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+2)^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:

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

Here we use the Neyman Pearson Lemma.

The left side of the last inequality does not depend on $x_1,\ldots,x_n$. So each critical region is "the best". The best critical region does not exist.