# How to test equal means when the observations are based on estimates?

We have two groups ($\mathbf{X}$ and $\mathbf{Y}$) to test the equality of the means. Suppose $$\mathbf{X}_i \sim MVN(\boldsymbol{\mu}_\mathbf{X},\boldsymbol{\Sigma}_\mathbf{X}), \qquad \mathbf{Y}_j \sim MVN(\boldsymbol{\mu}_\mathbf{Y},\boldsymbol{\Sigma}_\mathbf{Y})$$ where $1 \leq i \leq n_\mathbf{X}$ and $1 \leq j \leq n_\mathbf{Y}$. Here $MVN$ stands for Multi-Variate Normal. The means and covariance matrices are unknown, also $\boldsymbol{\Sigma}_\mathbf{X} \ne \boldsymbol{\Sigma}_\mathbf{Y}$. However, $\mathbf{X}_i$ and $\mathbf{Y}_j$ can only be estimated by $\mathbf{X}_i^*$ and $\mathbf{Y}_j^*$ respectively and $$\mathbf{X}_i^* \sim MVN(\mathbf{X}_i, \boldsymbol{\Sigma}_{\mathbf{X},i}), \qquad \mathbf{Y}_j^* \sim MVN(\mathbf{Y}_j, \boldsymbol{\Sigma}_{\mathbf{Y},j})$$ where $\boldsymbol{\Sigma}_{\mathbf{X},i}$ and $\boldsymbol{\Sigma}_{\mathbf{Y},j}$ are known. How can we test whether $\boldsymbol{\mu}_\mathbf{X}=\boldsymbol{\mu}_\mathbf{Y}$ or not?

• When you say that we only have estimates of X (etc), are you talking about having finite samples to estimate the population parameter (the standard statistical topic), or that your observations come w/ measurement error, or something else? – gung - Reinstate Monica Jan 25 '13 at 22:54
• I mean the observation come with measurement error. The covariance matrix of each error is known. – user20094 Jan 26 '13 at 13:58

Since $\mathbf{X}_i$ is MVN and $\mathbf{X}^*_i$ is MVN conditional on $\mathbf{X}_i$, $\mathbf{X}^*_i$ is itself marginally MVN with mean $\boldsymbol\mu_\mathbf{X}$. The same holds for $\mathbf{Y}^*_i$. Thus, you can test for equality of the means just as you would if you had observed $\mathbf{X}_i$ and $\mathbf{Y}_i$ directly, e.g. with Hotelling's $T^2$ test.
The only difference is that the test will have less power than if you had observed $\mathbf{X}_i$ and $\mathbf{Y}_i$ directly, due to the added variance.
• Thanks for the answer. But I want to make use of the known covariance matrices $\boldsymbol{\Sigma}_{\mathbf{X},i}$ and $\boldsymbol{\Sigma}_{\mathbf{Y},j}$ in testing. Any thoughts? – user20094 Jan 26 '13 at 14:07