Mean of Poisson Dist, given mode

[guthria]

Suppose a person takes data (say counts per minute of cars going past his window), for a long time. Then he loses his data, but knows that he counted 5 cars more often than any other number. What is the likely range for the average count rate?

I tried to solve this by saying the mode is 5, so the distribution peaks around 5. Which means that if I differentiate the Poisson distribution with respect to a parameter μ, which I assume is the mean of the distribution, then find the value at which the maximum occurs, I can find the mean in terms of the mode 5. So I know the mean, and the standard deviation, which gives me the range of the average count rate.

Carrying this out, I got the mean to be 5 also.

Is this correct? Can anyone help me? Or am I completely on the wrong track?

 

[haruspex]

Maximum Likelihood Estimation certainly gives that answer. The idea is to find the parameter value which maximises the likelihood of seeing a mode of 5.
But this is not always appropriate. In Bayesian analysis, you would need to plug in an a priori distribution for the parameter value (i.e. what seems reasonable for the possible parameter values), and see how that's modified by the observation.

 

[guthria]

Ok... Thanks.
Not sure if I understand this, or how to proceed...
Perhaps I will put up my attempt at a solution soon, that might help.

Thanks!

 

[haruspex]

Not sure if I understand this, or how to proceed...

The basic problem (here, and through much of practical statistics) is that if you presume the parameters of the distribution you can calculate the probability of the observation made; but what you want to know is the converse: the probability of the parameters given the observation.
Bayesian analysis admits that there is no absolute way to do this. Instead, you have to start with some gut feel ('a priori') notions of the likely values of the parameters, and use your observations to update them.
Classical hypothesis testing effectively does the same but hides it under the 'confidence level' carpet.
MLE brazenly says, just pick the parameters that maximise the likelihood of the observation. (I guess this is the same as treating all parameter values as equally likely.) This technique has the advantage of generally being relatively simple to apply.

 

[SW VandeCarr]

If you know the mean, the mode can be calculated from $\lfloor \lambda \rfloor , \lceil \lambda \rceil - 1$. You can estimate the approximate median from the floor function:

$$\lfloor \lambda + (1/3) - (0.02/\lambda) \rfloor$$

The bounds are given by $(\mu - ln2 )\leq \nu < \mu + (1/3)$

The derivation can be found here. The full PDF is free.

http://www.springerlink.com/content/h68un83g036h5p30/