# 非参数统计|STAT 8560/STAT 261/MATH335/PSY 610.01W/STAT 368/STAT 425/STAT 7610/MATH 494Nonparametric Statistics代写

If $f \in L_{2}(a, b)$ then 1
$$f(x)=\sum_{j=1}^{\infty} \theta_{j} \phi_{j}(x)$$
where
$$\theta_{j}=\int_{a}^{b} f(x) \phi_{j}(x) d x .$$
Furthermore,
$$\int_{a}^{b} f^{2}(x) d x=\sum_{j=1}^{\infty} \theta_{j}^{2}$$
which is known as Parseval’s identity.

## STAT 8560/STAT 261/MATH335/PSY 610.01W/STAT 368/STAT 425/STAT 7610/MATH 494 COURSE NOTES ：

where $c_{\alpha}$ is the upper $\alpha$ quantile of a $\chi_{1}^{2}$ random variable,
$$\ell(\theta)=2 \sum_{i=1}^{n} \log \left(1+\lambda(\theta) W_{i}\left(Y_{i}-\theta\right)\right),$$
$\lambda(\theta)$ is defined by
$$\begin{gathered} \sum_{i=1}^{n} \frac{W_{i}\left(Y_{i}-\theta\right)}{1+\lambda(\theta) W_{i}\left(Y_{i}-\theta\right)}=0, \ W_{i}=K\left(\frac{x-X_{i}}{h}\right)\left(s_{n, 2}-\frac{\left(x-X_{i}\right) s_{n, 1}}{h}\right), \end{gathered}$$
and
$$s_{n, j}=\frac{1}{n h} \sum_{i=1}^{n} \frac{K\left(\frac{x-X_{i}}{h}\right)\left(x-X_{i}\right)^{j}}{h^{j}} .$$