A hazard function approximation used in reliability theory
The cumulative hazard function H(n) should accumulate to infinity over the distribution support, because the survivor function is Sf(n) = exp(-H(n)). The widely used approximation for the cumulative hazard function, H(n) approximate to Sigma(k=1)(n) h(k), for a small value of the hazard function, h(k), can be useful and reasonably accurate for computing the survivor function. For the continuous case, assuming that pdf exists, the H(n) diverges as it should. For the discrete case, two examples show the use of the hazard function approximation. In example A for the uniform probability mass function, the approximation diverges. In example B for the geometric probability mass function, the approximation converges to the finite value, 1.606695, when it should be diverging. The result is surprising in light of the difference between the continuous case, pdf, and the discrete case, pmf. Thus in practice, the approximation must be used with caution.
hazard function cumulative hazard function survivor function