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Can we tell what happens to the limit as $x$ approaches $\pm \infty$ of a hazard rate $h(x)$ defined for unspecified or generalized density as: $$ h(x)=A/(1-B) $$ where $A=f(x)$ is the density function, and $B=F(x)$ is the CDF.

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The case $x\to-\infty$ is trivial, the case $x\to+\infty$ is undetermined. Here are some examples regarding the latter:

  • if $f(x)\sim cx^{-a-1}$ with $a$ and $c$ positive, then $h(x)\sim ax^{-1}\to0$,
  • if $f(x)\sim c\mathrm e^{-ax}$ with $a$ and $c$ positive, then $h(x)\to a$,
  • if $f(x)\sim c\mathrm e^{-ax^{1+b}}$ with $a$, $b$ and $c$ positive, then $h(x)\sim ax^b\to+\infty$.
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$\lim_{x\rightarrow\infty}F(x)=1$

$\lim_{x\rightarrow-\infty}F(x)=0$,

but that's about it, unless you know $f(x)\rightarrow 0$ for $x\rightarrow \pm\infty$ which only gives you $\lim_{x\rightarrow-\infty}h(x)=0$

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