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Details - The assumptions - and what goes wrong

The assumption

$\displaystyle \mathbf{y}\sim n \left (\mathbf{X}\boldsymbol{\beta},\sigma^2\boldsymbol{I}\right )
$

consists of several components: Normality, independence, linearity in the mean and equal variances.

The following sections give some examples of cases where general violations of these assumptions are handled using general nonlinear models.

Consider first the simplest deviations, namely of the variance structure:

  • If $ Vy_i=u_i \sigma^2$ where $ u_i$ are known, then we can define $ w_1=1/u_i$ and maximum likelihood is equivalent to $ min_{\mathbf{b}} \sum_i w_i \left ( \mathbf{y}_i - \mathbf{x}'_i \boldsymbol{\beta}\right ) ^2$ where $ \mathbf{x}'_i$ is the i'th row of $ \mathbf{X}$ . This is the traditional weighted linear regression.
  • If $ \Sigma_{\mathbf{y}}=\sigma^2 \mathbf{B}$ where $ \mathbf{B}$ is known, then we can write $ \mathbf{B}^{-1}=\mathbf{L}\mathbf{L}'$ where $ \mathbf{L}$ is lower triangular and define a new regression problem with $ \tilde{\mathbf{y}}=\mathbf{L}'\mathbf{y}$ , $ \tilde{\mathbf{X}}=\mathbf{L}'\mathbf{X}$ and it follows that $ E\tilde{\mathbf{y}}=\tilde{\mathbf{X}}\boldsymbol{\beta}$ , $ V\tilde{\mathbf{y}}=\ldots=\sigma^2\mathbf{I}$ so ordinary least squares can be used to estimate the parameters and for uncertainty estimation in the revised regression problem.
  • If $ \Sigma_{\mathbf{y}}$ contains ``a few'' unknown parameters, then these can be estimated as a part of maximum likelihood estimation of all parameters.


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