Derive the Hajek pojection of $T_n$.












1












$begingroup$


Let $X_1, dots , X_n$ i.i.d. copies of $X$ with distribution $F$ and density $f$. Let $(X_{1:n}, dots , X_{i:n}, dots , X_{n:n})$ be the order statistic. For a given $p in (0, 1)$ consider the Harrell-Davis estimator defined by



$$T_n = sum_{i=1}^n c_{ni}X_{n:i}$$



$$c_{ni} = frac{Gamma(n + 1)}{Gamma(k)Gamma(n − k + 1)} int^{i/n}_{(i−1)/n} u^{k−1}(1−u)^{n−k} du, i = 1,dots , n, k = [np].$$



I have three questions regarding this. I cannot seem to find any solid clear answers to them in my textbook



1st question: Which parameter $theta$ is estimated? I'm not sure if I have completely misinterpreted but it is rather unclear what we are estimating?



2nd question: What is the definition of the Hajek projection? I get that the Hajek projection has to do with the Hoeffding Decomposition, but I would really like a definition of the Hajek projection without delving into Hoeffding Decomposition.



3rd question: how does one derive the Hajek projection of $T_n$ from this example?






statistics statistical-inference order-statistics







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Let $X_1, dots , X_n$ i.i.d. copies of $X$ with distribution $F$ and density $f$. Let $(X_{1:n}, dots , X_{i:n}, dots , X_{n:n})$ be the order statistic. For a given $p in (0, 1)$ consider the Harrell-Davis estimator defined by



    $$T_n = sum_{i=1}^n c_{ni}X_{n:i}$$



    $$c_{ni} = frac{Gamma(n + 1)}{Gamma(k)Gamma(n − k + 1)} int^{i/n}_{(i−1)/n} u^{k−1}(1−u)^{n−k} du, i = 1,dots , n, k = [np].$$



    I have three questions regarding this. I cannot seem to find any solid clear answers to them in my textbook



    1st question: Which parameter $theta$ is estimated? I'm not sure if I have completely misinterpreted but it is rather unclear what we are estimating?



    2nd question: What is the definition of the Hajek projection? I get that the Hajek projection has to do with the Hoeffding Decomposition, but I would really like a definition of the Hajek projection without delving into Hoeffding Decomposition.



    3rd question: how does one derive the Hajek projection of $T_n$ from this example?






    statistics statistical-inference order-statistics







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $X_1, dots , X_n$ i.i.d. copies of $X$ with distribution $F$ and density $f$. Let $(X_{1:n}, dots , X_{i:n}, dots , X_{n:n})$ be the order statistic. For a given $p in (0, 1)$ consider the Harrell-Davis estimator defined by



      $$T_n = sum_{i=1}^n c_{ni}X_{n:i}$$



      $$c_{ni} = frac{Gamma(n + 1)}{Gamma(k)Gamma(n − k + 1)} int^{i/n}_{(i−1)/n} u^{k−1}(1−u)^{n−k} du, i = 1,dots , n, k = [np].$$



      I have three questions regarding this. I cannot seem to find any solid clear answers to them in my textbook



      1st question: Which parameter $theta$ is estimated? I'm not sure if I have completely misinterpreted but it is rather unclear what we are estimating?



      2nd question: What is the definition of the Hajek projection? I get that the Hajek projection has to do with the Hoeffding Decomposition, but I would really like a definition of the Hajek projection without delving into Hoeffding Decomposition.



      3rd question: how does one derive the Hajek projection of $T_n$ from this example?






      statistics statistical-inference order-statistics







      share|cite|improve this question









      $endgroup$




      Let $X_1, dots , X_n$ i.i.d. copies of $X$ with distribution $F$ and density $f$. Let $(X_{1:n}, dots , X_{i:n}, dots , X_{n:n})$ be the order statistic. For a given $p in (0, 1)$ consider the Harrell-Davis estimator defined by



      $$T_n = sum_{i=1}^n c_{ni}X_{n:i}$$



      $$c_{ni} = frac{Gamma(n + 1)}{Gamma(k)Gamma(n − k + 1)} int^{i/n}_{(i−1)/n} u^{k−1}(1−u)^{n−k} du, i = 1,dots , n, k = [np].$$



      I have three questions regarding this. I cannot seem to find any solid clear answers to them in my textbook



      1st question: Which parameter $theta$ is estimated? I'm not sure if I have completely misinterpreted but it is rather unclear what we are estimating?



      2nd question: What is the definition of the Hajek projection? I get that the Hajek projection has to do with the Hoeffding Decomposition, but I would really like a definition of the Hajek projection without delving into Hoeffding Decomposition.



      3rd question: how does one derive the Hajek projection of $T_n$ from this example?






      statistics statistical-inference order-statistics




      statistics statistical-inference order-statistics






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 7 at 17:13









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