Error when neglecting $\mu$ while computing Covariance-Matrix I would like to quantify the estimation error I have to accept when estimating the Covariance matrix based on $T$ observations of $x_i\in\mathbb{R}^N$ (multivariate normal distributed with mean vector $\mu\in\mathbb{R}^N$). Define $$X:=\left( \begin{array}{c} x_1' \\ \vdots \\ x_T'\end{array} \right)\in\mathbb{R}^{T \times N}$$

The standard approach is to compute $$\hat{\Sigma}=\frac{1}{T-1}X'X-\frac{T}{T-1}\hat{\mu}\hat{\mu}'.$$ where $\hat{\mu}\in\mathbb{R}^N$ is the empirical mean.

I make the following observation: I sampled 5000 times multivariate normal datapoints with $N=10$ and $T=500$ observations with mean around $0.005$ and high positive correlation. When computing the maximum relative difference between $\hat{\Sigma}$ and $\frac{1}{T-1}X'X$ I obtained on average a low error (around 0.05). Changing the sampling variance to the identity matrix ceteris paribus leads me towards a very different result with errors around 15.

Now I am wondering how the Correlation structure affects this estimation error. Every Idea is appreciated!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.