I read this paper: "Experiment Size and Power Comparisons for Two-Sample Poisson Tests", Wei-Kei Shiue and Lee J. Bain, Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 31, No. 2 (1982), pp. 130–134, about which I have a question.

Suppose two Poisson processes are observed for fixed $s_1$ and $s_2$ respectively,and let $x$ and $y$ denote the number of outcomes observed.

That is, $$X \sim \operatorname{Poisson}(\lambda_1), \quad Y \sim \operatorname{Poisson}(\lambda_2), \quad\lambda_i=s_ir_i$$Consider a test of $H_0:r_1=r_2$ at the $\alpha$ significance level against the one-sided alternative $H_a:r_2>r_1$.

I understand the conditional distribution of the variable $Y$ given the total $x+y=m$ is a binomial distribution,$$(Y\mid x+y=m) \sim \operatorname{Bin}(m,p)\,, \quad \text{where} \quad p=\frac{\lambda_2}{\lambda_1+\lambda_2}$$ for the unequal interval case the test of $H_0:r_1=r_2$ is equivalent to a binomial test of $H_0:p=\frac{s_2}{s_1+s_2}=p_0$.

If $B(x;n,p)$ denotes a cumulative binomial distribution with parameters $n$ and $p$ , then a $\text{UMPU}$ test of $H_0$ at level $1-B(c-1;m,p_0)$ is to reject $H_0$ if $y \geq c$.

Question: Why may this test also be expressed in terms of Snedecor's $F$-distribution as reject $H_0$ if $$\frac{(x+1)p_0}{y(1-p_0)}\leq f_\alpha(2y,2(x+1))\,,$$ where $f_\alpha$ denotes the $\alpha$ percentile of the $F$-distribution?

I can't understand what the rejection region is , and why it can expressed in terms of F-distribution.


closed as off-topic by gung, kjetil b halvorsen, Michael Chernick, John, Nick Cox Mar 26 '17 at 9:41

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There's a well-known connection* between the cdf for the binomial and the regularized incomplete beta function (which is the cdf for the beta distribution), as described in the Wikipedia page for the binomial. Also see it described at the corresponding article on Mathworld (cf the cdf for the beta)

(I do wish people would stop using $B()$ for the binomial, since that's the natural symbol for the beta - so that's confusing enough; and worse we often -- including right now - want to talk about both, in which case it becomes almost impossible not to be confused when you see capital Beta while you're talking about the beta, but it actually means something else, the binomial.)

In turn there's a connection between beta and F random variables. This means that binomial probabilities can be written in terms of F-distribution probabilities. See the third item here; also see here or the end of the article on the F-distribution at mathworld.

So since binomial cdf can be written in terms of the cdf of a beta which can be written in terms of the cdf of an F, we can write binomials in terms of F's.

* at least it used to be. It may be less well known nowadays. This is similar to (and related to) the connection between Poisson and chi-square (/gamma) which also seems to be becoming less well known.


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