Let the joint distribution of (X, Y) be bivariate normal with mean vector ( 0 0 ) and variance-covariance matrix ( 1 𝝆 𝝆 1 ) , where − 𝟏 &lt; 𝝆 &lt; 𝟏 . Let 𝚽 𝝆 ( 𝟎 , 𝟎 ) = 𝑷 ( 𝑿 ≤ 𝟎 , 𝒀 ≤ 𝟎 ) . Then what will be Kendall’s τ coefficient between X and Y equal to?

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Let the joint distribution of (X, Y) be bivariate normal with mean vector       (                            0                                      0                      )   and variance-covariance matrix      (                            1                                      𝝆                                                            𝝆                                    1                      )   , where   −      𝟏    &amp;lt;      𝝆    &amp;lt;      𝟏   . Let             𝚽                      𝝆              (      𝟎    ,      𝟎    )  =      𝑷    (      𝑿    ≤      𝟎    ,      𝒀    ≤      𝟎    ) . Then what will be Kendall’s   τ coefficient between X and Y equal to?

tilsjaskak6 · Accepted Answer

Step 1Originally, Kendall&#039;s tau, also called rank correlation, is a statistical measure that can be applied to a discrete set of observed data.In the more recent literature about dependency modelling with Copulas which became popular in mathematical finance the following definition of Kendall&#039;s tau is given.Let       Φ    ρ    (  x  ,  y  )  ,  Φ  (  x  )  ,  Φ  (  y  ) be the bivariate and the univariate CDFs of the standard normal distribution. Then the Gaussian Copula is defined as      C    ρ    (  x  ,  y  )  =      Φ    ρ    (      Φ          −      1        (  x  )  ,      Φ          −      1        (  y  )  )Kendall&#039;s tau is then defined as                              ρ          τ                                    =                  E                                      [                                    s          i          g          n                [        (        X        −                              X            ~                          )        (        Y        −                              Y            ~                          )        ]                              ]                                                            =                  P                                      [                          (        X        −                              X            ~                          )        (        Y        −                              Y            ~                          )        &amp;gt;        0                              ]                          −        P                              [                          (        X        −                              X            ~                          )        (        Y        −                              Y            ~                          )        &amp;lt;        0                              ]                                  .            where (X, Y) is bivariate standard normal, and   (            X      ~        ,            Y      ~        ) has the same distribution but is independent of (X, Y). It can be shown (see (1) and duplicate) that      ρ    τ    =  4      ∫    0    1        ∫    0    1        C    ρ    (  x  ,  y  )    d      C    ρ    (  x  ,  y  )  −  1  =      2    π    arcsin  ⁡  ρ    .

Let the joint distribution of (X, Y) be bivariate normal with mean vector ( <mtable r

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