这是一份nottingham诺丁汉大学MATH1054作业代写的成功案例
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
$$
\Sigma=\sigma^{2} R(d)
$$
MATH1054 COURSE NOTES ：
Among the most commonly used functions meeting these requirements are the exponential model
$$
r_{i j}=\exp \left(-3 d_{i j} / h\right)
$$
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