What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

Question

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

I am consider taking a undergraduate course in my college called mathematics of statistics and in the course description, a term comes up and it look so difficult to me.

What is a book that including a proof that is easy to read?

Please including an alternative proof that is easier if you want to! Appreciate in advance.

Answer 1

Problem 12 at this URL gives a proof of the Neyman--Pearson lemma. Maybe you'll find that more comprehensible. (Or maybe not?)

At any rate, the lemma says that for testing a point null hypothesis versus a point alternative, the likelihood ratio test is the unique most powerful test at any particular level (i.e. any particular tolerated probability of Type I error).

Answer 2

A more simple expression of the Neyman Pearson lemma is the following

A region $W$ maximises $$G(W) = \int_{w \in W} g(w) dw$$ subject to the constraints $$F_i(W) = \int_{w \in W} f_i(w) dw = c_i$$ if and only if for some constants $k_i$ we have that $g_i > \sum k_i f_i$ everywhere inside the region and $g_i < \sum k_i f_i$ everywhere outside the region. (And also given that such region exists)

The first occurrence of the lemma in 1933 (see here for some more history on stats.stackexchange) was in terms of a single constraint function $f_1$ and that makes the sufficiency part easy to prove.

Let $W_{\alpha}$ be the region that maximizes the integral. Then for some other region $W_\beta$ then we can consider three regions

• $W_{a}$ points in $W_{a}$ but not in $W_b$
• $W_{ab}$ points in both $W_{a}$ and in $W_b$
• $W_{b}$ points in not in $W_{a}$ but in $W_b$

Then you can split the integral up into parts like $F(W_\alpha) = F(W_{a}) + F(W_{ab})$ and the differences are

$$F(W_{\alpha}) - F(W_\beta) = F(W_{a}) - F(W_{b}) = 0$$

The part where the regions overlap cancel out. The difference $F(W_{\alpha}) - F(W_\beta)$ must be zero because the regions satisfy the same constraint.

$$G(W_{\alpha}) - G(W_\beta) = G(W_{a}) - G(W_{b})$$

where we know that $G(W_{a})>kF(W_{a})$ because it is a region inside $W_\alpha$ and $G(W_{b})<kF(W_{b})$ because it is a region outside $W_\alpha$. Thus

$$G(W_{\alpha}) - G(W_\beta) \geq k F(W_{a}) - k F(W_{b}) = 0$$