Let X be a random variable having the negative binomial distribution with P ( X = x ) = ( x − 1 r − 1 ) p r ( 1 − p ) x − r , x = r , r + 1 …where p ∈ ( 0 , 1 ) and r is a known positive integer.Find the UMVUE of p t ,, where t is a positive integer and t &lt; r.

Question

Let   X be a random variable having the negative binomial distribution with  P  (  X  =  x  )  =      (                            x          −          1                                      r          −          1                      )        p          r        (  1  −  p      )          x      −      r        ,  x  =  r  ,  r  +  1  …where  p  ∈  (  0  ,  1  ) and   r is a known positive integer.Find the UMVUE of       p          t        ,, where   t is a positive integer and   t  &amp;lt;  r.

Spielgutq1 · Accepted Answer

Since   X is a complete sufficient statistic for   p, all you need is an unbiased estimator of       p    t   based on   X. This estimator would be the UMVUE of       p    t   by Lehmann-Scheffé theorem.So take any function   g  (  X  ) which is unbiased for       p    t   for every   p  ∈  (  0  ,  1  ) and solve for   g.You have  E      [    g    (    X    )    ]    =      ∑          j      =      r        ∞    g  (  j  )                    (                            j        −        1                    r        −        1                            )                  p    r    (  1  −  p      )          j      −      r        =      p    t      ,  ∀    p  ∈  (  0  ,  1  )Taking   q  =  1  −  p, this implies                              (                      ⋆                    )                                      ∑                      j            =            r                    ∞                g        (        j        )                                            (                                                          j              −              1                                      r              −              1                                                          )                                                q          j                =                              q            r                                (            1            −            q                          )                              r                −                t                                                    =                  ∑                      k            =            r            −            t                    ∞                                                    (                                                          k              −              1                                      r              −              t              −              1                                                          )                                                q                      k            +            t                                  ,        ∀                q        ∈        (        0        ,        1        )            The infinite series expansion in the last step follows from the fact that      ∑          k        P  (  X  =  k  )  =  1    ⟹        ∑          k      =      r        ∞                      (                            k        −        1                    r        −        1                            )                  q    k    =            (              q                  1          −          q                    )        r  This can also be shown as a separate identity.Finally compare coefficients of       q    j   from both sides of   (  ⋆  ) to find   g  (  ⋅  ).

Let X be a random variable having the negative binomial distribution with P(X=x)=((x−1),(r−1))p^r(1−p)^(x−r),x=r,r+1… where p in (0,1) and r is a known positive integer. Find the UMVUE of p^t, where t is a positive integer and t<r.

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