# Testing equality of marginal means of a multivariate normal random vector given estimated means and covariances

$$x=(x_1,\dots,x_p)^\top$$ is a random vector the elements of which are estimators that are known to have an asymptotic joint normal distribution $$N(\mu,\Sigma)$$. I have the estimated mean vector $$\bar{x}$$ and covariance matrix $$\hat\Sigma$$, and the sample size permits the use of the asymptotic approximation without much concern. How would I test the hypothesis (namely, construct a test statistic and find its null distribution) $$H_0\colon \ \mu_1=\dots=\mu_p$$ where $$\mu_i:=\mathbb{E}(x_i)$$ for $$i=1,\dots,p$$?

My thoughts so far: I could do a Wald test focusing on $$p-1$$ pairs $$(\mu_1,\mu_2)$$ to $$(\mu_1,\mu_p)$$. But this is only one option of constructing pairs. I could do the $$p-1$$ pairs $$(\mu_2,\mu_1)$$ to $$(\mu_2,\mu_p)$$ (of course, omitting $$(\mu_2,\mu_2)$$) instead. Or I could do all possible pairs of which there are $$\frac{p(p-1)}{2}$$. Would the latter option be the most efficient? If so, should I pursue it if $$p$$ is large, or would doing a random selection of $$p-1$$ pairs be just fine?

• This is quite basic, so it might be a duplicate, but I could not find it myself. – Richard Hardy Sep 6 '19 at 10:07
• I would start with the likelihood ratio test for $H_0 \colon \mu_1=\dotso=\mu_p$ against a general alternative, unless you have a more specialized (maybe ordered?) alternative in mind. Theory can be found amazon.com/Aspects-Multivariate-Statistical-Theory-Muirhead/dp/… – kjetil b halvorsen Sep 6 '19 at 13:13
• @kjetilbhalvorsen, thank you for the reference. I wonder whether I can implement an LR test given only the information I have. Could you be more specific about the implementation and indicate more precisely where in the book I can read about it? I see Chapter 8 concerns hypothesis testing for multivariate normal distributions and there is a specific list of tests covered on p. 291. The test I care about is not among them. – Richard Hardy Sep 6 '19 at 14:39

Let the random vector $$X=(X_1, \dotsc, X_p)$$ be multivariate normal with mean vector $$\mu$$ and known covariance matrix $$\Sigma$$. Then we want to test the null $$H_0 \colon \mu=\mu_0 1_p$$ against a general alternative. First we need to find the mle of the scalar $$\mu_0$$, which is found to be $$\hat{\mu_0}= \frac{x^T \Sigma^{-1} 1_p}{1_p^T \Sigma^{-1} 1_p}$$ (assuming $$\Sigma$$ is posdef.)
Then we need the (generalized) likelihood ratio statistic of $$H_0$$ within $$H_1$$ (assuming the same known covariance matrix.) Twice the loglikelihood ratio is $$2\ell = (x-\hat{\mu_0} 1_p)^T \Sigma^{-1} (x-\hat{\mu_0} 1_p)$$ and, without going into details, this should be (approximately) chi-squared with $$p-1$$ df (degrees of freedom) under the null hypothesis.