Page 585 - Probability and Statistical Inference

562 12. Large-Sample Inference

suitable function g(.) such that the asymptotic variance of
becomes free from ρ. In other words, we want to have

2
that is g(ρ) = k ∫ 1/(1 − ρ )dρ. Hence, we rewrite

From (12.4.14), it is clear that we should look at the transformations

and consider the asymptotic distribution of
Now, in view of (12.4.1) we claim that

2
since with , one has g′(ρ) = 1/(1 − ρ ). That is, for large n,
we should consider the pivot

where U and ξ were defined in (12.4.15).
n
One may verify that the transformations given in (12.4.15) can be equiva-
lently stated as

which are referred to as Fisher’s Z transformations introduced in 1925.
-1
Fisher obtained the first four moments of tanh (r ) which were later up-
n
-1
dated by Gayen (1951). It turns out that the variance of tanh (r ) is approxi-
n
mated better by 1/n - 3 rather than 1/n when n is moderately large. Hence, in
applications, we use the pivot

whatever be ρ, − 1 < ρ < 1. We took n at least four in order to make
meaningful.
For large n, we use (12.4.19) to derive an approximate 100(1 − α)%
confidence interval for ρ. Also, to test a null hypothesis H : ρ = ρ , for large
0
0
n, one uses the test statistic