Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 29K (Principles of Statistics)

Provided $X$ and $Y$ have finite positive variances, their correlation coefficient isIt lies in $[-1,1]$ by the Cauchy--Schwarz inequality.

Write $Z_i=(X_i,Y_i{)}^T$ for the independent observations and put $\Omega ={\Sigma }^{-1}$. Apart from constants, the log-likelihood isUsing the stated matrix derivatives, its score function isAt a root of the score, ${\Omega }^{-1}=n^{-1}{\sum }_i{Z}_i{Z}_i^T$. Hence the maximum-likelihood covariance estimator for centered Gaussian data is

Set $a={\Sigma }_{11}$, $c={\Sigma }_{12}$, and $b={\Sigma }_{22}$. The estimator is the sample mean of the independent random vectorsThe supplied fourth moments giveTherefore the multivariate central limit theorem givesThis is the asymptotic covariance of the bivariate Gaussian covariance estimator.

LetIts gradient, in the coordinate order $(a,c,b)$, isApply the Delta method to the limit in part (c). Multiplying the displayed covariance matrix there on both sides by this gradient givesConsequently the asymptotic distribution of the Gaussian sample correlation is