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Asymmetric quasimedians: Remarks on an anomaly

Author

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  • Mudholkar, Govind S.
  • Hutson, Alan D.

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.

Suggested Citation

  • Mudholkar, Govind S. & Hutson, Alan D., 1997. "Asymmetric quasimedians: Remarks on an anomaly," Statistics & Probability Letters, Elsevier, vol. 32(3), pages 261-268, March.
    • Handle: RePEc:eee:stapro:v:32:y:1997:i:3:p:261-268
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    References listed on IDEAS

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    1. Oosterhoff, J., 1994. "Trimmed mean or sample median?," Statistics & Probability Letters, Elsevier, vol. 20(5), pages 401-409, August.
    2. Cabrera, J. & Maguluri, G. & Singh, K., 1994. "An odd property of the sample median," Statistics & Probability Letters, Elsevier, vol. 19(4), pages 349-354, March.
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    Cited by:

    1. Alan Hutson, 2000. "A composite quantile function estimator with applications in bootstrapping," Journal of Applied Statistics, Taylor & Francis Journals, vol. 27(5), pages 567-577.
    2. Huang, J. S., 1999. "Third-order expansion of mean squared error of medians," Statistics & Probability Letters, Elsevier, vol. 42(2), pages 185-192, April.

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