When is it true that if the compensator of a point process is invertible then the hazard function can be expressed as a function of the inverse of the compensator like so:
$$h(u) = \frac{\lambda(\Lambda^{-1}(u))}{u}$$
where λ is the conditional intensity function?
Answer: always
Given a closed-form expression for the inverse of the compensator, $\Lambda^{-1}(u)$, we aim to derive the inter-arrival density, hazard rate, and survival function. The compensator, $\Lambda(t)$, is defined as:
$$\Lambda(t) = \int_0^t \lambda(s) ds$$
where $\lambda(s)$ is the instantaneous intensity of the point process at time $s$. The inverse compensator, $\Lambda^{-1}(t)$, is the solution to the equation $\Lambda(\Lambda^{-1}(t)) = t$.
The inter-arrival density, $f(t)$, is the probability density of the time between two consecutive events. Using the inverse of the compensator, we can write:
$$f(t) = \frac{d}{dt} P(T \le t) = \frac{d}{dt} P(\Lambda(T) \le \Lambda(t))$$
By the definition of the compensator, we have:
$$\Lambda(T) = \int_0^T \lambda(s) ds$$
Therefore, we can rewrite the expression for $f(t)$ as:
$$f(t) = \frac{d}{dt} P\left(\int_0^T \lambda(s) ds \le \Lambda(t)\right)$$
Now, applying the change of variables, $u = \Lambda(s)$, and using the inverse of the compensator, we get:
$$f(t) = \lambda(\Lambda^{-1}(t)) \cdot \frac{d\Lambda^{-1}(t)}{dt} = \frac{\lambda(\Lambda^{-1}(t))}{\lambda(t)}$$
The hazard rate, $h(t)$, is the instantaneous probability of an event occurring at time $t$, given that no event has occurred up to time $t$. It can be expressed as:
$$h(t) = \frac{f(t)}{S(t)} = \frac{\lambda(\Lambda^{-1}(t))}{t}$$
The survival function, $S(t)$, represents the probability that no event has occurred up to time $t$. Using the hazard rate, we can derive the survival function as follows:
We know that:
$$h(t) = \frac{f(t)}{S(t)}$$
Then, differentiating the survival function with respect to $t$:
$$\frac{dS(t)}{dt} = -h(t)S(t)$$
Separating variables and integrating both sides, we get:
$$\int \frac{dS(t)}{S(t)} = -\int h(t) dt$$
Now, integrating both sides:
$$\ln S(t) = C - \int h(t) dt$$
Applying the initial condition $S(0) = 1$, we find that $C = 0$. Therefore, the survival function is:
$$S(t) = \exp\left(-\int h(t) dt\right) = 1 - F(t) = \exp\left(-\int_0^t \frac{\lambda(\Lambda^{-1}(s))}{\lambda(\Lambda^{-1}(t))} ds\right)$$
We have derived the inter-arrival density as a function of the compensator inverse, hazard rate, and survival function using the inverse of the compensator. To calculate these quantities for a specific point process, we need the compensator and its inverse, as well as the intensity function $\lambda(t)$.