这是一份uwa西澳大学STAT3063的成功案例

空间统计和建模|STAT3063 Spatial Statistics and Modelling代写 UWA代写

The distribution functions $D_{2}(r), D_{3}(r), \ldots$ of the distances to the $2 \mathrm{nd}, 3 \mathrm{rd}, \ldots$ nearest neighbours are
$$
D_{k}(r)=1-\sum_{j=0}^{k-1} \exp \left(-\lambda \pi r^{2}\right) \frac{\left(\lambda \pi r^{2}\right)^{j}}{j !} \quad \text { for } r \geq 0
$$
and the corresponding probability density functions are
$$
d_{k}(r)=\frac{2\left(\lambda \pi r^{2}\right)^{k}}{r(k-1) !} \exp \left(-\lambda \pi r^{2}\right) \quad \text { for } r \geq 0
$$The corresponding $j$ th moments are
$$
m_{k, j}=\frac{\Gamma\left(k+\frac{1}{2} j\right)}{(k-1) !(\lambda \pi)^{j / 2}} \quad \text { for } j=1,2, \ldots,
$$
and the position of the mode (maximum of density function) is
$$
r_{k}=\sqrt{\frac{k-\frac{1}{2}}{\lambda \pi}}
$$

STAT3063 COURSE NOTES ：

The so-called $L$-function is obtained by
$$
L(r)=\left(\frac{K(r)}{b_{d}}\right)^{\frac{1}{2}}
$$
as
$$
L(r)=r \quad \text { for } r \geq 0,
$$
and, similarly, the pair correlation function $g(r)$ is given by
$$
g(r)=1 \quad \text { for } r \geq 0,
$$
due to the general relation to $K(r)$,
$$
g(r)=\frac{\mathrm{d} K(r)}{\mathrm{d} r} / d b_{d} .
$$