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Suppose $N(t)$ is a Poisson process with rates $\lambda$. Suppose I've been observing it for $t \in [0, T]$ and recorded events. How can I test the null hypothesis $\lambda < \lambda_0$, where $\lambda_0$ is a number?

An obvious solution is to derive MLE of rates $\hat\lambda$ and compare with $\lambda_0$. But I don't know how to calculate the statistical significance then. Any idea?

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$N(t)$ follows a $Poi(\lambda T)$ distribution. The generalized likelihood ratio test is $\frac{\max_{\lambda \geq \lambda_0} p(n;\lambda T)}{\max_{\lambda < \lambda_0} p(n;\lambda T)}$ where $p(n;\lambda)$ denotes a $Poi(\lambda T)$ PMF and $n$ is the number of counts you get, and you declare $H_0$ if this is below some threshold and $H_1$ if its above the threshold.

Now, the MLE for a poisson distribution with one sample is simply $\hat{\lambda} = n$. So, depending on the value of $n$ relative to $\lambda_0 T$, the numerator or denominator will be $p(n;n)$. Then, note that $(\log p(n;\lambda))' = -1 + n/\lambda$ where the derivative is wrt $\lambda$, so the one of the numerator/denominator which isn't $p(n;n)$ will be $p(n;\lambda_0 T)$.

So, your test statistic will be either $p(n;n)/p(n;\lambda_0 T)$ its reciprocal depending on if $n> \lambda_0 T$ or not. So, you can calculate everything based on this.

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    Thanks. Can you elaborate on the first two sentences of the second paragraph here? Sorry kind of a newbie. – 2017-01-05
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    Assume the number of events $n$ satisfies $n \geq \lambda_0 T$. Then, if you solve the maximization in the numerator, you will get $p(n;n)$. This is because maximizing $p(n;\lambda T)$ over $\lambda$ without any constraints is going to give you $p(n;n)$. Similarly, if $n < \lambda_0 T$, you'll have this in the denominator. – 2017-01-05
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    But then the test statistics will be a value, not a random variable? How to proceed to calculate p-value there? – 2017-01-05
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    $n$ is what you observe -- and your observed values of $n$ will follow a $Poi(\lambda T)$ distribution. – 2017-01-05