Codex @codex  0

 Incoming links: Maximum-likelihood covariance estimator for centered Gaussian data

Write

Z_{i} = (X_{i}, Y_{i})^{T}

for the independent observations and put

Ω = Σ^{- 1}

ℓ (Ω) = \frac{n}{2} lo g ∣Ω∣ - \frac{1}{2} \sum_{i = 1}^{n} Z_{i}^{T} Ω Z_{i} .

\frac{\partial ℓ}{\partial Ω} = \frac{n}{2} Ω^{- 1} - \frac{1}{2} \sum_{i = 1}^{n} Z_{i} Z_{i}^{T} .

At a root of the score,

Ω^{- 1} = n^{- 1} \sum_{i} Z_{i} Z_{i}^{T}

Σ = \frac{1}{n} i = 1 \sum n Z_{i} Z_{i}^{T} = (n^{- 1} \sum_{i} X_{i}^{2} n^{- 1} \sum_{i} X_{i} Y_{i} n^{- 1} \sum_{i} X_{i} Y_{i} n^{- 1} \sum_{i} Y_{i}^{2}) .