Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 29K / d

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- Solution d

Solution

Let

g (a, c, b) = \frac{c}{ab} = ρ .

Its gradient, in the coordinate order

(a, c, b)

, is

\nabla g = (- \frac{ρ}{2 a}, \frac{1}{ab}, - \frac{ρ}{2 b})^{T} .

Apply the Delta method to the limit in part (c). Multiplying the displayed covariance matrix there on both sides by this gradient gives

(\nabla g)^{T} V \nabla g = (1 - ρ^{2})^{2} .

Consequently the asymptotic distribution of the Gaussian sample correlation is

n (ρ - ρ_{X, Y}) d N (0, (1 - \frac{Σ _{12}^{2}}{Σ _{11} Σ _{22}})^{2}) .

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