# What is the unbiased estimator of covariance matrix of N-dimensional random variable?

Suppose $$x$$ is a random vector in $$\mathbb{R}^n$$ which is distributed according to $$D$$.

What is the unbiased estimator of covariance matrix of an N-dimensional random variable?

When $$y$$ is a i.i.d. random variable and we have access to $$(y_1,y_2,\cdots,y_n)$$, the sample mean is an unbiased estimator of $$\hat{\mu}=\frac{\sum_{i=1}^N}{N}$$ and $$\hat{\sigma}^2=\frac{1}{N-1}\sum_{i=1}^N(y_i-\hat{\mu})^2$$ is an unbiased estimator of variance.

By going to higher dimension in addition to variance we have covariance between each element of the random vector. My question is

$$\hat{C}=?$$ where $$\hat{C}$$ is an unbiased estimator of $$C = \mathbb{E}[(x-\mu)(x-\mu)^T]$$.

• Essentially what you might expect, with $\hat{\vec{\mu}}= \frac1n \sum \vec{x}_i$ for the estimator of the mean vector and $\frac1{n-1} \sum (\vec{x}_i - \hat{\vec{\mu}})(\vec{x}_i - \hat{\vec{\mu}})^T$ for the estimator of the covariance matrix. See math.stackexchange.com/questions/2019122/… – Henry Dec 26 '18 at 18:12
• @Henry: That answer is for the $(X,Y)$ random vector, I need a proof in terms of vector. – Saeed Dec 26 '18 at 19:33