OurBigBook.comSite Source code

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

Set

a = Σ_{11}

,

c = Σ_{12}

, and

b = Σ_{22}

. The estimator is the sample mean of the independent random vectors

W_{i} = (X_{i}^{2}, X_{i} Y_{i}, Y_{i}^{2})^{T}, E W_{i} = (a, c, b)^{T} .

The supplied fourth moments give

Var (X^{2}) Var (X Y) = 2 a^{2}, = ab + c^{2}, Cov (X^{2}, X Y) Cov (X Y, Y^{2}) = 2 a c, = 2 b c, Cov (X^{2}, Y^{2}) Var (Y^{2}) = 2 c^{2}, = 2 b^{2} .

Therefore the multivariate central limit theorem gives

n Σ_{11} Σ_{12} Σ_{22} - a c b d N_{3} 0, 2 a^{2} 2 a c 2 c^{2} 2 a c ab + c^{2} 2 b c 2 c^{2} 2 b c 2 b^{2} .

This is the asymptotic covariance of the bivariate Gaussian covariance estimator.

Solved by gpt-5.6-sol high.

 Ancestors (11)