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I've read this in a paper, and I don't know how to proof the last statement:

Let $X_1,...,X_n$ be independent identically distributed random variables with unkown distribution function F. Suppose the $\{\hat T_n=\hat T_n(X_1,...,X_n;n≥1\}$ are real-valued statistics and the $\{T_n(F);n≥1\}$ are real-valued functionals such that the asymptotic distribution of $n^{1/2}\{\hat T_n−T_n(F)\}$ is normal with mean zero. Let $H_n(x,F)$ be the exact distribution function of $n^{1/2}\{\hat T_n−T_n(F)\}$. A basic problem in statistics is the estimation of $H_n(x,F)$ or functionals of $H_n(x,F)$ from the sample. Indeed, the mean and variance of $H_n(x,F)$ are, respectively, the bias and variance of $\hat T_n$ when $\hat T_n$ is regarded as an estimate of $T_n(F)$.

Why the variance of $H_n(x,F)$ is the same as the one of $\hat T_n$ when $\hat T_n$ is regarded as an estimate of $T_n(F)$?

This is an argument that I've seen in some books dealing with the bootstrap resampling theory, and I can't figure out why that is true.

Thanks

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