Maximum-likelihood Estimation For The Multivariate Normal Distribution
A random vector X ∈ Rp (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ Rp × p is a positive-definite matrix and the probability density function of X is
where μ ∈ Rp×1 is the expected value of X. The covariance matrix Σ is the multidimensional analog of what in one dimension would be the variance, and normalizes the density so that it integrates to 1.
Suppose now that X1, ..., Xn are independent and identically distributed samples from the distribution above. Based on the observed values x1, ..., xn of this sample, we wish to estimate Σ.
Read more about this topic: Estimation Of Covariance Matrices
Famous quotes containing the words estimation, normal and/or distribution:
“... it would be impossible for women to stand in higher estimation than they do here. The deference that is paid to them at all times and in all places has often occasioned me as much surprise as pleasure.”
—Frances Wright (1795–1852)
“You have promise, Mlle. Dubois, but you must choose between an operatic career and what is usually called “a normal life.” Though why it is so called is beyond me.”
—Eric Taylor, Leroux, and Arthur Lubin. M. Villeneuve (Frank Puglia)
“There is the illusion of time, which is very deep; who has disposed of it? Mor come to the conviction that what seems the succession of thought is only the distribution of wholes into causal series.”
—Ralph Waldo Emerson (1803–1882)