Page 293 - Probability and Statistical Inference

270 5. Concepts of Stochastic Convergence

using the expansion of e . Thus, one gets
x

which is the Cornish-Fisher expansion of F ν for large ν .
2, 2,α 2
One may ask the following question: How good is the expression in (5.4.18)
as an approximation for F ν when ν is small or moderate? Let us write
2, 2,α 2

The Table 5.4.2 provides the values of log(α), correct up to five decimal
places, and those of F2,ν , correct up to three decimal places. We have
2,α
computed F2,ν 2,α,approx , correct up to five decimal places, for α = .05, .10 and
ν = 10, 20, 30. The values of F2,ν are obtained from Lindley and Scott
2,α
2
(1995) and the F Table 14.3.4. The entries in the Table 5.4.2 show clearly that
the approximation (5.4.19) works well.

Table 5.4.2. Comparing F ν with F ν
2, 2,α 2, 2,α,approx
α = .10 log(α) = –2.30259 α = .05 log(α) = –2.99573

ν F ν F ν F ν F ν
2 2, 2,α 2, 2,α,approx 2, 2,α 2, 2,α,approx
10 2.9245 2.91417 4.1028 4.07240
20 2.5893 2.58803 3.4928 3.48926
30 2.4890 2.48836 3.3160 3.31479

5.5 Exercises and Complements

5.2.1 Let X , ..., X be iid having the following common discrete probabil-
1
ity mass function: n
X values: –2 0 1 3 4
Probabilities: .2 .05 .1 .15 .5
Is there some real number a such that as n → ∞? Is there some
positive real number b such that as n ? ∞? {Hint: Find the values of µ,
σ and then try to use the Weak WLLN and Example 5.2.11.}
5.2.2 Prove Khinchine’s WLLN (Theorem 5.2.3). {Hint: For specific ideas,
see Feller (1968, pp. 246-248) or Rao (1973, p.113).}