# Find the $\alpha$-level uniformly most powerful test (UMPT) of Poisson distribution

Let $$(X_1, X_2, X_3)$$ be a sample from $$Poisson(\lambda)$$. Here we test $$H_0: \lambda\le 1$$ v.s. $$H_1: \lambda>1$$. Fix $$\alpha=0.05$$. Find the $$\alpha$$-level uniformly most powerful test (UMPT).

My work: note that the density function has monotone likelihood ratio on $$T=\sum_{i=1}^3 X_i$$.

By Karlin-Rubin theorem, the UMPT for test $$H_0: \lambda\le 1$$ v.s. $$H_1: \lambda>1$$ is given by $$$$\Phi(X_1,\dots,X_n)= \begin{cases} 1 & \text{if } T>k\\ \gamma & \text{if } T=k\\ 0 & \text{if } T where the rejection region is the likelihood ratio $$\Lambda>k$$, and $$k$$ and $$\gamma$$ are decided by $$\alpha=P_{\lambda_0}(T>k)+\gamma P_{\lambda_0}(T=k).$$

Now my problem is, it seems that I can't solve the gamma. I read this theorem, and some people can't write this gamma, especially for continuous functions. Do we need to consider gamma? I mean the test function is: $$$$\Phi(X_1,\dots,X_n)= \begin{cases} 1 & \text{if } T>k\\ 0 & \text{if } T\le k \end{cases}$$$$ where the rejection region is the likelihood ratio $$\Lambda>k$$, and $$k$$ and $$\gamma$$ are decided by $$\alpha=P_{\lambda_0}(T>k).$$

If this one holds, we can solve $$k$$...

• $P_{\lambda_0}(T>k)$ is not a continuous function of $k$ for a discrete distribution - it jumps at each integer - so for most $\alpha$ you cannot find a $k$ satisfying $\alpha=P_{\lambda_0}(T>k)$. What did you get for $k$ when you tried to find the critical region in this particular case? Apr 5, 2023 at 7:29

An apt quote ($$\rm [I]$$):

... First $$k$$ is adjusted so that $$P_0(L(X) > k)$$ and $$P_0(L(X) \geq k)$$ bracket $$\alpha,$$ and then a value $$\gamma\in [0, 1]$$ is chosen for $$\varphi(X)$$ when $$L(X) = k$$ to achieve level $$\alpha.$$

$$T(\mathbf X)\sim \mathrm{Poi}(3).$$ A quick look at $$\mathtt R:$$

> qpois(0.95, 3)
[1] 6
> ppois(6, 3)
[1] 0.9664915
> ppois(7, 3)
[1] 0.9880955
> ppois(5, 3)
[1] 0.916082


Now can you choose a particular $$k$$ from above and as per the quote, then calculate the necessary $$\gamma$$?

## Reference:

$$\rm [I]$$ Theoretical Statistics: Topics for a Core Course, Robert W. Keener, Springer Science$$+$$Business Media, $$2010,$$ p. $$224.$$

• So the value of $\gamma$ and $K$ are not unique? Apr 5, 2023 at 12:05
• MP tests agree with each other except on a set of measure zero. Re-read the quote. You won't find multiple $k$s. Apr 5, 2023 at 15:21
• straight to the point (+1) also for the reference to one of my preferred books! Apr 6, 2023 at 19:59