Asymmetric quasimedians: Remarks on an anomaly

Let X₁ ≤ X₂ ≤ ⋯ ≤ X_n denote the order statistics of a random sample of size n and M the sample median defined conventionally as the middle X_m for n = 2m + 1 and the average (X_m + X_m+1)/2 for n = 2m. Hodges (1967) observed that for the normal populations the asymptotic efficiency 2/π of the sample median is approached consistently through higher values for the even sample sizes, n = 2m, than for the odd samples sizes, n = 2m + 1. Hodges and Lehmann (1967) explained this even-odd anomaly in terms of the O(n^-2)-term in the large sample variance of M, and extended it to quasimedians M_r of arbitrary symmetric populations. We obtain the large sample bias and variance of the asymmetric average M_r,s(h) = hX_m+1-r + h̄X_m+1+s, h̄ = 1 - h, consider various tradeoffs, construct modifications M⁽¹⁾_r and M⁽²⁾_r, of M_r for asymmetric distributions. Also, the observation due to Hodges and Lehmann (1967), which is often interpreted as an anomaly, is examined in the asymmetric case.