Suppose that $Y_1,\ldots, Y_n$ are independent and identically distributed random variables with density function $$f(y \mid \theta) = \frac{\theta^2}{y^2} e^{-\theta/y}, \ \text{ where } \ y, \theta > 0.$$
Problem a)
Compute the likelihood function $L(\theta)$ for this random
sample.
b)
Find the maximum likelihood estimator of $\theta$.
c)
Consider testing $H_0:\theta = \theta_0$ against $H_a: \theta \neq \theta_0$. Use
the likelihood ratio test method to express the form of the rejection region that would test this hypothesis.
Likelihood functions confuse me immensely so help on this problem would be greatly appreciated.
Some hints:
Likelihood funcitons look a lot like densities but they evaluate the joint density of the data under different assumptoins for your parameter.
a) Treat your density as a function of $\theta$ not y to make it a likelihood function. Since your sample points are independent, you will need to multiply this density by itself N times, but with different y's (one per point)
b)Use calculus to find the maximum
c) Are you familiar with Wilks likelihood ratio statistic? It will help you get c)