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Example 1.3. Linear regression in semiparametric form. In Example 1.2 replace the assumption of normality by an assumption that the $Y_k$ are uncorrelated with constant variance. This is semiparametric in that the systematic part of the variation, the linear dependence on $z_k$, is specified parametrically and the random part is specified only via its covariance matrix, leaving the functional form of its distribution open. A complementary form would leave the systematic part of the variation a largely arbitrary function and specify the distribution of error parametrically, possibly of the same normal form as in Example 1.2. This would lead to a discussion of smoothing techniques.

Example 1.4. Linear model. We have an $n \times 1$ vector $Y$ and an $n \times q$ matrix $z$ of fixed constants such that
$$E(Y)=z \beta, \quad \operatorname{cov}(Y)=\sigma^2 I,$$
where $\beta$ is a $q \times 1$ vector of unknown parameters, $I$ is the $n \times n$ identity matrix and with, in the analogue of Example 1.2, the components independently normally distributed. Here $z$ is, in initial discussion at least, assumed of full rank $q<n$. A relatively simple but important generalization has $\operatorname{cov}(Y)=$ $\sigma^2 V$, where $V$ is a given positive definite matrix. There is a corresponding semiparametric version generalizing Example 1.3.

Both Examples 1.1 and 1.2 are special cases, in the former the matrix $z$ consisting of a column of $1 \mathrm{~s}$.

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