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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.