Unbiased sufficient statistic for 1/p of geometric distributionSuppose that a random variable X has the geometric distribution with unknown parameter p, where the geometric probability mass function is: f ( x ; p ) = p ( 1 − p ) x , x = 0 , 1 , 2 , … ; 0 &lt; p &lt; 1.Find a sufficient statistic T(X) that will be an unbiased estimator of 1/p.Now I know the population mean is 1 − p p and the sufficient statistic for p is a function of ∑ i = 1 n X i . But I am unsure on how to proceed any help greatly appreciated!

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Unbiased sufficient statistic for 1/p of geometric distributionSuppose that a random variable X has the geometric distribution with unknown parameter p, where the geometric probability mass function is:  f  (  x  ;  p  )  =  p  (  1  −  p      )    x    ,    x  =  0  ,  1  ,  2  ,  …  ;    0  &amp;lt;  p  &amp;lt;  1.Find a sufficient statistic T(X) that will be an unbiased estimator of 1/p.Now I know the population mean is             1      −      p        p   and the sufficient statistic for p is a function of       ∑          i      =      1              n            X    i  . But I am unsure on how to proceed any help greatly appreciated!

Holly Crane · Accepted Answer

Step 1Assuming       X    1    ,      X    2    ,  …  ,      X    n   are i.i.d with pmf f.Step 2You have       E    p    (      X    1    +  1  )  =      1    p   for all   p  ∈  (  0  ,  1  ), so       E    p        [          1      n              ∑              i        =        1            n        (          X      i        +    1    )    ]    =      E    p        [          1      n              ∑              i        =        1            n              X      i        +    1    ]    =      1    p      ,    ∀    p  ∈  (  0  ,  1  )

Suppose that a random variable X has the geometric distribution with unknown parameter p, where the geometric probability mass function is: f(x;p)=p(1-p)^x, x=0, 1, 2,…;0<p<1.

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