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In multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. Although no one is surprised that the estimator of the population covariance matrix is simply the sample covariance matrix, the mathematical derivation is perhaps not widely known and is surprisingly subtle and elegant.

1 The multivariate normal distribution

A random vector X ∈ R^p×1 (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix V precisely if V ∈ R^{p × p} is a positive definite matrix and the probability density function of X is

where μ ∈ R^p×1 is the expected value. The matrix V is the higher-dimensional analog of what in one dimension would be the variance.

2 Maximum-likelihood estimation

Suppose now that X₁, ..., X_n are independent and identically distributed with the distribution above. Based on the observed values x₁, ..., x_n of this sample, we wish to estimate V (we adhere to the convention of writing random variables as capital letters and data as lower-case letters).

2.1 First steps

It is fairly readily shown that the maximum-likelihood estimate of the expected value μ is the "sample mean"

See the section on estimation in the article on the normal distribution for details; the process here is similar.

Since the estimate of μ does not depend on V, we can just substitute it for μ in the likelihood function

and then seek the value of V that maximizes this.

We have

2.2 The trace of a 1 × 1 matrix

Now we come to the first surprising step.

Regard the scalar as the trace of a 1×1 matrix!

This makes it possible to use the identity tr(AB) = tr(BA) whenever A and B are matrices so shaped that both products exist. We get

(so now we are taking the trace of a p×p matrix!)

where

2.3 Using the spectral theorem

It follows from the spectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can of linear algebraLinear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is wi that a positive-definite symmetric matrix S has a unique positive-definite symmetric square root S^1/2. We can again use the "cyclic property" of the trace to write

Let B = S^1/2 V⁻¹ S^1/2. Then the expression above becomes

The positive-definite matrix B can be diagonalized, and then the problem of finding the value of B that maximizes

reduces to the problem of finding the values of the diagonal entries λ₁, ..., λ_p that maximize

This is just a calculus problem and we get λ_i = n, so that B = n I_p, i.e., n times the p×p identity matrix.

2.4 Concluding steps

Finally we get

i.e., the p×p "sample covariance matrix"

is the maximum-likelihood estimator of the "population covariance matrix" V. At this point we are using a capital X rather than a lower-case x because we are thinking of it "as an estimator rather than as an estimate", i.e., as something random whose probability distribution we could profit by knowing. This random matrix can be shown to have a Wishart distribution with n − 1 degrees of freedom.

StatisticsStatistics is the science and practice of developing human knowledge through the use of empirical data. It is based on statistical theory which is a branch of applied mathematics. Within statistical theory, randomness and uncertainty are modelled by proba

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Topics: Estimation Of Covariance Matrices, The Multivariate Normal...

EstimationEstimation is an approximate calculation of a result, often based on approximate input(s). Let's says you get 80% of the result by 20% input and efforst. For an account of statistical estimation, see estimator. The ability to accurractly estimate the time

Estimation of covariance matricesIn multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. Although no one

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