Am I not understanding this right or is there a contradiction between what is discussed in the Hazard rate discussion on Page <28> enclosed below and Question 16.5 discussion on Limiting Density Ratios. The answer for 16.5 states that Limiting Density Ratio of ~Gamma(alpha, lamda) (given alpha>1) and ~Exp(lamda) results in Gamma distribution having fatter tails than exponential distribution. I find this contracting the discussion on Hazard rate where for alpha>1, Gamma has a lighter tail. Can anyone explain this to me?
If you look at the limit value to the answer in question 16.5, you can see that as long as alpha > 1, then the limit is always positive. But if the limit is positive, then it takes it longer and longer to reach 0, as if it reaches 0 then it is decreasing at the same rate as the exponential distribution. For the gamma distribution to have the heavier tail, then its limit must be > 0, as this implies that the gamma distribution takes longer to reach the x-axis compared to the exponential distribution. However, since the ratio of the gamma distribution compared to the exponential distribution goes to infinity, and not negative infinity as x goes to infinity, then this implies that the value of the gamma distribution at each specific x value on the graph of the gamma distribution must be larger than the corresponding x value for the exponential distribution. Due to this, this proves that the gamma distribution has heavier tails than the exponential distribution, as it decreases at a slower rate due to its value on the graph consistently being higher than the corresponding exponential distribution point at the same x value. Hope that makes sense
Appreciate the response, however I don't think I understand your explanation, or else, from what I understood I have to respectfully disagree with the explanation. It is quite clear computationally, that the limit density ratio spits out an infinite value thereby implying ~Gamma(alpha,lamda) has fatter tails. That's not the problem! My question is why does this contradict the explanation provided in the Hazard rate discussion where for alpha>1, gamma distributions has a thinner tail than if alpha=1 (which is the equivalent of the exponential distribution ~Exp(lamda)). 16.5 is an ACTED question and not a past-exam question, so I would appreciate a response on the logic behind this question and why it contradicts (if this contradiction is in fact real?) with the Hazard rate discussion on Page 28. Further to your explanation, my disagreement is as follows: Firstly, we are looking at limiting density ratios which are always going to be non-negative irrespective of alpha greater or less than one (they are ratios of PDFs which are always non-negative), so not sure what you are trying to say here. Secondly, this ratio can never go to negative infinity (for the same reason above), which makes it harder to really get at what you are trying to explain here. Thanks very much again
No worries I think I've jammed this one up as I don't have the Core Reading on me at all, so I'd love for someone else to help out if possible.
The limit of the hazard rate of the Gamma(alpha, lambda) distribution as x tends to infinity is lambda (if you wish to see this then you may, for example, start with definition of hazard rate given and use L’Hopital). It does not tend to 0 as x-> infinity. You can also see this visually with some of the example graphs in the Core Reading. Thus (under certain conditions met here that allows us to equate this comparison with the pdf ratio test) comparing a Gamma(alpha, lambda_1) distribution to an Exponential (lambda_2) distribution comes down to comparing lambda_1 with lambda_2: 1. If lambda_1 is bigger than lambda_2 then the Gamma is lighter (larger long term hazard and eventually a lighter tail) 2. If lambda_2 is bigger than lambda_1 then the Gamma is heavier (smaller long term hazard and eventually heavier tail) 3. If lambda_1 = lambda_2 (as in 16.5) then if alpha > 1 Gamma is heavier, if alpha < 1 then Gamma is lighter. (If alpha = 1 then in this case the distributions are the same). Indeed, if you perform the pdf ratio limit comparison in the above cases, you should come to the same conclusion. The difference you noticed is from comparing a generic statement about how hazard rates may increase vs. decrease vs. be constant for different types of distributions (and how this affects the shape of the distributions and the tails) and an actual comparison between two 'specific' distributions (i.e. with particular ranges given for lambda and alphas). Recall that alpha is the shape parameter for the Gamma distribution, have a look at pdfs for alpha above and below 1 and how this affects the tails. You should conclude that the generic statement about tail weight appears reasonable in comparison to the shape of the Exponential distribution. Note in comparison to, for example, the Pareto distribution, as the Pareto hazard -> 0 as x -> infinity then if you use the pdf ratio limit test you should find that all Pareto distributions will have a heavier tail than all Exponential distributions.
