I am trying to prove the Lemma(3.4.2)(1) of book "Testing statistical hypotheses" by Lehmann and Romano, its statement is " If $\{p_{\theta}(x)\}$ be a family of densities on the real line with monotone likelihood ratio in $x$ and if $\phi(x)$ is a non-decreasing function of $x$, then $E_{\theta}(\phi(X))$ is a non-decreasing function of $\theta$."
In the proof, let $\theta^{'}>\theta$ and define $A$ and $B$ are the set of $x$ such that $p_{\theta^{'}}(x)<p_{\theta}(x)$ and $p_{\theta^{'}}(x)>p_{\theta}(x)$ respectively. Let $a=sup_{A}\phi(x)$ and $b=sup_{B}\phi(x)$ then it is clear that $b\ge a$.
Now consider that\begin{equation}\begin{split} \int \phi(x)(p_{\theta^{'}}(x)-p_{\theta}(x))dx&=\int_{A}\phi(x)(p_{\theta^{'}}(x)-p_{\theta }(x))dx+\int_{B}\phi(x)(p_{\theta^{'}}(x)-p_{\theta}(x))dx \\&\ge a\int_{A}(p_{\theta^{'}}(x)-p_{\theta }(x))dx+b\int_{B}(p_{\theta^{'}}(x)-p_{\theta}(x))dx \\&= (b-a)\int_{B}(p_{\theta^{'}}(x)-p_{\theta }(x))dx \\&\ge 0\end{split}\end{equation}My doubt is that it is nowhere mentioned in the statement of lemma that $\phi(x)$ is non-negative than how is he writing\begin{equation}\int_{A}\phi(x)(p_{\theta^{'}}(x)-p_{\theta }(x))dx\ge a\int_{A}(p_{\theta^{'}}(x)-p_{\theta }(x))dx.\end{equation}Am i trying it wrongly?, Is there any other way through which without taking $\phi(x)$ non-negative it can be proved.
Thank you for any hint/explanation.
