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

Question

$x=(x_1,\dots,x_p)^\top$ 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 $i=1,\dots,p$?

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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?

Answer 1

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.