Lemma 3.1: asymptotic representation of the kernel quantile estimator

A Bahadur-type representation – an i.i.d. average plus a smaller-order remainder – which is what makes normality, the asymptotic variance and the Edgeworth expansion all follow from standard theory for sums.

Usage

morie_fauzi_lemma_3_1(x, p, h = NULL, q_true = NULL)

Arguments

x

numeric sample.

p

probability level strictly in (0, 1).

h

bandwidth passed to the quantile estimator.

q_true

the population quantile, when known.

Value

list: influence, linear_term, estimate, remainder, density_at_quantile, centre, centred_at, asymptotic_variance, representation, n, method.

Details

The lemma expands about the POPULATION quantile. Centring on the sample quantile instead makes the linear term vanish identically, because the empirical df at its own p-quantile is p up to 1/n; the decomposition then carries no information and the variance it licenses is unsupported. When q_true is not supplied the sample quantile is used and centred_at says so.

References

Fauzi and Maesono (2023), Lemma 3.1. From the PDF.

Examples

set.seed(1)
morie_fauzi_lemma_3_1(stats::rexp(200), 0.5, q_true = log(2))$linear_term
#> [1] 0.1213494