Consider a random vector with two components $X$ and $Y$. Denote the cumulative distribution function (cdf) as $F_{X, Y}(x, y)$, and the marginal cdf as $F_{X}(x)$ and $F_{Y}(y)$, respectively.

i) Show, using first principles, that

$$
F_{X}(x)+F_{Y}(y)-1 \leq F_{X, Y}(x, y) \leq \sqrt{F_{X}(x) F_{Y}(y)}
$$

always holds.

ii) Suppose that the $X$ and $Y$ are components of continuous random vector with a density $f_{X, Y}(x, y)=c x y, 0<y<x, 0<x<2$ (and zero else). Here $c$ is a normalizing constant.

a) Show that $c=\frac{1}{2}$.

b) Find the marginal density $f_{X}(x)$ and $F_{X}(x)$.

c) Find the marginal density $f_{Y}(y)$ and $F_{Y}(y)$.

d) Find the conditional density $f_{Y \mid X}(y \mid x)$.

e) Find the conditional expected value $a(x)=E(Y \mid X=x)$.

Make sure that you show your working and do not forget to always specify the support of the respective distribution.

At a critical stage, a fund manager has to make a decision about investing or not investing in certain company stock. He intends to apply a statistical decision theory approach to work out the appropriate decision based on the potential long-term profitability of the investment. He uses two independent advisory teams with teams of experts and each team should provide him with an opinion about the profitability. Data $X$ represents the number of teams recommending investing in the stock (due, of course, to their belief in its profitability).

If the investment is not made and the stock is not profitable, or when the investment is made and the stock turns out profitable, nothing is lost. In the manager’s judgement, if the stock turns out to be not profitable and decision is made to invest in it, the loss is three time higher than the cost of not investing when the stock turns out profitable.

The two independent expert teams have a history of forecasting the profitability as follows. If a stock is profitable, each team will independently forecast profitability with probability $4 / 5$ (and no profitability with $1 / 5)$. On the other hand, if the stock is not profitable, then each team predicts profitability with probability $1 / 2$. The fund manager will listen to both teams and then make his decisions based on the data $X$.

a) There are two possible actions in the action space $\mathcal{A}=\left{a_{0}, a_{1}\right}$ where action $a_{0}$ is to invest and action $a_{1}$ is not to invest. There are two states of nature $\Theta=\left{\theta_{0}, \theta_{1}\right}$ where $\theta_{0}=0$ represents “profitable stock” and $\theta_{1}=1$ represents “stock not profitable”. Define the appropriate loss function $L(\theta, a)$ for this problem.

b) Compute the probability mass function (pmf) for $X$ under both states of nature.

c) The complete list of all the non-randomized decisions rules $D$ based on $x$ is given by:

\begin{tabular}{l|llllllll}
\hline & $d_{1}$ & $d_{2}$ & $d_{3}$ & $d_{4}$ & $d_{5}$ & $d_{6}$ & $d_{7}$ & $d_{8}$ \
\hline$x=0$ & $a_{0}$ & $a_{1}$ & $a_{0}$ & $a_{1}$ & $a_{0}$ & $a_{1}$ & $a_{0}$ & $a_{1}$ \
$x=1$ & $a_{0}$ & $a_{0}$ & $a_{1}$ & $a_{1}$ & $a_{0}$ & $a_{0}$ & $a_{1}$ & $a_{1}$ \
$x=2$ & $a_{0}$ & $a_{0}$ & $a_{0}$ & $a_{0}$ & $a_{1}$ & $a_{1}$ & $a_{1}$ & $a_{1}$ \
\hline
\end{tabular}

For the set of non-randomized decision rules $D$ compute the corresponding risk points.

d) Find the minimax rule(s) among the non-randomized rules in $D$.

e) Sketch the risk set of all randomized rules $\mathcal{D}$ generated by the set of rules in $D$. You might want to use R (or your favorite programming language) to make this sketch more precise.

f) Suppose there are two decisions rules $d$ and $d^{\prime}$. The decision $d$ strictly dominates $d^{\prime}$ if $R(\theta, d) \leq R\left(\theta, d^{\prime}\right)$ for all values of $\theta$ and $R(\theta, d)<\left(\theta, d^{\prime}\right)$ for at least one value $\theta$. Hence, given a choice between $d$ and $d^{\prime}$ we would always prefer to use $d$. Any decision rules which is strictly dominated by another decisions rule (as $d^{\prime}$ is in the above) is said to be inadmissible. Correspondingly, if a decision rule $d$ is not strictly dominated by any other decision rule then it is admissible. Show on the risk plot the set of randomized decisions rules that correspond to the fund manager’s admissible decision rules.

g) Find the risk point of the minimax rule in the set of randomized decision rules $\mathcal{D}$ and determine its minimax risk. Compare the two minimax risks of the minimax decision rule in $D$ and in $\mathcal{D}$. Comment.

h) Define the minimax rule in the set $\mathcal{D}$ in terms of rules in $D$.

i) For which prior on $\left{\theta_{1}, \theta_{2}\right}$ is the minimax rule in the set $\mathcal{D}$ also a Bayes rule?

j) Prior to listening to the two teams, the fund manager believes that the stock will be profitable with probability $1 / 2$. Find the Bayes rule and the Bayes risk with respect to his prior.

k) For a small positive $\epsilon=0.1$, illustrate on the risk set the risk points of all rules which are $\epsilon$-minimax.

In a Bayesian estimation problem, we sample $n$ i.i.d. observations $\mathbf{X}=\left(X_{1}, X_{2}, \ldots, X_{n}\right)$ from a population with conditional distribution of each single observation being the geometric distribution

$$
f_{X_{1} \mid \Theta}(x \mid \theta)=\theta^{x}(1-\theta), x=0,1,2, \ldots ; 0<\theta<1 .
$$

The parameter $\theta$ is considered as random in the interval $\Theta=(0,1)$.

i) If the prior on $\Theta$ is given by $\tau(\theta)=3 \theta^{2}, 0<\theta<1$, show that the posterior distribution $h\left(\theta \mid \mathbf{X}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right)$ is in the Beta family. Hence determine the Bayes estimator of $\theta$ with respect to quadratic loss.

Hint: For $\alpha>0$ and $\beta>0$ the beta function $B(\alpha, \beta)=\int_{0}^{1} x^{\alpha-1}(1-x)^{\beta-1} d x$ satisfies $B(\alpha, \beta)=\frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha+\beta)}$ where $\Gamma(\alpha)=\int_{0}^{\infty} \exp (-x) x^{\alpha-1} d x$. A Beta $(\alpha, \beta)$ distributed random variable $X$ has a density $f(x)=\frac{1}{B(\alpha, \beta)} x^{\alpha-1}(1-x)^{\beta-1}, 00.80$. (You may use the integrate function in $\mathrm{R}$ or another numerical integration routine from your favourite programming package to answer the question.)

Let $X_{1}, X_{2}, \ldots, X_{n}$ be i.i.d. uniform in $(0, \theta)$ and let the prior on $\theta$ be the Pareto prior given by $\tau(\theta)=\beta \alpha^{\beta} \theta^{-(\beta+1)}, \theta>\alpha$. (Here $\alpha>0$ and $\beta>0$ are assumed to be known constants). Show that the Bayes estimator with respect to quadratic loss is given by $\hat{\theta}{\text {Bayes }}=\max \left(\alpha, x{(n)}\right) \frac{n+\beta}{n+\beta-1}$. Justify all steps in the derivation.