Asymmetric quasimedians: Remarks on an anomaly

Let X(1) less than or equal to X(2) less than or equal to ... less than or equal to X(n) denote the order statistics of a random sample of size it and M the sample median defined conventionally as the middle X(n) for n = 2m + 1 and the average (X(m) + X(m+1))/2 for n = 2m. Hedges (1967) observed that for the normal populations the asymptotic efficiency 2/eta 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. Hedges and Lehmann (1967) explained this even-odd anomaly in terms of the 0(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) + hX(m+1+s) h = 1 - h, consider various tradeoffs, construct modifications M(r)((1)) and M(r)((2)), of M(r) for asymmetric distributions. Also, the observation due to Hedges and Lehmann (1967), which is often interpreted as an anomaly, is examined in the asymmetric case.