Define the sample ????-quantile of x1,..., xn – with ???? ∈ (1∕n, 1∕1∕n) – as x(k), where k is the smallest integer ≥ n???? and x(i) are the order statistics (1.2). Let

????(x) = ????I(x > 0)−(1 − ????)I(x < 0)

Show that ???? = x(k) is a solution (not necessarily unique) of (2.19). Use this fact to derive the asymptotic distribution of sample quantiles, assuming that (xi)
has a unique ????-quantile. Note that this ???? is not odd.