The fewer positive raw moments that exist, the greater the tail weight.

The fewer positive raw moments that exist, the greater the tail weight.

• A. True
• B. False
• C. It implies lighter tails
• D. It has no effect on tail weight

Answer: (A)

Tail weight reflects how quickly probabilities fade as values get large. The existence of positive raw moments hinges on that tail decay: the kth moment exists only if the distribution’s tail is light enough for x^k to be integrable. If the tail is heavy, large values of x contribute too much and higher-order moments fail to converge. So when fewer positive raw moments exist, it signals that the tail is heavier. Conversely, lighter tails allow more moments to be finite (for example, the normal distribution has all moments finite). An example is the Pareto family, where the nth moment exists only if the shape parameter exceeds n; otherwise that moment diverges, indicating a heavier tail. Thus the statement is true.

Tail weight reflects how quickly probabilities fade as values get large. The existence of positive raw moments hinges on that tail decay: the kth moment exists only if the distribution’s tail is light enough for x^k to be integrable. If the tail is heavy, large values of x contribute too much and higher-order moments fail to converge. So when fewer positive raw moments exist, it signals that the tail is heavier. Conversely, lighter tails allow more moments to be finite (for example, the normal distribution has all moments finite). An example is the Pareto family, where the nth moment exists only if the shape parameter exceeds n; otherwise that moment diverges, indicating a heavier tail. Thus the statement is true.