Let (Yi, Zi) be i.i.d. bivariate random vectors in the plane, with both Yi and Zi assumed to have finite nonzero variances. Let µY = E(Y1) and

µZ = E(Z1), let ρ denote the correlation between Y1 and Z1, and let ˆρn denote the sample correlation, as defined in (11.29).

(i). Under the assumption ρ = 0, show directly (without appealing to Example 11.2.10) that n1/2ρˆn is asymptotically normal with mean 0 and variance

τ 2 = V ar[(Y1 − µU )(Z1 − µZ )]/V ar(Y1)V ar(Z1).

(ii). For testing that Y1 and Z1 are independent, consider the test that rejects when n1/2|ρˆn| > z1− α

2 . Show that the asymptotic rejection probability is α, without assuming normality, but under the sole assumption that Y1 and Z1 have arbitrary distributions with finite nonzero variances.

(iii). However, for testing ρ = 0, the above test is not asymptotically robust.
Show that there exist bivariate distributions for (Y1, Z1) for which ρ = 0 but the limiting variance τ 2 can take on any given positive value.
(iv). For testing ρ = 0 against ρ > 0, define a denominator Dn and a critical value cn such that the rejection region n1/2ρˆn/Dn ≥ cn has probability tending to α, under any bivariate distribution with ρ = 0 and finite, nonzero marginal variances.