Write the relevant mathematical arguments in a premake inferences on the basis of a fitted model and recognise the assumptions underlying these inferences and possible limitations to their accuracy.

这是一份Bath巴斯大学MA50260作业代写的成功案

$$
Q(p)=\lambda+(\eta / 2)[(1+\delta) S(p)-(1-\delta) S(1-p)],-1 \leq \delta \leq 1
$$
The distribution lies in the range of $(\lambda-a(\eta / 2)(1-\delta), \lambda+a(\eta / 2)(1+\delta))$, which for $a=\infty$ is $(-\infty, \infty)$. The quantile density, using $s(p)$ for the quantile density of $S(p)$, is
$$
q(p)=(\eta / 2)[(1+\delta) s(p)+(1-\delta) s(1-p)]
$$
A study of the quantile statistics of this distribution will show why this form has been adopted. Notice that using $R$ to denote the reflected distribution we have that $M_{R}=-M_{S}, U Q_{R}=-L Q_{S}$ and $L Q_{R}=-U Q_{S}$. Substituting $p=0.5$ gives
$$
\begin{aligned}
M=\lambda+(\eta / 2) & {\left[(1+\delta) M_{S}+(1-\delta) M_{R}\right] } \
=& \lambda+\eta \delta M_{\delta}
\end{aligned}
$$

MA50260 COURSE NOTES ：

value, $x$, which for each one is $p$. This probability is, by the multiplication law of probability, $p^{n}$. Hence
$$
p_{(n)}=p^{n} \text { so } p=p_{(n)}^{1 / n} \text { and } F(x)=p_{(n)}^{1 / n}
$$
Finally, therefore, inverting the CDF, $F(x)$, to get the quantile function, we have the equivalent statements that the $p_{(n)}$ quantile of the largest value is given by both $Q_{(n)}\left(p_{(n)}\right)$ and $Q\left(p_{(n)}^{1 / n}\right)$.
Hence
$$
Q_{(n)}\left(p_{(n)}\right)=Q\left(p_{(n)}^{1 / n}\right) .
$$
The quantile function of the largest observation is thus found from the original quantile function in the simplest of calculations.