Distribution of the mean difference
Section of Estimation and inference for the multivariate normal, part of Statistical learning
Distribution of the mean difference
| Type | Section |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
The difference of the sample means is a linear image of the stacked data, so the linear-image proposition applies inside each sample and the covariances add. Stack the n_l observations of sample l as a vector of length n_l p. Independence makes its covariance block diagonal, with \Sigma repeated n_l times. The sample mean of sample l is therefore
\overline{\mathbf{X}}^{(l)}\sim\operatorname{N}_p(\boldsymbol{\mu}_l,\,\Sigma/n_l).
Under H_0 with \boldsymbol{\delta}_0=\mathbf{0}, independence of the two samples adds the covariances:
\overline{\mathbf{X}}^{(1)}-\overline{\mathbf{X}}^{(2)} \sim\operatorname{N}_p\Bigl(\mathbf{0},\,\Bigl(\frac1{n_1}+\frac1{n_2}\Bigr)\Sigma\Bigr) =\operatorname{N}_p\Bigl(\mathbf{0},\,\frac{n_1+n_2}{n_1 n_2}\Sigma\Bigr).
The factor 1/n_1+1/n_2 is (n_1+n_2)/(n_1 n_2). It is the sum of the sampling-variance coefficients of two independent sample means. A single sample mean contributes the coefficient 1/n.
Used in
See also
Invariance of the maximum likelihood estimator · Wishart distribution · Sampling law of the normal mean and scatter · Multivariate central limit theorem · The limit of Hotelling’s statistic · Null distribution of Hotelling’s · Two-sample Hotelling test · Likelihood of a multivariate normal sample