Assume we have Y 1 , . . . , Y n iid sample from the uniform distribution U ( 0 , θ ). Assume T ( Y 1 , . . . , Y n ) = M i n 1 &lt;= i &lt;= n ( Y i )

Question

Assume we have       Y    1    ,  .  .  .  ,      Y    n   iid sample from the uniform distribution   U  (  0  ,  θ  ). Assume   T  (      Y    1    ,  .  .  .  ,      Y    n    )  =  M  i      n          1      &amp;lt;=      i      &amp;lt;=      n        (      Y    i    )

Arely Davila · Accepted Answer

t      y    ″    −  t      y    ′    +  y  =  1  −            (              s        2                    L            (      y      )      −      s      y      (      0      )      −              y        ′            (      0      )      )        ′    +            (      s              L            (      y      )      −      y      (      0      )      )        ′    +      L    (  y  )  =            1      s        (  s  −      s    2    )            L        ′    (  y  )  +  (  2  −  2  s  )      L    (  y  )  =            1      s                  L        ′    (  y  )  +            2      s            L    (  y  )  =            1                        s          2                (        1        −        s        )            with integrating factor       s    2   we have                    (                  s          2                          L                (        y        )                  )          ′                                    =                              1                          1              −              s                                                          y                            =                              L                                −            1                                                              −              ln              ⁡              (              1              −              s              )                                      s              2                                      +        C        t                                          =                  ∫          0          t                                                    e              x                        x                          (        t        −        x        )        d        x        +        C        t                                          =        t                  ∫          0          t                                                    e              x                        x                          d        x        −                  e          t                +        1        +        C        t                                          =        t        Ei        ⁡        (        t        )        −                  e          t                +        1        +        C        t

Assume we have Y_1,...,Y_n iid sample from the uniform distribution U(0,theta). Assume T(Y_1,...,Y_n)=M_in_(1<=i<=n)(Y_i)

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