Answer to Question #104479 in Statistics and Probability for BARUN YADAV

Question

Examine whether a best critical region exists for testing the null hypothesis
H0 : θ =θ0 against the alternative hypothesis H1 :θ >θ0 for the parameter θ of the distribution:
f(x,θ)= 1+θ/(x+θ)^2,1≤x<∞.

Answer 1

Suppose $x_1,\ldots,x_n$ is a random sample from a population. We will find the test with the best critical region, that is, find the uniformly most powerful test, with a sample size n and a significance level $\alpha$ to test the simple $H_0: \theta=\theta_0$ against the composite $H_1: \theta>\theta_0$. Let $\alpha=P(C;\theta_0)$.

For each simple $H_1: \theta=\theta_a$​, say, the ratio of the likelihood functions is:

$\frac{L(\theta_0)}{L(\theta_a)}=\frac{(1+\frac{\theta_0}{(x_1+\theta_0)^2})\ldots(1+\frac{\theta_0}{(x_n+\theta_0)^2})}{(1+\frac{\theta_a}{(x_1+\theta_a)^2})\ldots(1+\frac{\theta_a}{(x_n+\theta_a)^2})}\leq k, k=const.$

In order to be the most powerful test, the ratio of the likelihoods should be small for sample points X inside the critical region C ("less than or equal to some constant k") and large for sample points X outside of the critical region ("greater than or equal to some constant k") (the Neyman Pearson Lemma).

From the left side of the last inequality we see that the best critical region exists.