这是一份非参数统计作业代写的成功案

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
$$
s_{n, j}=\frac{1}{n h} \sum_{i=1}^{n} \frac{K\left(\frac{x-X_{i}}{h}\right)\left(x-X_{i}\right)^{j}}{h^{j}} .
$$