Assume X is a continuous random variable which have a density function. Assume Y _ is a random vector which also have a density function. And finally assume we have a joint density function of the random variable and the random vector.How can I find a function ϕ, such that E [ | X − ϕ ( Y _ ) | | Y _ ] would be minimal?I'm not sure where to start. I do know that the asnwer should be that ϕ ( Y _ ) should be a median of F X | Y _ , but Im not sure how to prove it. Any help would be appreciated.Thanks in advance.

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Assume   X is a continuous random variable which have a density function. Assume       Y    _   is a random vector which also have a density function. And finally assume we have a joint density function of the random variable and the random vector.How can I find a function   ϕ, such that       E        [          |      X      −      ϕ              (                  Y          _                )            |              |              Y      _        ]   would be minimal?I&#039;m not sure where to start. I do know that the asnwer should be that   ϕ  (      Y    _    ) should be a median of       F          X              |                    Y        _            , but Im not sure how to prove it. Any help would be appreciated.Thanks in advance.

Blaine Foster · Accepted Answer

For a fixed   y,                    ∫                  |                x        −        m                  |                          f                      X            ∣            Y                          (        x        ∣        y        )                d        x                            =                  ∫                      −            ∞                                m                          (        m        −        x        )                  f                      X            ∣            Y                          (        x        ∣        y        )                d        x                                                  +                  ∫                      m                                ∞                          (        x        −        m        )                  f                      X            ∣            Y                          (        x        ∣        y        )                d        x        .            Differentiating the RHS w.r.t.   m and equating to zero yields      ∫          −      ∞                      m        ∗                  f          X      ∣      Y        (  x  ∣  y  )    d  x  =      ∫                  m        ∗                    ∞            f          X      ∣      Y        (  x  ∣  y  )    d  x  ,or      F          X      ∣      Y        (      m    ∗    ∣  y  )  =  1  −      F          X      ∣      Y        (      m    ∗    ∣  y  )  .That is m∗ is the conditional median of   X given   Y.

Assume X is a continuous random variable which have a density function. Assume <munder>

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