# How to test $\mu_1 = \mu_2 = \mu_3$ when the covariance matrix is known/unknown?

I need to verify the hypothesis H0: $$\mu_1 = \mu_2 = \mu_3$$ vs the alternative(at least one mean doesn't equal to the others). Where $$N=20$$ and the estimated mean $$\overline{x} = (1, 0, -1)^T$$. Popluation comes from normal distribution $$N_3(μ,Σ)$$.

a) with known covariance matrix $$\begin{bmatrix} 3 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 4 \end{bmatrix}$$. I wrote the hypothesis in the form of $$C\mu$$ where $$C = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ \end{bmatrix}$$. Then I calculated the statistics (I found the formula in the book, is it ok?) in such a way

$$T^2 = N(C\overline{x})^T(CSC^T)^{-1}(C\overline{x})$$ but then I cannot find one good critical value to compare it to (I found this formula, but again could you please confirm/deny is it correct): $$T^2_{n-1,N-1,\alpha} = \frac{(N-1)(n-1)}{N-n+1}, \quad \text{where n is the dimension of the mean distribution, here it's 3.}$$ Is this formula correct? (I saw other formulas as well and I am confused.)

B) What will be the difference in the calculations if we assume that the covariance matrix is unknown?

Note: I don't need any coding solutions (R, python, etc.); I want to understand it on the theoretical level.

• @StubbornAtom Yes, indeed the population is from normal distribution N_3(μ,Σ) Jan 23, 2022 at 9:58

You want to test a null hypothesis of the form $$H_0:C\begin{pmatrix} \mu_1\\ \mu_2\\ \vdots\\ \mu_q \end{pmatrix}=c\in\mathbb{R}^p, C\in\mathbb{R}^{p\times q},\mathrm{rank}(C)=p.$$ Under $$H_0$$, and assuming that the sample mean $$\bar{x}$$ is calculated from an i.i.d. random sample from a $$\mathcal{N}_q(\mu,\Sigma)$$ population, we have $$N(C\bar{x}-c)^\mathsf{T}(C\Sigma C^\mathsf{T})^{-1}(C\bar{x}-c)\mathrel{=:}T_a\sim\chi^2(p)$$ if $$\Sigma$$ is known, as well as $$(N-1)(C\bar{x}-c)^\mathsf{T}(CSC^\mathsf{T})^{-1}(C\bar{x}-c)\mathrel{=:}T_b\sim T^2(p,N-1)$$ if $$\Sigma$$ is unknown (and estimated based on $$\frac{N}{N-1}S$$, the unbiased version of the sample covariance matrix $$S$$) with $$S=\frac{1}{N}\sum_{i=1}^{N}(x_i-\bar{x})(x_i-\bar{x})^\mathsf{T},$$ where $$x_i$$ denotes the $$i$$-th observation of the $$q$$-variate normal random variable.
In your question we have $$p=2$$, $$c=(0,0)^\mathsf{T}$$, and $$N=20$$.
Therefore, at a $$(\alpha\cdot100)\%$$ significance level, your critical value in a) is the $$(1-\alpha)$$-quantile of the chi-squared distribution with $$2$$ degrees of freedom and you want to compare the realization of $$T_a$$ to this critical value.
Your critical value in b) is the $$(1-\alpha)$$-quantile of the Hotelling's T-squared distribution with degrees of freedom $$2$$ and $$19$$.
Since $$\frac{N-p}{(N-1)p}T_b\mathrel{=:}T_b^*\sim F(p,N-p) \iff T_b\sim T^2(p,N-1),$$ you can equivalently compare the realization of $$T_b^*$$ to the $$(1-\alpha)$$-quantile of the $$F$$-distribution with degrees of freedom $$2$$ and $$18$$, which should be easy to get in a statistical software of your choice.