Suppose we have X 1 , … , X n i.i.d, X i ∼ Exp ( 1 , μ ) (pdf is f μ ( x ) = e − ( x − μ ) for x ≥ μ and 0 for x &lt; μ). Is there any one dimensional (i.e. T : R n → R ) sufficient statistic for parameter μ? Obvious two dimensional sufficient statistic is T ( x 1 , … , x n ) = ( ∑ x i , min x i )

Question

Suppose we have       X    1    ,  …  ,      X    n   i.i.d,       X    i    ∼  Exp  (  1  ,  μ  ) (pdf is       f    μ    (  x  )  =      e          −      (      x      −      μ      )       for   x  ≥  μ and   0 for   x  &amp;lt;  μ). Is there any one dimensional (i.e.   T  :            R        n    →      R  ) sufficient statistic for parameter   μ? Obvious two dimensional sufficient statistic is   T  (      x    1    ,  …  ,      x    n    )  =      (    ∑          x      i        ,    min          x      i        )

Davian Nguyen · Accepted Answer

The likelihood of an i.i.d. sample   x  =  (      x    i        )          1      ⩽      i      ⩽      n       of distribution       E    (  1  ,  μ  ) is  L  (  μ  ,  x  )  =  exp  ⁡  (  −  s  (  x  )  +  n  μ  )  ⋅  [  t  (  x  )  ⩾  μ  ]  ,where   s  (  x  )  =      ∑          i      =      1        n        x    i   and   t  (  x  )  =      min          1      ⩽      i      ⩽      n            x    i  . This is  L  (  μ  ,  x  )  =  h  (  x  )  g  (  μ  ,  t  (  x  )  )  ,with   h  (  x  )  =  exp  ⁡  (  −  s  (  x  )  ) and   g  (  m  ,  u  )  =  exp  ⁡  (  n  m  )  ⋅  [  u  ⩾  m  ] hence   t  :  x  ↦  t  (  x  )  =      min          1      ⩽      i      ⩽      n            x    i   is a sufficient statistic for   μ.

Elias Heath · Accepted Answer

We have  L  (      x    1    ,  …  ,      x    n    ,  μ  )  =      ∏          j      =      1        n        e          −      (              x        j            −      μ      )                  1              [      μ      ,      +      ∞      [        (      x    j    )  =      e          −      n                        x          ¯                      ⋅      e          n      μ                  1              [      μ      ,      +      ∞      [            (          min              1        ≤        j        ≤        n                    x      k        )    ,so by Fischer-Neyman&#039;s theorems,   φ  (      x    1    ,  …  ,      x    n    )  =      min          1      ≤      j      ≤      n            x    j   is a sufficient statistic.

Suppose we have X_1,…,X_n i.i.d, X_i∼Exp(1,μ) (pdf is f_μ(x)=e^(−(x−μ)) for x≥μ and 0 for x<μ). Is there any one dimensional (i.e. T:R^n→R) sufficient statistic for parameter μ? Obvious two dimensional sufficient statistic is T(x_1,…,x_n)=(∑_(xi),min x _i)

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