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I have a set of data from my workplace where we note how many support calls we receive. I have been playing around with it in my spare time just to see if I could predict anything interesting.

I have been trying to fit the amount of calls to the Poisson distribution and I know the MLE of the $\lambda$ parameter is the mean of the count of calls. What I would like to know is the following.

  1. What is the best confidence interval for the $\lambda$ parameter? I know of a normal approximation and a few others but couldn't find any good derivation or proof that the significance value really is $(1-\alpha)$.

  2. What can I say about the likelihood of an observation? How do I evaluate the likelihood of getting 0 calls for example? I'm aware that with the MLE I can use the PMF of the Poisson to get an estimate for the likelihood of the observation but could I make an interval of probabilities with a certain significance?

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  • $\begingroup$ Confidence intervals for a Poisson parameter can be constructed using the relationship between the Poisson and chi-squared (or the gamma) distribution. However, with something like calls to a call center, the Poisson rate wouldn't be expected to be constant across days (Sunday is probably not like Tuesday) nor across time of day (9am, 1pm and 6pm would not have similar call rates), making estimation by straight averages (rather than a more sophisticated model) likely to be misleading. $\endgroup$ – Glen_b Feb 4 '16 at 0:58
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(1) Your first question is What is the best confidence interval for the $λ$ parameter? Before we get into that we should think about what the best estimate of the parameter $λ$ is. We do this because we often invert a Test Statistic into a confidence interval and often desirable test statistics are based on a specific estimator, such as the Likelihood Ratio Test and the MLE. As you stated the MLE is the sample mean. MLE's have some very nice qualities, they are asymptotically efficient and consistent, meaning that as the sample gets larger your estimate converges to the "correct" value and the variability reaches the Cramer Rao Lower Bound(lowest possible variance for an estimator). These are a couple of the reasons MLE's are so widely used, not only are they in simple cases often intuitive(in your example the sample mean), they are usually not to difficult to find either by hand or numerically and they have desirable properties such as consistency.

With that in mind we proceed to your first question. When considering multiple confidence intervals, if they all had the same confidence level, we would be interested in which one had the shortest interval, because this would imply greater precision with no loss in confidence. There is some theory you might be interested in concerning uniformly most accurate level alpha confidence sets. Unfortunately as Casella and Berger note they "exist() only in rather rare circumstances" pg 445, second edition. Usually there won't be a great difference in confidence lengths and for Poisson data you should just use the Normal approximation assuming its assumptions are valid. As for why "the significance value really is (1−$α$)" this comes from the the Normality of the test statistics that the confidence interval is based on and the Normality of that test statistic comes from the Central Limit Theorem.

(2) Then you ask, what can I say about the Likelihood of an observation? There are two ways I could see this question being interpreted. First, if you are asking how likely an observed value is, you would need to have formulated some hypotheses about $λ$ so that you could quantify how extreme it was. Secondly, if you are using the definition of Likelihood function, then you would be maximizing the the Likelihood function in relation to the data in order to find the most likely value of the parameter, which is the MLE which we have already discussed. So I assume the first way was correct interpretation which ties in better with the rest of your question.

If you want to see how likely it was that you observed 0 calls in a time interval that depends on what you assume about how many calls you should receive. Let's say you assume(Hypothesize) in an hour you will receive 10 calls, then you record 15 calls during the next hour. Then you would find out the probability of getting 15 or more calls during an hour given that your Hypothesis was correct, this is a p-value. We use samples to create statistics which estimate parameters which describe distributions by which we can make inferences about the state of our world. In order to test how likely your data are you must have a hypothesis, otherwise the MLE is the best description of the phenomenon because it estimates the parameter which defines your data.

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