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For which joint distributions is a conditional expectation an additive

  1. Jun 5, 2014 #1
    I know that, for a random vector (X,Y,Z) jointly normally distributed, the conditional expectation E[X|Y=y,Z=z] is an additive function of y and z

    For what other distributions is this true?
     
  2. jcsd
  3. Jun 5, 2014 #2

    statdad

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    Homework Helper

    The class of elliptical distributions (see ``Multivariate Statistical Theory'' by Robb Muirhead for an introductory discussion) has the property I believe you want. Specifically, if
    [tex]
    \mathbf{X} = \begin{bmatrix} \mathbf{X_1} \\ \mathbf{X_2} \end{bmatrix},%
    \mathbf{\mu} = \begin{pmatrix} \mathbf{\mu_1} \\ \mathbf{\mu_2} \end{pmatrix},%
    V = \begin{bmatrix} V_{11} & V_{12} \\ V_{21} & V_{22} \end{bmatrix}
    [/tex]

    then

    [tex]
    E[\mathbf{X_1} \mid \mathbf{X_2}] = \mathbf{\mu_1} + V_{12} V_{22}^{-1} \left(%
    \mathbf{X_2} - \mathbf{\mu_2}\right)
    [/tex]

    - the same type of relationship demonstrated by multivariate normal distributions.
     
    Last edited: Jun 5, 2014
  4. Jun 5, 2014 #3
    Yes. That's exactly what I need. Thanks a lot!
     
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