X 1 , . . . X n iid with f ( x ; θ ) = 1 θ ( 1 + x ) − θ + 1 θ For the statistic T ( x ) = ( X 1 , . . . , X n ) (which I guess just means compute the MLE in a general sense for the sample)

Question

X    1    ,  .  .  .      X    n    iid with   f  (  x  ;  θ  )  =      1    θ    (  1  +  x      )                            −          θ          +          1                θ            For the statistic   T  (  x  )  =  (      X    1    ,  .  .  .  ,      X    n    ) (which I guess just means compute the MLE in a general sense for the sample)

Sauppypefpg · Accepted Answer

L                (        x        ;                  θ                )                            =                  ∏                      k            =            1                    n                f        (                  x          k                ;                  θ                )        =                  1                                    θ                        n                                    ∏                      k            =            1                    n                (        1        +                  x          k                          )                      −            1            +                          1                              θ                                                                                  l                (        x        ;                  θ                )        =        ln        ⁡                  (                      L                    (          x          ;                      θ                    )          )                                    =        −        n        ln        ⁡                              θ                          +                  (                      1                          θ                                −          1          )                          ∑                      k            =            1                                n                          ln        ⁡                  (          1          +                      x            k                    )                                              d                      d                          θ                                                l                (        x        ;                  θ                )                            =        −                  n                      θ                          −                  1                                    θ                        2                                    ∑                      k            =            1                                n                          ln        ⁡                  (          1          +                      x            k                    )                          =          !                0                ⟺                                                    θ                        ^                          =        −                  1          n                          ∑                      k            =            1                                n                          ln        ⁡                  (          1          +                      x            k                    )

X_1,...X_n iid with f(x;theta)=1/theta (1+x) (−theta+1)/theta. For the statistic T(x)=(X_1,...,X_n) (which I guess just means compute the MLE in a general sense for the sample)

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