Likelihood estimator for inhomgeneous Poisson process

Question

Assume, i have an inhomogeneous Poisson process $N(t)$ with time-dependent intensity $\lambda$, i.e.

• $ N(0) = 0$
• $ N(t)-N(s) \sim \operatorname{Poiss}(\Lambda(s,t))$
• $N$ has independent increments.

where $$\Lambda(s,t) = \int_s^t \lambda(\tau) d \tau.$$

Now assume $\lambda$ is of known to be of form $\lambda(\tau)=f(\tau, \boldsymbol\beta)$, for a known function $f$ and unknown vector $\boldsymbol \beta$.

I have a sample of realizations of $N$ given in an interval $[0,T_{\max}]$: ($N(t,\omega_1), \ldots, N(t,\omega_n))$.

Now here is my question: If I want to get a maximum likelihood estimator for $\boldsymbol \beta$, I can chose a set $\{t_0,\ldots,t_k\}$ with $0=t_0 < t_1 < \ldots t_k=T$ and maximize the likelihood

$$\boldsymbol \beta = \arg\max_{\boldsymbol \beta} \prod_{i=0}^{k-1} \prod_{j=1}^n \mathbb P\big(N(t_i)-N(t_{i+1}) = N(t_i, \omega_j) - N(t_{i+1}, \omega_j) \big).$$

So now I am a bit confused by the following questions:

• How does the partition of the interval $[0,T]$ affect my estimation of $\boldsymbol \beta$. Both the number of $t_j$ and also how do I distribute them on the interval?
• Is that a stupid question to ask, because I forgot something trivial?

Thanks for any advice

Answer 1

Letting $t\downarrow s$, we see that the conditions $Cov(N(s),N(t))=\Lambda(s,t)$ and $\Lambda(s,t)=\int_s^t \lambda(\tau)\, d\tau$ imply $Var\,N(s)=0$ for all $s$, whence $\Lambda(s,t)=0$ for all $s<t$ and thus $f(\tau,\beta)=\lambda(\tau)=0$ for all $\tau$ and $\beta$ (assuming, for instance, that $\lambda(\cdot)$ is continuous). So, there is no problem estimating $\beta$.