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I have a step hazard function $h(t)$ (a simple function that jumps at specific times and it is constant between those jumps). Something like:

Example piecewise constant hazard function

How can I simulate survival times from this hazard function? Is there a clever way for doing so?

I was thinking of using the probability integral transform, by solving $$\exp\left[-\int_0^t h(s)ds \right] = u, $$ where $u\sim U(0,1)$, but I started wondering if there is a more efficient way for simulating these times.

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