Let X 1 , X 2 , X 3 be a iid sample of the Bernoulli p distribution. Consider the statistic T = X 1 X 2 + X 3 . Show that T is not sufficient for p.

Question

Let       X    1    ,      X    2    ,      X    3   be a iid sample of the Bernoulli   p distribution. Consider the statistic   T  =      X    1        X    2    +      X    3  . Show that   T is not sufficient for   p.

scapatofc · Accepted Answer

Use Law of Total Probability, partitioned on       X    3                                P                (        T                  =                t        )                 =                                       P                (                  X          3                          =                0        )                           P                (                  X          1                          X          2                          =                t        )        +                  P                (                  X          3                          =                1        )                           P                (                  X          1                          X          2                          =                t                  −                1        )                            =                             (        1        −        p        )                           P                (                  X          1                          X          2                          =                t        )        +        p                           P                (                  X          1                          X          2                          =                t                  −                1        )                                      P                (                  X          1                          X          2                          =                z        )                 =                                       {                                                    1                −                                  p                  2                                                            :                z                =                0                                                                                      p                  2                                                            :                z                =                1                                                                    0                                            :                                  otherwise                                                                                                            P                (        T                  =                t        )                 =                                       {                                                    (                1                −                p                )                (                1                −                                  p                  2                                )                +                0                                            :                t                =                0                                                                    (                1                −                p                )                                  p                  2                                +                p                (                1                −                                  p                  2                                )                                            :                t                =                1                                                                                  0                +                                  p                  3                                                            :                t                =                2                                                                    0                                            :                                  otherwise                                                                                                  =                                       {                                                    1                −                p                −                                  p                  2                                +                                  p                  3                                                            :                t                =                0                                                                    p                +                                  p                  2                                −                2                                  p                  3                                                            :                t                =                1                                                                                                    p                  3                                                            :                t                =                2                                                                    0                                            :                                  otherwise                                                                                                            P                (                  X          1                          =                x        ,                  X          2                          =                y        ,                  X          3                          =                t        −        x        y        )                 =                                       {                                                    (                1                −                p                                  )                  3                                                            :                (                x                ,                y                ,                t                )                =                (                0                ,                0                ,                0                )                                                                    (                1                −                p                                  )                  2                                p                                            :                (                x                ,                y                ,                t                )                ∈                {                (                1                ,                0                ,                0                )                ,                (                0                ,                1                ,                0                )                ,                (                0                ,                0                ,                1                )                }                                                                    (                1                −                p                )                                  p                  2                                                            :                (                x                ,                y                ,                t                )                ∈                {                (                1                ,                0                ,                1                )                ,                (                0                ,                1                ,                1                )                ,                (                1                ,                1                ,                1                )                }                                                                                      p                  3                                                            :                (                x                ,                y                ,                t                )                =                (                1                ,                1                ,                2                )                                                                    0                                            :                                  otherwise

Let X_1,X_2,X_3 be a iid sample of the Bernoulli (p) distribution. Consider the statistic T=X_1X_2+X_3. Show that T is not sufficient for p.

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