Test estimated against true values

I am generating a series of numbers using a Poisson Distribution with mean rate lambda.

I am then determining/estimating the long term average of these numbers over different periods of time.

The smaller the period of time the more focus is on the current numbers and vice versa. It is a decaying averaging function.

I am looking at various periods of time to determine the one that gives me the best results. I am looking at Hypothesis testing and factorial designs in comparing the output averages for the different period of times.

However, I would like to test the estimated average rates against the true mean rate lambda. How can this be done using ANOVA? What tests should I use?