Bootstrapping the Sample Quantile: A Survey

Abstract

Let X ₁,…, X _n be independent and identically distributed (≡ iid) random variables (≡ rvs) with common distribution function (≡ df) F and let T(F) be an unknown parameter of interest. The natural nonparametric estimator of T(F) is T(F _n ), where ${F_n}(t): = {n^{{ - 1}}}{\sum\nolimits_{{i = 1}}^n 1_{{\left( { - \infty, t} \right]}}}\left( {{X_i}} \right),t \in \mathbb{R}$, denotes the empirical df pertaining to the sample X ₁ ,…, X _n Our target function is now the df of the estimator T(F _n ), centered at the unknown parameter T(F), i.e.

$$ {G_n}(x): = {P_F}\left\{ {{n^{{\frac{1}{2}}}}\left( {T\left( {{F_n}} \right) - T(F} \right) \leqslant x} \right\},x \in \mathbb{R} $$

(1)

In a great many of cases, the factor n ^1/2 is the corrected standardization constant to ensure the nondegenerate limit distribution of n ^1/2 (T(F _n) -T(F)) as n increases.