How to get a Standard Error for a Poisson Variable with Exposure?

Question

I'm looking over some work performed by a consultancy and I'm unsure whether the standard error formula they have used is correct, and the subsequent conclusions they have drawn are erroneous.

They have provided a Standard Error between the difference in observed claim frequencies and modelled claim frequencies, and then checked whether the modelled frequency lies between the observed 95% Confidence Interval.

The Standard Error Formula they have used is: $\sqrt{\frac{Observerd Frequency * (1 - Modelled Frequency)}{Exposure}}$.

However I would have thought a more appropriate Standard Error would be along the lines of the usual Standard Error for a Proportion: $\sqrt{\frac{Observerd Frequency * (1 - Observed Frequency)}{Exposure}}$.

My problem is I don't know how appropriate is it to use the Modelled Frequency in the Standard Error Calculation?

The data looks like this:

1. Sum Insured Band: \$0 - \$5,000
2. Exposure (Years): 27,233
3. Observed Claims: 1271
4. Modelled Claims: 1433
5. Observed Frequency: 4.7
6. Modelled Frequency: 5.3

I've had a look around at disease incidence and claims frequency standard error literature on google, but haven't really found anything concrete.

Thanks for any help.

Answer 1

You state the data are Poisson.

Why do you try to use a standard error for a binomial proportion?

Where do the modelled numbers come from? If they're based on a model applied to data they have their own standard error which you can't just ignore. If it's from the same data, the two estimates will also be dependent in some way.

When you say "observed claims" you mean "observed claim numbers" or "observed claim count", right?

Let's take the Poisson model as a given. So the observed number of claims in exposure period $i$ will be $Y_i \sim \text{Pois}(\mu_i)$, where $\mu_i = \lambda m_i$.

Hence $\text{Var}(Y_i) = \mu_i = \lambda m_i$.

Therefore, $\text{Var}(Y_i/m_i) = 1/m_i^2 \text{Var}(Y_i) = \lambda / m_i$.

(The usual estimate of that variance is just the sample frequency itself.)

Our only issue now is to figure out what the variance of the modelled value is, and its covariance with the observed, and then we can compute a standard error. You haven't given enough information to work that out.