(a) Let \(f_{X}(x)\) be the PDF of a random variable \(X\). Suppose that for some \(x_{u}<0\), the form of \(f_{X}(x)\) over \(x \leq x_{u}\) is

\[ f_{X}(x)=c(-x)^{-(\alpha+1)} \]

where \(c>0\) and \(\alpha>1\)

Show that the conditional PDF for \(x \leq x_{u}\) is \(f_{X \leq x_{u}}(x)=\) \(\alpha\left|x_{u}\right|^{\alpha}|x|^{-(\alpha+1)}\).

(b) Consider a random sample of size \(n\) from the distribution with PDF \(f_{X}(x)=c|x|^{-(\alpha+1)}\). Let the \(n_{x_{u}}\) be the number of order statistics in the sample that are are less than or equal to \(x_{u}\). Show that the \(\log\)-likelihood of \(\alpha\), given the smallest \(n_{x_{u}}\) order statistics is

\[ l_{L}(\alpha)=n_{x_{u}} \log (\alpha)-n_{x_{u}} \alpha \log \left(\left|x_{u}\right|\right)-(\alpha+1) \sum_{i=1}^{n_{x_{u}}} \log \left(\left|x_{(i: n)}\right|\right) \]

Show that the MLE of \(\alpha\) is

\[ \widehat{\alpha}=\frac{n_{x_{u}}}{\sum_{i=1}^{n_{x_{u}}} \log \left(x_{(i: n)} / x_{u}\right)} \]

(c) Show that Hill's estimator of the tail index is the reciprocal of the mean of the logs of the absolute values of the first \(k-1\) order statistics minus the log of the absolute value of the \(k^{\text {th }}\) order statistic. (The log of the absolute value of the \(k^{\text {th }}\) order statistic is subtracted before computing the mean.)

(d) The discussion in Exercises 4.14a through 4.14c concerned the index of the lower tail of a distribution. That discussion also assumed that the lower tail was over the negative reals.

What modifications must be made for a similar method to estimate the index of the upper tail?

What modifications must be made for a similar method to estimate the tail index if that tail may be over both negative and positive reals?