I`m trying to find an answer, but i have some problems, help.Let P θ = U [ 0 , θ ].For h , θ 0 &gt; 0 and Z ∼ e x p ( 1 θ 0 ) I have to show that: d P θ 0 − h / n n d P θ 0 n ⟶ d , P θ 0 n e h θ 0 1 { Z ≥ h } I already proved that for Z n = n ( θ 0 − max { X 1 , … , X n } ) with X 1 , … X n ∼ P θ 0 holds Z n → D Z and this task seems like I have to prove that the pdf is converging too. I'm not sure which technical steps I need to show this and I'm not sure which kind of convergence is meant by d , P θ 0 n .

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I`m trying to find an answer, but i have some problems, help.Let             P        θ    =  U  [  0  ,  θ  ].For   h  ,      θ    0    &amp;gt;  0 and   Z  ∼      e    x    p        (          1              θ        0              )   I have to show that:                    d                              P                                      θ            0                    −          h                      /                    n                n                            d                              P                                      θ            0                          n                  ⟶          d      ,                        P                                      θ            0                          n                  e          h              θ        0                        1              {      Z      ≥      h      }      I already proved that for       Z    n    =  n  (      θ    0    −  max  {      X    1    ,  …  ,      X    n    }  ) with       X    1    ,  …      X    n    ∼            P                      θ        0             holds       Z    n        →    D    Z and this task seems like I have to prove that the pdf is converging too. I&#039;m not sure which technical steps I need to show this and I&#039;m not sure which kind of convergence is meant by   d  ,            P                      θ        0              n  .

Govorei9b · Accepted Answer

The convergence follows with Radon-Nikodym, Slutzky and continuous mapping theorem.                                                        d                                                      P                                                              θ                  0                                −                h                                  /                                n                            n                                                          d                                                      P                                                              θ                  0                                            n                                                          =                                            d                                                      P                                                              θ                  0                                −                h                                  /                                n                            n                                                          d                        λ                                                (                                                            d                                                                      P                                                                              θ                      0                                                        n                                                                              d                                λ                                      )                                −            1                                                            =                              (                                          θ                                  0                                                                              θ                                      0                                                  −                                  h                  n                                                      )                                n                                                              1                                      {              0              ≤                              X                i                            ≤                              θ                0                            −              h                              /                            n                             ∀              i              }                                                          1                                      {              0              ≤                              X                i                            ≤                              θ                0                                           ∀              i              }                                                                        =                              (                                          θ                                  0                                                                              θ                                      0                                                  −                                  h                  n                                                      )                                n                                                1                                {                          Z              n                        ≥            h            }                                                                      →          Slutzky                          e          x          p                          (                      h                          θ              0                                )                                      1                                {            Z            ≥            h            }

I`m trying to find an answer, but i have some problems, help. Let <mrow class="MJX-TeXAtom-

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