A CURIOUS LIKELIHOOD IDENTITY FOR THE MULTIVARIATE T-DISTRIBUTION
A CURIOUS LIKELIHOOD IDENTITY FOR THE MULTIVARIATE T-DISTRIBUTION
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DOI：
10.1080/03610919408813180
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发表时间：
1994-01-01
影响因子：
0.9
通讯作者：
VARDI, Y
中科院分区：
文献类型：
--
作者：
KENT, JT;TYLER, DE;VARDI, Y
本文证明了自由度为nu-1 > 0的p维多元t分布的位置向量和散布矩阵的极大似然估计可以与自由度为nu-1 > 0的（p+1）维多元t分布的仅散布估计问题的极大似然估计相一致。t-分布是唯一一个可以使用这种对偶公式的分布。由于最大似然估计的存在性和唯一性性质对于一般的仅散射问题是直接证明的，因此我们能够立即推导出在t分布的特殊情况下更棘手的位置散射问题的存在性和唯一性结果。这两个公式中的每一个都产生了一个EM算法来最大化可能性，尽管这两个算法略有不同。极限柯西情况nu = 1需要一些特殊的处理。
It is shown that maximum likelihood estimates of the location vector and scatter matrix for a multivariate t-distribution in p dimensions with nu greater-than-or-equal-to 1 degrees of freedom can be identified with the maximum likelihood estimates for a scatter-only estimation problem from a (p+1)-dimensional multivariate t-distribution with nu-1 > 0 degrees of freedom. The t-distribution is the 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 nu = 1 requires some special treatment.