Exercise 5

Question 1.

Let $Y \sim N(\mu, \sigma^2)$, where $|\mu| \leq a$ and $\sigma$ is known.
  1. Find the MLE for $\mu$.
  2. Consider the family of linear estimators $\hat{\mu}_c=cY$.
    1. Find the minimax estimator for $\mu$ within this family w.r.t. the quadratic loss. What is the corresponding minimax risk?
    2. Consider the following two-point prior on $\mu: \pi(a)=\pi(-a)=0.5$ and zero otherwise. Show that this is the least favorable prior. Is the corresponding Bayesian estimator admissible?

Question 2.

  1. Show that the minimax estimator for the probability of success $p$ from $Y \sim B(n,p)$ w.r.t. the quadratic loss $L(p,a)=(p-a)^2$ is $\frac{Y+.5 \sqrt{n}}{n+\sqrt{n}}$.
    (hint: consider the $Beta(\alpha,\beta)$ family of priors on $p$ and show that for a specific choice of $\alpha$ and $\beta$ this estimator is a Bayes rule whose risk does not depend on $p$).
  2. Is the above estimator admissible? What is the corresponding least favorable prior?
  3. Whether a "usual" (MLE, UMVUE, etc.) estimator $Y/n$ is admissible w.r.t the quadratic loss?
    (hint: use the fact that $Y/n$ is a Bayes rule w.r.t. to the loss function $L(p,a)=\frac{(p-a)^2}{p(1-p)}$)

Question 3.

Assume we want to test two simple hypotheses $H_0:\theta=\theta_0$ vs. $H_1:\theta=\theta_1$ based on a random sample $Y_1,\ldots,Y_n \sim f(y|\theta)$, where the prior probabilities of the hypotheses are $\pi_0$ and $\pi_1=1-\pi_0$ respectively. The loss is $L_0$ for erroneous rejection of $H_0$, $L_1$ for erroneous rejection of $H_1$ and zero for a correct decision ("$0-L_i$" loss).
  1. Derive the resulting Bayes testing rule.
  2. Interpretate the Bayes rule from the previous paragraph in terms of the frequentist approach. Is it a most powerful test? What is the critical value for the test statistic?
  3. Let $\alpha$ and $\beta$ be the probabilities of the I and II Type Errors respectively for the above test. Show that if $L_0\alpha = L_1\beta$, then it is also a minimax test.
  4. Let $Y_1,\ldots,Y_{100}$ be a random sample from a $N(\mu,25)$ distribution. Obtain the minimax test for testing $H_0:\mu=0$ vs. $H_1:\mu=2$ under "$0-L_i$" loss, where $L_0=25$ and $L_1=10$.

Question 4.

A device has been created to classify type of blood: A, B, AB or O. The device measures a certain quantity X, which has a density $f(x|\theta)=e^{-(x-\theta)},\; x \geq \theta$. If $0 < \theta < 1$, the blood is of type AB; if $1 < \theta < 2$, the blood is of type A; if $2 < \theta < 3$, the blood is of type B; and if $\theta>3$, the blood is of type O. It is known that in the population as a whole, $\theta \sim exp(1)$.

The loss in misclassifying the blood is given in the following table:

Classified As
AB A B O
True AB 0 1 1 2
Blood A 1 0 2 2
Type B 1 2 0 2
O 3 3 3 0
A patient has been tested and x=4 is observed. What is the Bayes action?

Question 5.

Children are given an intelligence test. The test result $X \sim N(\mu,100)$, where $\mu$ is the true IQ (intelligence) level of child. It is known that in the population as a whole, $\mu$ is distributed $N(100,225)$. A young Genius got 115 in the test.
  1. Find the Bayes estimate for the Genius' IQ w.r.t. the quadratic loss $L(\mu,a)=(\mu-a)^2$.
  2. In estimating IQ, it is deemed to be twice harmful to underestimate as to overestimate and the following loss is felt appropriate: $L(\mu,a)=2(\mu;-a), \mu \geq a$ and $L(\mu,a)=(a-\mu), \mu < a$. Find the Bayes estimate for the Genius' IQ w.r.t. this loss.
  3. Some people say that it is important to detect particularly high or low IQs and use the weighted quadratic loss $L(\mu,a)=(\mu-a)^2 e^{(\mu-100)^2/900}$ (note that this means that detecting an IQ of 145 or 55 is about nine times as important as detecting an IQ of 100). Find the Bayes estimate for Genius' IQ.
  4. Genius is to be classified as having below average IQ (less than 90), average (90 to 110), or above average IQ (over 110). Find the corresponding Bayes action and classify Genius to one of these three groups.