Author: Eiko

Time: 2025-01-03 11:35:04 - 2025-01-03 12:34:16 (UTC)

Reference:

• Theoretical Statistics by Robert W. Keener

• Probability Theory by Eiko

• Foundations of Modern Probability by Olav Kallenberg

Uniformly Most Powerful Tests

• A test ${\phi }^{\ast }$ with level ${\alpha }_{{\phi }^{\ast }}=\alpha$ is called uniformly most powerful if for any $\theta \in {\mathrm{\Omega }}_1$, we have

  $${\mu }_{\theta }({\phi }^{\ast })\ge \underset{{\alpha }_{\phi }\le \alpha }{sup}{\mu }_{\theta }(\phi )\mathrm{\forall }\theta \in {\mathrm{\Omega }}_1.$$

  This means we are not just optimizing for one ${\theta }_1$ nor for the power function, we are optimizing for all ${\theta }_1\in {\mathrm{\Omega }}_1$. It might not exist.

• A family of densities $p_{\theta }(x)$, $\theta \in \mathrm{\Omega }\subset \mathbb{R}$ has monotone increasing likelihood ratios if there exists a statistic $T=T(x)$ such that whenever ${\theta }_1<{\theta }_2$, the likelihood ratio $p_{{\theta }_2}(x)/p_{{\theta }_1}(x)$ is a non-decreasing function of $T(x)$.

  For the decreasing case it suffices to replace $T$ by $-T$.

Example. For densities of the form of exponential families (of single parameter)

$${\mu }_{\theta }\sim \exp (\eta (\theta )T(x)-B(\theta ))h(x)\mathrm{d}x,$$

if $\eta (\theta )$ is non-decreasing / increasing, the likelihood ratio

$$L(x)=\frac{\mathrm{d}{\mu }_{{\theta }_2}}{\mathrm{d}{\mu }_{{\theta }_1}}=\exp ((\eta ({\theta }_2)-\eta ({\theta }_1))T(x)-B({\theta }_2)+B({\theta }_1))$$

is a non-decreasing / increasing function of $T(x)$. This includes the example of the exponential distribution, for which $\eta (\theta )=\theta$ is increasing and $T(x)=-x$.

Monotone Likelihood Gives Uniformly Most Powerful Tests

For this concept of monotone likelihood, we assume the parameter space $\theta \in \mathrm{\Omega }\subset \mathbb{R}$ is one-dimensional.

Suppose the family of density $\mathbb{P}(X|\theta )$ has monotone likelihood ratio (with respect to a test statistic $T(x)$). Then

• The monotone $T$ test

  $${\phi }^{\ast }(x)=\{\begin{array}{ll}1,& T>c\\ \gamma ,& T=c\\ 0,& T<c\end{array}$$

  is a uniformly most powerful test for the hypothesis $H_0:\theta \le {\theta }_0$ and $H_1:\theta >{\theta }_0$, with level

  $$\alpha ={\mu }_{{\theta }_0}{\phi }^{\ast }.$$

  $c$ and $\gamma$ can be adjusted to achieve any desired level $\alpha$.

• The power function $\beta (\theta )={\mu }_{\theta }({\phi }^{\ast })$ for this test is strictly increasing when it is in $(0,1)$, and non-decreasing overall.