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Consider the random sample X from the multivariate normal distribution where xi are i.i.d as N(µ,Σ). *Show that the sample mean x̄ and Sample covariance matrix S are jointly complete and sufficient statistics for µ and Σ

In my attempt I tried to justify that it belongs to the exponential family but I still don't know how to trigger out the sample mean x̄ and Sample covariance matrix S enter image description here

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enter image description here

Secondly, it can been shown that X is distributed according to a full rank exponential family with minimal sufficient statistics T(X)=(x̄,S). In an exponential family, it turns out that not only is the statistic minimal sufficient but it is also complete.

Therefore x̄ and S are jointly complete and sufficient statistics for µ and Σ

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