What are the limitations of using the "limiting density ratios" method to compare the tail weight of two distributions? (when will this measure not work ?)

What are the disadvantage of using the method of moments approach to compare the tails of distributions? (especially if both distributions are light tailed and for both distributions all the moments exist?)

2 other measures of tail weight that I came across are: The hazard rate and the mean residual life. What are the limitations of these 2 measures? (when will they not be useful to compare the the tails weight of distributions?)

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