# 统计学 Statistics MATH1054

For instance, suppose the goal is, as before, to estimate a regression
$$Y=x \beta \quad e$$
where the joint dependence between errors might be described by a multivariate Normal prior
$$e \sim N_{n}(0, \Sigma)$$

such that the dispersion matrix $\Sigma$ reflects the spatial interdependencies within the data. Outcomes may also be discrete, and then one might have, for binomial data, say,
$$\begin{gathered} Y_{i} \sim \operatorname{Bin}\left(\pi_{i}, N_{i}\right) \quad i=1, \ldots n \ \operatorname{logit}\left(\pi_{i}\right)=x_{i} \beta \quad e_{i} \end{gathered}$$
where again, the errors may be spatially dependent. Let the $n \times n$ covariance matrix for $e$ be
$$\Sigma=\sigma^{2} R(d)$$

$$r_{i j}=\exp \left(-3 d_{i j} / h\right)$$
where $h$ is the range, or inter-point distance at which spatial correlation ceases to be important $^{12}$. The Gaussian correlation function has
$$r_{i j}=\exp \left(-3 d_{i j}^{2} / h^{2}\right)$$
and the spherical (Mardia and Marshall, 1984) has $r_{i j}=0$ for $d_{i j}>h$ and
$$r_{i j}=\left(1-3 d_{i j} / 2 h \quad d_{i j}^{3} / 2 h^{3}\right)$$
for $d_{i j}<h$; see Example $7.5$ for an illustration. In each of these functions, $h$ is analogous to the bandwidth of kernel smoothing models. If $\Sigma=\sigma^{2} R(d)$, then the covariance tends