Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Covariance ▷ Sample covariance matrix

Definition: Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a sample from a random vector $X \in \mathbb{R}^{p \times 1}$. Then, the sample covariance matrix of $x$ is given by

\[\label{eq:covmat-samp} \hat{\Sigma} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x}) (x_i - \bar{x})^\mathrm{T}\]

and the unbiased sample covariance matrix of $x$ is given by

\[\label{eq:covmat-samp-unb} S = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}) (x_i - \bar{x})^\mathrm{T}\]

where $\bar{x}$ is the sample mean.

 
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Metadata: ID: D153 | shortcut: covmat-samp | author: JoramSoch | date: 2021-05-20, 07:46.