Let X 1 , . . . , X n be independent and identically distributed with density P θ ( x ) = { 2 x / θ 2 for 0 ≤ x &lt; θ 0 else Demonstrate that m n = m a x ( x 1 , . . . , x n ) is a sufficient statistic .

Question

Let   X  1  ,  .  .  .  ,  X  n be independent and identically distributed with density       P          θ        (  x  )  =      {                            2          x                      /                                              θ                        2                                    for           0          ≤          x          &amp;lt;                      θ                                                0                          else                        Demonstrate that       m    n    =  m  a  x  (      x    1    ,  .  .  .  ,      x    n    ) is a sufficient statistic .

Teagan Raymond · Accepted Answer

You need to express all constraints inside your function, resulting in a density of the form      f          X        (  x  )  =  h  (  x  )  g  (  θ  ,  T  (      x    )  )  ,where   T is the sufficient statistic, in our case  T  (  x  )  =  max  (      x    1    ,  …  ,      x    n    )  .To do this, rewrite the expression for the density as follows:                              f                      X                          (        x        )                            =                  (                      ∏                          j              =              1                        n                                              2                              x                j                                                              1                                                  {                                      x                    j                                    &amp;gt;                  0                  }                                                                    θ              2                                )                                      1                                {            max            (                          x              1                        ,            …            ,                          x              n                        )            ≤            θ            }                                                            =                                                            (                                  ∏                                      n                    =                    1                                    n                                2                                  x                  j                                                                      1                                                        {                                          x                      j                                        &amp;gt;                    0                    }                                                  )                            ⏟                                            =            h            (            x            )                          ×                                                                              θ                                      −                    2                    n                                                  ×                                                      1                                                        {                    max                    (                                          x                      1                                        ,                    …                    ,                                          x                      n                                        )                    ≤                    θ                    }                                                              ⏟                                            =            g            (            θ            ,            T            (            x            )            )

Let X_1,...,X_n be independent and identically distributed with density Ptheta(x)= {2x/theta^2 for 0<=x<theta, 0 else Demonstrate that mn=max(x_1,...,x_n) is a sufficient statistic .

Answered question

Answer & Explanation

New Questions in College Statistics