Measures Of Tail Weight

[Actuary_11]

In the topic of Hazard rate in Measures of Tail weight in Chapter 16 : Extreme Value Theory

Core Reading says that if the force of mortality increases as a person’s age increases, relatively few people will live to old age (corresponding to a light tail). If, on the other hand, the force of mortality decreases as the person’s age increases, there is the potential to live to a very old age (corresponding to a heavier tail).
So this means that when the hazard rate is decreasing function of x , it has the heavier tail .

But in section 4.3 , page no 29 :-

If alpha < 1, the hazard rate is a decreasing function of x , which tends to lambda from above for large x . This indicates a lighter tail than the exponential distribution with parameter lambda above any fixed threshold x , but the same tail thickness in the limit.
If alpha > 1, the hazard rate is an increasing function of x , which tends to lambda from below for large x . This indicates a heavier tail than the exponential distribution with parameter lambda above any fixed threshold x , but the same tail thickness in the limit.

From the above explanation it means that when we are having decreasing hazard rate , it has the lighter tail , which contradicts the explanation given above .

What is the explanation for the tail weights from the hazard rate function of that distribution ..? Like it will have a lighter tail or heavy tail when the hazard rate is the decreasing function of x and vice versa.

 

[Dave Johnson]

Hi

I agree there is an apparent contradiction here, and the reason is that the first statement where the parallel is drawn with the force of mortality is not completely correct.

What the technical example with the gamma/exponential shows is that just because one distribution has an increasing hazard function and another has a decreasing hazard function, it doesn't necessarily mean it will have the lighter tail. This is because the hazard function could be lower in absolute terms for all x even though it is increasing.

We could make the first paragraph more accurate by saying: "if the force of mortality increases to infinity as a person’s age increases, relatively few people will live to old age (corresponding to a light tail). If, on the other hand, the force of mortality decreases to zero as the person’s age increases, there is the potential to live to a very old age (corresponding to a heavier tail)", however even this isn't perfect, and it is already getting less intuitive.

It's best to interpret the first paragraph as broadly indicating how tail weight relates to the force of mortality, and treat the second block of text as a technical example.

Hope that helps

Dave