simulating hazard

A rather straightforward X validated question that however leads to an interesting simulation question: when given the hazard function h(·), rather than the probability density f(·), how does one simulate this distribution? Mathematically h(·) identifies the probability distribution as much as f(·),

1-F(x)=\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}=\exp\{H(x)\}

which means cdf inversion could be implemented in principle. But in practice, assuming the integral is intractable, what would an exact solution look like? Including MCMC versions exploiting one fixed point representation or the other.. Since

f(x)=h(x)\,\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}

using an unbiased estimator of the exponential term in a pseudo-marginal algorithm would work. And getting an unbiased estimator of the exponential term can be done by Glynn & Rhee debiasing. But this is rather costly… Having Devroye’s book under my nose [at my home desk] should however have driven me earlier to the obvious solution to… simply open it!!! A whole section (VI.2) is indeed dedicated to simulations when the distribution is given by the hazard rate. (Which made me realise this problem is related with PDMPs in that thinning and composition tricks are common to both.) Besides the inversion method, ie X=H⁻¹(U), Devroye suggests thinning a Poisson process when h(·) is bounded by a manageable g(·). Or a generic dynamic thinning approach that converges when h(·) is non-increasing.

2 Responses to “simulating hazard”

  1. Fabrizio Leisen Says:

    Jim Griffin and I have something similar a BNP paper. We use a pseudo-marginal approach with a Poisson estimator, see Theorem 1 in:
    We deal with quantities like e^{-\int \phi(x)dx}, \phi(x)>0.

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