# How to check if a data is poisson sampled?

I was reading one article which develops a theory for the Poisson sampled data. That is the data is collected over time-points $$\{T_k, k>1\}$$, which are jump-moments of a homogeneous Poisson process $$\{N(t), t>0\}$$.

My question is: if I have a dataset collected over irregular time-points, how do I check that it is Poisson sampled?

• Interarrival times between Poisson events are exponential. – BruceET Apr 14 at 16:32

Extract the time intervals $$x_k=(T_k-T_{k-1})$$ for $$k=1,\ldots,N$$, since the intervals between events in a Poisson process are distributed according to some exponential distribution $$Exp(\lambda)$$.
Calculate the mean $$\bar{x}=\frac{1}{N}\sum_{k=1}^Nx_k$$ and set $$\lambda=1/\bar{x}$$, since the expected value of an exponential distribution $$Exp(\lambda)$$ is $$1/\lambda$$.
Now you could calculate the Kolmogorov-Smirnov statistic $$D_N=\sup_{x\geq0}|F_N(x)-F(x)|$$, where $$F(x)=1-\exp(-\lambda x)$$ is the distribution function of the exponential distribution, and $$F_N(x)=\frac{1}{N}\cdot\#\lbrace x_k \leq x\rbrace$$ and assess how well your empirical distribution $$F_N$$ fits the exponential distribution $$F$$.
Unfortunately, you estimated the parameter $$\lambda$$ (better: $$\hat{\lambda}$$) from the data itself... It has been shown that assessing the significance with the Kolmogorov-Smirnov test directly is not valid anymore in this case, but improved tables are available (the test is now called Lilliefors test for an exponential distribution with unknown mean). You could also use the package KScorrect in R.