×

Estimating the asymptotic variance of generalized \(L\)-statistics. (English) Zbl 1075.62534

Summary: Strong convergence of the estimated asymptotic variance \((\sigma^2(T_a(H_N)))\) of generalized L-statistics is demonstrated for smooth and discrete weighting functions. For smooth weighting functions that are trimmed, such as the 25%-trimmed mean, and for discrete functions, such as the median, minimal conditions are required for strong convergence of \(\sigma^2(T_a(H_N))\). In a simulation study, \(\sigma^2(T_a(H_N))\) for the trimmed mean, but not the median, appeared to converge to the sample variance of the statistic for samples from three distributions (normal, contaminated normal and Cauchy). For the smallest sample in the study (\(n=16\)), \(\sigma^2(T_a(H_N))\) tended to underestimate the sample variance of the GL-statistics.

MSC:

62F12 Asymptotic properties of parametric estimators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1214/aop/1176993518 · Zbl 0514.60040 · doi:10.1214/aop/1176993518
[2] DOI: 10.1214/aos/1176346393 · Zbl 0538.62015 · doi:10.1214/aos/1176346393
[3] Lee A.J., Number 110 in Statistics, Textbooks and Monographs, in: U-Statistics: Theory and Practice (1990)
[4] DOI: 10.2307/2669765 · Zbl 1018.62099 · doi:10.2307/2669765
[5] Helmers R., Scan. J. of Stat. 17 pp 65– (1990)
[6] DOI: 10.1111/1467-9469.00161 · Zbl 0939.62044 · doi:10.1111/1467-9469.00161
[7] Senn S., Encyclopedia of Biostatistics 2 pp pp. 1033– (1998)
[8] Vonesh E.F., Linear and Nonlinear Models for the Analysis of Repeated Measurements (1997) · Zbl 0893.62077
[9] DOI: 10.1002/sim.4780080412 · doi:10.1002/sim.4780080412
[10] DOI: 10.1016/0378-3758(94)90191-0 · Zbl 0825.62649 · doi:10.1016/0378-3758(94)90191-0
[11] DOI: 10.1177/096228029400300405 · doi:10.1177/096228029400300405
[12] DOI: 10.1002/sim.4780131008 · doi:10.1002/sim.4780131008
[13] Friedman L.M., Fundamentals of Clinical Trials, (1998) · Zbl 1136.62072 · doi:10.1007/978-1-4757-2915-3
[14] DOI: 10.2307/2285666 · Zbl 0305.62031 · doi:10.2307/2285666
[15] van der Vaart A.W., Cambridge Series in Statistical and Probabilistic Mathematics, in: Asymptotic Statistics (2000)
[16] Billingsley P., Probability and Measure,, 3. ed. (1995) · Zbl 0822.60002
[17] DOI: 10.1214/aop/1176995853 · Zbl 0362.60019 · doi:10.1214/aop/1176995853
[18] DOI: 10.1007/BF00319105 · Zbl 0631.60032 · doi:10.1007/BF00319105
[19] Casella G., Statistical Inference (1990)
[20] DOI: 10.1016/0378-3758(88)90036-5 · Zbl 0662.62051 · doi:10.1016/0378-3758(88)90036-5
[21] DOI: 10.1016/0378-3758(90)90048-Y · Zbl 0693.62022 · doi:10.1016/0378-3758(90)90048-Y
[22] Kahaner D., Numerical Methods and Software (1989) · Zbl 0744.65002
[23] Bickel P., Holden-Day Series in Probability and Statistics, in: Mathematical Statistics (1977)
[24] Putt M.E., Doctoral Dissertation, in: Aspects of the Analysis of Crossover Studies (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.