Let's go through Question Two thoroughly.

Question Two (15 Marks)

(a) Show that the family of $N(0, 16)$ distributions has Monotone Likelihood Ratio (MLR)

To show that a family of distributions has the Monotone Likelihood Ratio (MLR) property, we need to demonstrate that the likelihood ratio $\frac{f(x|\theta_2)}{f(x|\theta_1)}$ is monotonic (either non-increasing or non-decreasing) in $x$ for all $\theta_2 > \theta_1$.

1. The distribution is $N(0, 16)$, which is a normal distribution with mean 0 and variance 16.
2. The probability density function of $N(0, 16)$ is: $f(x) = \frac{1}{\sqrt{2\pi \cdot 16}} e^{-\frac{x^2}{2 \cdot 16}} = \frac{1}{4\sqrt{2\pi}} e^{-\frac{x^2}{32}}.$
3. Since the variance is fixed and we are only dealing with changes in the mean (which is zero in this case), the family $N(0, 16)$ meets the MLR property because any shift in $x$ will maintain a monotonic form.

(b) Derive the uniformly most powerful test for $H_0 : \theta = 25$ versus $H_1 : \theta > 25$

Given that the sample $X_1, X_2, \ldots, X_n$ comes from $N(\theta, 16)$, we have:

1. Since we want to test $H_0 : \theta = 25$ versus $H_1 : \theta > 25$, we look at the likelihood ratio test.
2. The test statistic for this problem, given the normal distribution, will be based on the sample mean $\bar{X}$: $\bar{X} \sim N(\theta, \frac{16}{n}).$
3. For the UMP test, we reject $H_0$ if $\bar{X}$ is sufficiently large.
4. The critical value $c$ is chosen such that: $P(\bar{X} \geq c | \theta = 25) = \alpha.$

(c) Generalized likelihood ratio test of size $\alpha = 0.05$ for testing $H_0 : \theta = 0$ versus $H_1 : \theta \neq 0$

Given a random sample $X_1, X_2, \ldots, X_n$ with the density $f(x; \theta) = \theta x^2 e^{-\theta x^2}, x > 0$:

1. The likelihood function is: $L(\theta) = \prod_{i=1}^n \theta x_i^2 e^{-\theta x_i^2}.$
2. To construct the generalized likelihood ratio, we compare: $\lambda = \frac{\sup_{\theta = 0} L(\theta)}{\sup_{\theta} L(\theta)}.$
3. For this exponential-type distribution, we find the maximized likelihoods under both hypotheses and use the ratio to construct the test.

Would you like further steps on each part of Question Two? Here are related questions you might explore:

1. What is the Monotone Likelihood Ratio property and how does it apply to normal distributions?
2. How do we determine the critical region for hypothesis tests involving normal distributions?
3. Why is the uniformly most powerful test often based on the sample mean in normal distribution cases?
4. What are the general principles of constructing a generalized likelihood ratio test?
5. How does setting an alpha level influence the power of a hypothesis test?

Tip: When dealing with hypothesis tests, always check if the distribution has a known or easily computable likelihood ratio test form, as it can simplify finding the UMP test.