×

zbMATH — the first resource for mathematics

Spline confidence bands for variance functions. (English) Zbl 1165.62317
Summary: Asymptotically exact and conservative confidence bands are obtained for possibly heteroscedastic variance functions, using piecewise constant and piecewise linear spline estimation, respectively. The variance estimation is as efficient as an infeasible estimator when the conditional mean function is known, and the widths of the confidence bands are of optimal order. Simulation experiments provide strong evidence that corroborates the asymptotic theory while the computing is extremely fast. A slower bootstrap band is also proposed, with much higher accuracy. As illustration, the bootstrap spline band has been applied to test for heteroscedasticity in fossil data and in motorcycle data.

MSC:
62G15 Nonparametric tolerance and confidence regions
62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1214/aos/1176350260 · Zbl 0634.62032 · doi:10.1214/aos/1176350260
[2] Hall P., J. Roy. Statist. Soc. Ser. B 51 pp 3– (1989)
[3] DOI: 10.2307/1271131 · Zbl 0891.62029 · doi:10.2307/1271131
[4] DOI: 10.1093/biomet/85.3.645 · Zbl 0918.62065 · doi:10.1093/biomet/85.3.645
[5] Yao Q., Statist. Sinica 10 pp 751– (2000)
[6] DOI: 10.1093/biomet/77.2.415 · Zbl 0711.62035 · doi:10.1093/biomet/77.2.415
[7] DOI: 10.1016/j.spl.2006.05.018 · Zbl 1107.62109 · doi:10.1016/j.spl.2006.05.018
[8] DOI: 10.1016/j.csda.2005.08.001 · Zbl 1445.62082 · doi:10.1016/j.csda.2005.08.001
[9] DOI: 10.1214/009053607000000145 · Zbl 1126.62024 · doi:10.1214/009053607000000145
[10] DOI: 10.1016/0047-259X(88)90127-3 · Zbl 0664.62046 · doi:10.1016/0047-259X(88)90127-3
[11] DOI: 10.1016/0047-259X(89)90022-5 · Zbl 0667.62028 · doi:10.1016/0047-259X(89)90022-5
[12] DOI: 10.1111/1467-9868.00155 · Zbl 0909.62043 · doi:10.1111/1467-9868.00155
[13] DOI: 10.1214/aos/1074290329 · Zbl 1042.62044 · doi:10.1214/aos/1074290329
[14] DOI: 10.1214/aos/1176342558 · Zbl 0275.62033 · doi:10.1214/aos/1176342558
[15] DOI: 10.1007/BF00539840 · Zbl 0495.62046 · doi:10.1007/BF00539840
[16] DOI: 10.1214/aop/1176995945 · Zbl 0369.62028 · doi:10.1214/aop/1176995945
[17] DOI: 10.1007/BF02018047 · Zbl 0386.60006 · doi:10.1007/BF02018047
[18] DOI: 10.1214/aos/1024691356 · Zbl 0929.62052 · doi:10.1214/aos/1024691356
[19] Wang J., Statist. Sinica 19 pp 325– (2009)
[20] DOI: 10.1214/aos/1176325361 · Zbl 0827.62038 · doi:10.1214/aos/1176325361
[21] DOI: 10.1214/aos/1065705120 · Zbl 1042.62035 · doi:10.1214/aos/1065705120
[22] Xue L., Statist. Sinica 16 pp 1423– (2006)
[23] DOI: 10.1111/1467-9868.00149 · Zbl 0909.62035 · doi:10.1111/1467-9868.00149
[24] Bissantz N., J. Royal Statist. Soc. Ser. B 71 pp 28– (2009) · Zbl 1231.62060 · doi:10.1111/j.1467-9868.2008.00670.x
[25] de Boor C., A Practical Guide to Splines (2001) · Zbl 0987.65015
[26] DOI: 10.1007/978-3-642-57292-0 · Zbl 0963.62001 · doi:10.1007/978-3-642-57292-0
[27] Silverman B. W., Density Estimation for Statistics and Data Analysis (1986) · Zbl 0617.62042
[28] Fan J., Local Polynomial Modelling and Its Applications (1996) · Zbl 0873.62037
[29] Gantmacher F. R., Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme (1960)
[30] Zhang F., Matrix Theory. Basic Results and Techniques (1999)
[31] Yang L., J. Data Sci 6 pp 207– (2008)
[32] DOI: 10.1214/aos/1024691254 · Zbl 0935.62049 · doi:10.1214/aos/1024691254
[33] DOI: 10.2307/2669996 · Zbl 1072.62556 · doi:10.2307/2669996
[34] DOI: 10.1017/CBO9780511755453 · Zbl 1038.62042 · doi:10.1017/CBO9780511755453
[35] Leadbetter M. R., Extremes and Related Properties of Random Sequences and Processes (1983) · Zbl 0518.60021 · doi:10.1007/978-1-4612-5449-2
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.