Suppose that $X_1,X_2,…,X_n$ is a random sample from the exponential distribution with rate $\lambda\gt0$.

Derive a uniformly most powerful hypothesis test of size $\alpha$ for

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda\gt\lambda_0\]

The uniformly most powerful testof size $\alpha$ for testing

\[H_0:\lambda\in\Theta_0\quad\text{vs.}\quad H_1:\lambda\in\Theta\backslash\Theta_0\]

is a test defined by a rejection region $R^*$ such that

  1. It has size $\alpha$.

i.e. $\underset{\theta\in\Theta_0}{\text{max}}P(\stackrel{\rightharpoonup}{X}\in R^*;\theta)=\alpha$

  1. It has higher power for all $\theta\in\Theta\backslash\Theta_0$

i.e. $\gamma_{R^*}(\theta)\ge\gamma_{R}(\theta)$ for all $\theta\in\Theta\backslash\Theta_0$.

i.e. $P(\stackrel{\rightharpoonup}{X}\in R^*;\theta)\ge P(\stackrel{\rightharpoonup}{X}\in R;\theta)$ for all $\theta\in\Theta\backslash\Theta_0$.

Step One:

Consider the simple versus simple hypothesis

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda=\lambda_1\]

for some fixed $\lambda_1\gt\lambda_0$.

Step Two, Three, and Four:

Find the best test of size $\alpha$ for

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda=\lambda_1\]

for some fixed $\lambda_1\gt\lambda_0$.

This test is to reject $H_0$, in favor of $H_1$ if

\[\overline{X}\lt\frac{\chi^2_{1-\alpha,2n}}{2n\lambda_0}\]

Note that this test does not depend on the particular value of $\lambda_1$. It does, however, depend on the fact that $\lambda_1\gt\lambda_0$.

It is the uniformly most powerful (best) test for

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda\gt\lambda_0\]

Suppose we’ve looked at a different hypotheses

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda\lt\lambda_0\]

Reject $H_0$ in favor of $H_1$, if

\[\bigg(\frac{\lambda_0}{\lambda_1}\bigg)^ne^{-(\lambda_0-\lambda_1)\sum_{i=1}^{n}X_i}\le c\\ \huge{.}\\ \huge{.}\\ \huge{.}\\ -(\lambda_0-\lambda_1)\sum_{i=1}^{n}X_i\le c\\ \sum_{i=1}^{n}X_i\color{red}{\ge} c\\ (\color{red}{\text{if }\lambda_1\lt\lambda_0})\\\]

The “UMP” test for

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda\gt\lambda_0\]

is to reject $H_0$, in favor of $H_1$ if

\[\overline{X}\lt\frac{\chi^2_{1-\alpha,2n}}{2n\lambda_0}\]

The “UMP” test for

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda\lt\lambda_0\]

is to reject $H_0$, in favor of $H_1$ if

\[\overline{X}\gt\frac{\chi^2_{1-\alpha,2n}}{2n\lambda_0}\]

Does there exist a “UMP” test for

\[H_0:\lambda=\lambda_0\quad\text{vs.}\quad H_1:\lambda\neq\lambda_0\]

In this case, the answer is No

For any $\lambda_1\neq\lambda_0$,

  • The best test if $\lambda_1\gt\lambda_0$ is to reject $H_0$ if

    \[\overline{X}\lt\frac{\chi^2_{1-\alpha,2n}}{2n\lambda_0}\]

  • The best test if $\lambda_1\lt\lambda_0$ is to reject $H_0$ if

    \[\overline{X}\gt\frac{\chi^2_{1-\alpha,2n}}{2n\lambda_0}\]

There is no one best test that we can use for all $\lambda_1\neq\lambda_0$!