Let X, Y, Z be zero-mean, unit-variance random variables whose pairwise correlations are all equal to ρ. Find the tight lower bound on ρ such that the 3×3 correlation matrix is positive semidefinite. Show your work by analyzing eigenvalues of the equicorrelation matrix. Then generalize: for n variables with common off-diagonal correlation ρ, derive the feasible interval of ρ as a function of n.

This question evaluates understanding of positive semidefiniteness for equicorrelation matrices, eigenvalue analysis, and parameter constraints in multivariate statistics.

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Derive lower bound for equicorrelation rho | Citadel Interview Question

Equicorrelation Matrix PSD Condition

Setup

Consider zero-mean, unit-variance random variables whose pairwise correlations are all equal to a common value ρ. The corresponding n×n correlation matrix has 1’s on the diagonal and ρ on every off-diagonal entry (an equicorrelation or compound symmetry matrix):

Σ(ρ) = [σ_ij] where σ_ii = 1 and σ_ij = ρ for i ≠ j.

Tasks

For n = 3, find the tight lower bound on ρ such that the 3×3 correlation matrix is positive semidefinite (PSD). Show your work by analyzing the eigenvalues of the equicorrelation matrix.
Generalize: for n variables with common off-diagonal correlation ρ, derive the full feasible interval of ρ (as a function of n) for which Σ(ρ) is PSD.

Equicorrelation Matrix PSD Condition

Setup

Σ(ρ) = [σ_ij] where σ_ii = 1 and σ_ij = ρ for i ≠ j.

Tasks

For n = 3, find the tight lower bound on ρ such that the 3×3 correlation matrix is positive semidefinite (PSD). Show your work by analyzing the eigenvalues of the equicorrelation matrix.
Generalize: for n variables with common off-diagonal correlation ρ, derive the full feasible interval of ρ (as a function of n) for which Σ(ρ) is PSD.

Derive lower bound for equicorrelation rho

Quick Overview

Equicorrelation Matrix PSD Condition

Setup

Tasks

Solution

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Derive lower bound for equicorrelation rho

Quick Overview

Equicorrelation Matrix PSD Condition

Setup

Tasks

Solution

Comments (0)