Derive the Hajek pojection of $T_n$.
$begingroup$
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
asked Jan 7 at 17:13
L200123L200123
855
$endgroup$
add a comment |
$begingroup$
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
asked Jan 7 at 17:13
L200123L200123
855
$endgroup$
add a comment |
$begingroup$
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
asked Jan 7 at 17:13
L200123L200123
855
$endgroup$
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
asked Jan 7 at 17:13
L200123L200123
855
asked Jan 7 at 17:13
L200123L200123
855
asked Jan 7 at 17:13
L200123L200123
855
asked Jan 7 at 17:13
L200123L200123
855
asked Jan 7 at 17:13
L200123L200123
855
855
add a comment |
add a comment |
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