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M18 Upper tail dependence

Discussion in 'SP9' started by ALEX_AK, Jul 3, 2019.

  1. ALEX_AK

    ALEX_AK Member

    Hello I have a question on the upper tail dependence formula.

    From the notes,
    P(F(x)>u, F(y)>u)=C bar(1-u, 1-u)

    May I know the meaning of C bar(u, u)? From the above, I have derived the below. Is the equation below correct and what does it mean?
    C bar(u, u) = P(F(x)>1-u, F(y)>1-u)
     
    ALEX_AK, Jul 3, 2019
    #1
  2. Anna Bishop

    Anna Bishop ActEd Tutor Staff Member

    Hi Alex

    My brain is sore from thinking about this!

    As an example, let's take u = 0.05

    u = F(x) = F(y)

    0.05 = u = P(X < x) = P(Y < y)
    0.95 = 1-u = P(X > x) = P(Y > y)

    C(u, u) = P(F(x)<u, F(y)<u)
    This is the probability that X and Y take values in the lower 5% of the distribution.

    C(1-u, 1-u) = P(F(x)<1-u, F(y)<1-u)
    This is the probability that X and Y take values in the lower 95% of the distribution.

    C bar(1-u, 1-u) = P(F(x)>u, F(y)>u) {Result from Page 19 of Module 18)
    C bar is the survival probability.
    This is the probability both X and Y take values in the upper 95% of the distribution, ie that they survive the first 5%!

    C bar(u, u) = P(F(x)>1-u, F(y)>1-u)
    C bar is the survival probability.
    This is the probability both X and Y take values in the upper 5% of the distribution, ie that they survive the first 95%!

    Is this OK?
    Anna
     
    Anna Bishop, Jul 4, 2019
    #2
  3. ALEX_AK

    ALEX_AK Member

    Hello Anna,

    It is very clear. Thanks a lot for your help. :)
     
    ALEX_AK, Jul 5, 2019
    #3
    Anna Bishop likes this.

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