I've been reading up on Neyman Pearson Lemma but don't understand it to its full extent. Could someone please explain to me how to obtain the most powerful test for a sample of Bernoulli variables?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Given a random sample $(X_1, \ldots, X_n)$ from a $\mathrm{Bernoulli}(p)$ distribution, the Neyman-Pearson Lemma tests the null hypothesis $H_0 : p = p_0$ against the alternative $H_1 : p = p_1$. The test rejects $H_0$ if $$\frac{L(p_0)}{L(p_1)}=\frac{\prod_{i=1}^np_0^{x_i}(1-p_0)^{1-x_i}}{\prod_{i=1}^np_1^{x_i}(1-p_1)^{1-x_i}}=\frac{p_0^{\sum_{i=1}^n x_i}(1-p_0)^{n-\sum_{i=1}^n x_i}}{p_1^{\sum_{i=1}^n x_i}(1-p_1)^{n-\sum_{i=1}^n x_i}}<k$$ for some $k$. This condition can also be written as $$\left(\frac{p_0(1-p_1)}{p_1(1-p_0)}\right)^{\sum_{i=1}^n x_i}<k'$$ for some $k'$. Thus, when $p_0 < p_1$, we must have $\sum_{i=1}^n x_i > c$ for some $c$ and when $p_0 > p_1$,we must have $\sum_{i=1}^n x_i < c$ for some $c$.
