I will talk about the estimation of an unknown covariance matrix from a sample (of N points) when:
- The mean vector (MV) is known.
- The mean vector is unknown.
In the case of when the mean vector is known, the estimation of the covariance matrix is given by the following equation:(the SCM is = the "Sample Covariance Matrix" is:) $$\frac1N \sum_{i=1}^N (x_i - \bar x)(x_i - \bar x)^T$$
In the case of when the mean vector is unknown, the covariance matrix should be estimated by the following equation: (SCM = Sample Covariance Matrix) $$\frac1{N+1} \sum_{i=1}^N (x_i - \bar x)(x_i - \bar x)^T$$
In that case, there are N degrees of freedom in a sample of N points, and simultaneous estimation of mean and covariance means one degree of freedom goes to the sample mean and the remaining N-1 degrees of freedom (the residuals) go to the sample covariance.
In other terms, when the mean vector is unknown, the Bessel's correction is applied to the estimated covariance matrix: multiplying (1/N) by (N/N-1) in order to get the fraction (1/N-1). So the fraction (n/n-1) is called "Bessel's correction".
Unfortunately I didn't understand the above bolded statements. Kindly can someone explain to me in details how to get the N degrees of freedom and what is the motivation of multiplying 1/n by n/n-1 ? I will really appreciate if you can give a brief simple example..