Gumbel distribution是什么分布

问题如下：

The CRO of a regional bank expresses concern in a risk team meeting that the bank’s internal risk models are

not adequate in assessing potential random extreme losses. A risk analyst suggests that implementing a model

based on extreme value theory (EVT) could address this concern. Which of the following is correct when

applying EVT and examining distributions of losses exceeding a threshold value?

选项：

A.

As the threshold value is increased, the distribution of losses over a fixed threshold value converges to a

generalized Pareto distribution.

B.

If the tail parameter value of the generalized extreme-value (GEV) distribution goes to infinity, then the

GEV essentially becomes a normal distribution.

C.

To apply EVT, the underlying loss distribution must be either normal or lognormal.

D.

The number of exceedances decreases as the threshold value decreases, which causes the reliability of

the parameter estimates to increase.

解释：

中文解析：

A是正确的。EVT的一个关键基础是，随着阈值的增加，极端损失的分布趋近于广义帕累托分布。假设阈值足够高，极端损失可以用广义帕累托来建模。这被称为GPBdH定理，并且在阈值峰值（POT）方法中被广泛使用。

B是不正确的。如果尾部参数的广义极限参数值（GEV）分布趋于零，而不是无穷大，那么原始数据的分布(不是GEV)可以是一个轻尾分布，如正态分布或对数正态分布。换句话说，对应的GEV分布为Gumbel分布。

C是不正确的。要应用EVT，损失分布可以是任何一种常见的分布类型：正态分布、对数正态分布、t分布等。

D不正确。随着阈值的减小，超出值的数量也会增加。

A is correct. A key foundation of EVT is that as the threshold value is increased, the

distribution of loss exceedances converges to a generalized Pareto distribution. Assuming

the threshold is high enough, excess losses can be modeled using the generalized Pareto

distribution. It is known as the Gnedenko–Pickands–Balkema–deHaan (GPBdH) theorem

and is heavily used in the peaks-over-threshold (POT) approach.

B is incorrect. If the tail parameter value of the generalized extreme-value (GEV)

distribution goes to zero, and not infinity, then the distribution of the original data (not

the GEV) could be a light-tail distribution such as normal or log-normal. In other words,

the corresponding GEV distribution is a Gumbel distribution.

C is incorrect. To apply EVT, the underlying loss distribution can be any of the commonly

used distributions: normal, lognormal, t, etc.

D is incorrect. As the threshold value is decreased, the number of exceedances increases.