Let us say we observed one event on 1 Jan 2005, and we were observing the events since 2000. What's the best estimate of Poisson distribution?
One approach is to set intensity $\lambda=1/18$ per year. This is a Poisson MLE estimate, since we had 18 years passed and only one event observed.
Another approach seems to be fitting exponential distribution to time to arrival. Will they lead to the same estimate?
In my problem the estimation of Poisson intensity is conditional on the observation of a single event. This is an important factor, because it's not obvious that it doesn't mess up the usual MLE for Poisson.
I ran a simulation where random intensities are drawn, then random frequencies are drawn from these intensities. We look only at frequencies equal to one in 18 years. Finally, we use intensity estimator that is equal to observed frequency, like in a usual MLE for Poisson, and compare to the true intensity.
I suspected that for rare events when event was observed precisely once in a sample period, there could be a better estimator of the Poisson intensity than the observed frequency. However, this simulation shows zero bias of the usual Poisson MLE.
import numpy as np import matplotlib.pyplot as plt import scipy.stats as stats ns = 100000 T = 10 lam = np.random.gamma(1,1/T,ns) # random intensities ~1/18 (per year) obsf = np.random.poisson(lam*T,ns) # observe frequencies for 18 years plt.hist(obsf) plt.title('random intensities') plt.show() sidx = np.where(obsf==1) # we only look at trials where 1 event happened in 18 years err = (obsf[sidx]/T - lam[sidx]) # error when intensity estimator is set to observed frequency print(stats.describe(err)) plt.hist(err) plt.title('errors') plt.show()
DescribeResult(nobs=25146, minmax=(-0.30917753984994045, 0.055486021558327742), mean=-0.00044121203771743959, variance=0.0015661042110224508, skewness=-1.434031238643681, kurtosis=2.993433694497429)
I used Gamma as the simulated parameter distribution. It's unfair, because Gama is a prior to Poisson. I'll use some other distribution.