is it true that fν ν (x) →
1 2 1, 2