Let X 1 , X 2 , . . . , X n be a random sample from a N ( θ 1 , θ 2 ) distribution. Find the uniformly minimum variance unbiased estimator of 3 θ 2 2 .

Question

Let       X    1    ,      X    2    ,  .  .  .  ,      X    n   be a random sample from a   N  (      θ    1    ,      θ    2    ) distribution. Find the uniformly minimum variance unbiased estimator of   3                    θ        2              2  .

Kristen Garza · Accepted Answer

Lets       X    i    ∼  N  (  μ  ,      σ    2    ). We want to show   (  ∑      X    i    ,  ∑      X    i    2    ) is complete for   (  μ  ,      σ    2    ).It is enough to show   (            X      ¯        ,  S  =  ∑  (      X    i    −            X      ¯            )    2    ) is complete. We know             X      ¯       and   S are independent and             X      ¯        ∼  N  (  μ  ,            σ      2        n    ),   S  ∼  G  a  m  m  a  (            n      −      1        2    ,  2      σ    2    ).We should show if    ∀  (  μ  ,      σ    2    )  E  (  g  (            X      ¯        ,  S  )  )  =  0  ⇒  P  (  g  (            X      ¯        ,  S  )  =  0  )  =  1  0  =  E  (  g  (            X      ¯        ,  S  )  )  =      ∫    0    ∞        ∫          −      ∞              +      ∞        g  (            x      ¯        ,  s  )  f  (            x      ¯        )  f  (  s  )  d            x      ¯          d  s  =      1          Γ      (                        n          −          1                2            )      (              σ        2                    )                                            n              −              1                        2                                    ∫    0    ∞        (          ∫              −        ∞                    +        ∞              g    (                  x        ¯              ,    s    )    f    (                  x        ¯              )          s                                    n            −            1                    2                −        1                    e              −                  s                      σ            2                                d                  x        ¯                  )    d  s  =      1          Γ      (                        n          −          1                2            )      (              σ        2                    )                                            n              −              1                        2                                    ∫    0    ∞        (          ∫              −        ∞                    +        ∞              g    (                  x        ¯              ,    s    )    f    (                  x        ¯              )          s                                    n            −            1                    2                −        1              d                  x        ¯                  )        e          −              s                  σ          2                      d  s  =      1          Γ      (                        n          −          1                2            )      (              σ        2                    )                                            n              −              1                        2                                    ∫    0    ∞        (    h    (    s    )    )        e          −              s                  σ          2                      d  sThe above is a Laplace transform of   h  (  s  ), which implies   h  (  s  )  =  0, a.e.So  0  =      ∫          −      ∞              +      ∞        g  (            x      ¯        ,  s  )  f  (            x      ¯        )  d            x      ¯        =      ∫          −      ∞              +      ∞        g  (            x      ¯        ,  s  )      1          2      π                        σ          2                n                  e          −              n                  2                      σ            2                              (                        x          ¯                    −      μ              )        2              d            x      ¯        =      ∫          −      ∞              +      ∞            (    g    (                  x        ¯              ,    s    )          1              2        π                              σ            2                    n                            e              −                  n                      2                          σ              2                                                                          x              ¯                                2                            e              −                  n                      2                          σ              2                                                μ          2                      )        e                  n                  2                      σ            2                              2                        x          ¯                    μ        d            x      ¯      The above is a Two-sided Laplace transform.So   g  (            x      ¯        ,  s  )  =  0 a.e.

Let X_1,X_2,...,X_n be a random sample from a N(theta1,theta2) distribution. Find the uniformly minimum variance unbiased estimator of 3theta_2^2.

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