Asymmetric quasimedians: Remarks on an anomaly

Govind S. Mudholkar and Alan D. Hutson

Statistics & Probability Letters, 1997, vol. 32, issue 3, 261-268

Abstract: Let X1 [less-than-or-equals, slant]X2 [less-than-or-equals, slant] ··· [less-than-or-equals, slant]Xn denote the order statistics of a random sample of size n and M the sample median defined conventionally as the middle Xm for n = 2m + 1 and the average (Xm + Xm+1)/2 for n = 2m. Hodges (1967) observed that for the normal populations the asymptotic efficiency 2/[pi] 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 Mr of arbitrary symmetric populations. We obtain the large sample bias and variance of the asymmetric average , consider various tradeoffs, construct modifications Mr(1) and Mr(2), of Mr 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.

Keywords: Asymptotic; expansions; Bias-variance; tradeoff (search for similar items in EconPapers)
Date: 1997
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Citations: View citations in EconPapers (2)

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