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12. Large-Sample Inference 561
(12.4.19) why we have assumed that the sample size is at least four. Let all
five parameters be unknown with (µ , σ ) ∈ ℜ×ℜ ,l = 1,2 and −1 < ρ < 1. In
+
l
l
Section 11.4.2, α level a likelihood ratio test was given for H : ρ = 0 against
0
H : ρ ≠ 0. In Exercise 11.4.5, one verified that the MLEs for µ , µ ,
2
1
1
and ρ were respectively given by
These stand for the customary sample means, sample variances (not unbi-
ased), and the sample correlation coefficient.
The level α LR test for H : ρ = 0 against H : ρ ≠ 0 is this:
0 1
We recall that under H , the pivot has the Students t
0
distribution with n − 2 degrees of freedom.
Figure 12.4.1. Two-Sided Students t Rejection Region
n-2
But, now suppose that we wish to construct an approximate 100(1 − α)%
confidence interval for ρ. In this case, we need to work with the non-null
distribution of the sample correlation coefficient r . The exact distribution of
n
r , when ρ ≠ 0, was found with ingenious geometric techniques by Fisher
n
(1915). That exact distribution being very complicated, Fisher (1915) pro-
ceeded to derive the following asymptotic distribution when ρ ≠ 0:
For a proof of (12.4.12), one may look at Sen and Singer (1993, pp. 134-136)
among other sources.
Again, one should realize that a variance stabilizing transformation may be
useful here. We invoke Mann-Wald Theorem from (12.4.1) and require a
