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Say $\{x_i\}_{i=1}^m$ are i.i.d observations from a multivariate normal distribution $N(\mu, \Sigma$).

How could I test the hypothesis $H_0:A \mu + b=0$ against $H_0:A \mu + b\neq0$, such that A is some invertible $2\times 2$ matrix and b is a 2D vector?

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Under $H_0$, the quadratic form $$ (A\bar x+b)^T (A\Sigma A^T)^{-1}(A\bar x+b) $$ is chi-square with 2 degrees of freedom. If you substitute $\Sigma$ by its estimate $\hat\Sigma=\frac1{m-1} \sum_{i=1}^m (x_i-\bar x)(x_i-\bar x)^T $, then $$ (A\bar x+b)^T (A\hat\Sigma A^T)^{-1}(A\bar x+b) $$ is instead Hotelling T-squared distributed with $2$ and $m-1$ degrees of freedom.

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  • $\begingroup$ Can you explain why we are using the quadratic form to get inferences of $H_0$? We simply reject $H_0$ if $T>F$ right? $\endgroup$
    – CCZ23
    Sep 11 '20 at 2:54
  • $\begingroup$ @CCZ23 See en.wikipedia.org/wiki/Mahalanobis_distance $\endgroup$ Sep 11 '20 at 8:27

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