A curious likelihood identity for the multivariate t-distribution
- 1 January 1994
- journal article
- research article
- Published by Taylor & Francis in Communications in Statistics - Simulation and Computation
- Vol. 23 (2) , 441-453
- https://doi.org/10.1080/03610919408813180
Abstract
It is shown that maximum likelihood estimates of the location vector and scatter matrix for a multivariate t-distribution in p dimensions with v≥1 degrees of freedom. can be identified with the maximum likelihood estimates for a scatter-only estimation problem from a (p+1)-dimensional multivariate the t-distribution with v−1>0 degrees of freedom. The t-distribution is only distribution for which this dual formulation is possible. Since the existence and uniqueness properties of maximum likelihood estimates are straightforward to prove for general scatter-only problems. we are able to immediately deduce existence and uniqueness results for the trickier location-scatter problem in the special case of the t-distribution. Each of these two formulations gives rise to an EM algorithm to maximize the likelihood. though the two algorithms are slightly different. The limiting Cauchy case v=1 requires some special treatment.Keywords
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