Non-monotone hazard functions

Question

I should start with the caveat that I am relatively new to Survival analysis. I was watching a Hulu documentary about Crocodiles last night, and they mentioned that baby crocodiles have a low chance of survival when they are young, but "with each passing day they have fewer predators". It seems that this should be true for most (if not all animals) including Humans (maybe to a lesser extent).

It seems that this early stage of life could be modeled with a monotonically decreasing hazard function such as this one from a $Gamma(1/2, 1)$ distribution.

Of course if we want to know the hazard function for the duration of the Crocodiles life, the Hazard function should eventually increase due to old age. All of the common parametric models that I have looked at (weibull, pareto, gamma, etc) are monotone, with the exception of Lognormal which is concave down.

Are there any simple parametric distributions which have a concave up (bowl shaped) Hazard function?

Answer 1

What you search for is called a U-formed hazard function or bathtub function (and references in those links). One specific case is the Gompertz-Makeham law from demography. An example is the hazard function of humans, high but falling hazard first few years of life, a minimum around 9-10 years of life, then slowly increasing.

Googling with those terms will lead to much information. Much of interest here

EDIT

Some more information. This paper is a good starting point. They discuss a new extension of the Weibull, which they call EMWE (Exponentiated Modified Weibull Extension distribution) with four parameters, which permits bathtube shaped hazard with form close to hazard functions seen in practice. A plot from that paper is

with pdf's on the left and corresponding hazard rates on the right.

For reference I will give the cdf and pdf functions: $$ f(x;\alpha,\beta,\lambda,\gamma)=\lambda\beta\gamma(x/\alpha)^{\beta-1}\exp\left\{(x/\alpha)^\beta+\lambda\alpha(1-e^{(x/\alpha)^\beta} \right\}\cdot \left\{1-e^{\lambda\alpha(1-e^{(x/\alpha)^\beta}}\right\}^{\gamma-1}\\ F(x;\alpha,\beta,\lambda,\gamma)=\left\{1-\exp[\lambda\alpha(1-e^{(x/\alpha)^\beta}]\right\}^\gamma $$ and the hazard rate is $$ h(x;\alpha,\beta,\lambda,\gamma)=\frac{\lambda\beta\gamma(x/\alpha)^{\beta-1}\exp[(x/\alpha)^\beta+\lambda\alpha(1-e^{(x/\alpha)^\beta})]}{[1-\exp[\lambda\alpha(1-e^{(x/\alpha)^\beta})]]^{1-\gamma}+\exp\left\{ \lambda\alpha(1-e^{(x/\alpha)^\beta})\right\} -1} $$ Estimation can be done with maximum likelihood.