Top

Published in:

2019 | OriginalPaper | Chapter

Novel Test for the Equality of Continuous Curves with Homoscedastic or Heteroscedastic Measurement Errors

Authors : Zhongfa Zhang, Yarong Yang, Jiayang Sun

Published in: Pharmaceutical Statistics

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Testing equality of two curves occurs often in functional data analysis. In this paper, we develop procedures for testing if two curves measured with either homoscedastic or heteroscedastic errors are equal. The method is applicable to a general class of curves. Compared with existing tests, ours does not require repeated measurements to obtain the variances at each of the explanatory values. Instead, our test calculates the overall variances by pooling all of the data points. The null distribution of the test statistic is derived and an approximation formula to calculate the p value is developed when the heteroscedastic variances are either known or unknown. Simulations are conducted to show that this procedure works well in the finite sample situation. Comparisons with other test procedures are made based on simulated data sets. Applications to our motivating example from an environmental study will be illustrated. An R package was created for ease of general applications.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Project Data Sphere and the Applications of Historical Patient Level Clinical Trial Data in Oncology Drug Development

next chapter Quality Control Metrics for Extraction-Free Targeted RNA-Seq Under a Compositional Framework

Available only for authorised users

ATSDR: The nature and extent of lead poisoning in children in the united states: a report to congress. Technical report, Agency for Toxic Substances and Disease Registry, Atlanta: US Department of Health and Human Services, Public Health Service(1988)

Besse, P., Ramsay, J.O.: Principle components analysis of sampled functions. Psychometrika 51(2), 285–311 (1986). https://doi.org/10.1007/bf02293986MathSciNetCrossRefMATH

Chakravarti, I.M., Laha, R.G., Roy, J.: Handbook of Methods of Applied Statistics, Vol. I. John Wiley and Sons (1967)

Clive, R.: Loader: Bandwidth selection: classical or plug-in? Ann. Statist. 27(2), 415–438 (1999)MathSciNetMATH

Cleveland, W., Devin, S.: Locally weighted regression: an approach to regression analysis by local fitting. J. Am. Stat. Assoc. 83(403), 596–610 (1988). https://doi.org/10.1080/01621459.1988.10478639CrossRefMATH

Dai, J., Sperlich, S.: Simple and effective boundary correction for kernel densities and regression with an application to the world income and Engel curve estimation. Comput. Stat. Data Anal. Elsevier 54(11), 2487–2497 (2010)MathSciNetCrossRef

Fan, J., Lin, S.: Test of significance when data are curves. J. Am. Stat. Assoc. 93, 1007–1021 (1998). https://doi.org/10.1080/01621459.1998.10473763MathSciNetCrossRefMATH

Hotelling, H.: Tubes and spheres in n-spaces, and a class of statistical problems. Am. J. Math. 61, 440–460 (1939)MathSciNetCrossRef

James, G., Hastie, T.: Functional linear discriminant analysis for irregularly sampled curves. J. R. Stat. Soc. Ser. B 63(3), 533–550 (2001). https://doi.org/10.1111/1467-9868.00297MathSciNetCrossRefMATH

10.

James, W., Stein, C.: Estimation with quadratic loss. In: Proceedings of Fourth Berkeley Symposium on Mathematical Statistics and Probability Theory University of California Press, pp 361–380 (1961)

11.

Johansen, S., Johnstone, I.: Hotelling’s theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis. Ann. Stat. 18, 652–684 (1990)MathSciNetCrossRef

12.

Kitska, D.J.: Simultaneous inference for functional linear models. Ph.D. thesis, Case Western Reserve University (2005)

13.

Knowles, M., Siegmund, D.: On hotelling’s approach to testing for a nonlinear parameter in regression. Int. Stat. Rev. 57(3), 205–220 (1989). https://doi.org/10.2307/1403794CrossRefMATH

14.

Leurgans, S.E., Moyeed, R.A., Silverman, B.W.: Canonical correlation analysis when the data are curves. J. R. Stat. Soc. Ser. B 55(3), 725–740 (1993)MathSciNetMATH

15.

Loader, C.: Local Regression and Likelihood. Springer, New York (1999)

16.

Naiman, D.Q.: Simultaneous confidence bounds in multiple regression using predictor variable constraints. J. Am. Stat. Assoc. 82, 214–219 (1987). https://doi.org/10.2307/2289156MathSciNetCrossRefMATH

17.

Naiman, D.Q.: On volumes of tubular neighborhoods of spherical polyhedra and statistical inference. Ann. Stat. 18(2), 685–716 (1990)MathSciNetCrossRef

18.

Parzen, E.: An approach to time series analysis. Ann. Math. Stat. 32(4), 951–989 (1961)CrossRef

19.

Ramsay, J., Daizell, C.: Some tools for functional data analysis. J. R. Stat. Soc. Ser. B 53(3), 539–572 (1991). https://doi.org/10.2307/2345586MathSciNetCrossRef

20.

Robbins, N., Zhang, Z., Sun, J., Ketterer, M., Lalumandier, J., Shulze, R.: Childhood lead exposure and uptake in teeth in the Cleveland area during the era of leaded gasoline. Sci. Total Environ. 408(19), 4118–27 (2010). https://doi.org/10.1016/j.scitotenv.2010.04.060CrossRef

21.

Shapiro, S., Wilk, M.B.: An analysis of variance test for normality (complete samples). Biometrika 52(3 & 4), 591–611 (1965)MathSciNetCrossRef

22.

Sun, J., Loader, C.: Simultaneous confidence bands for linear regression and smoothing. Ann. Stat. 22, 1328–1345 (1994). https://doi.org/10.1214/aos/1176325631MathSciNetCrossRefMATH

23.

Sun, J.: Tail probabilities of the maxima of gaussian random fields. Ann. Probab. 21(1), 34–71 (1993). https://doi.org/10.1214/aop/1176989393MathSciNetCrossRefMATH

24.

Sun, J.: Multiple comparisons for a large number of parameters. Biom. J. 43, 627–643 (2001). https://doi.org/10.1002/1521-4036(200109)43:53.3.CO;2-6MathSciNetCrossRefMATH

25.

Weyl, H.: On the volume of tubes. Am. J. Math. 61(2), 461–472 (1939)MathSciNetCrossRef

26.

Wang, J.-L., Chiou, J.-M., Mller1, H.-G.: Functional data analysis. Ann. Rev. Stat. Appl. 3, 257–295 (2016). https://doi.org/10.1146/annurev-statistics-041715-033624CrossRef

27.

Xintaras, C.: Impact of lead-contaminated soil on public health (1992). http://www.cdc.gov/search.do. (Technical report, Agency for Toxic Substances and Disease Registry)

Title: Novel Test for the Equality of Continuous Curves with Homoscedastic or Heteroscedastic Measurement Errors
Authors: Zhongfa Zhang
Yarong Yang
Jiayang Sun
Publisher: Springer International Publishing
Book: Pharmaceutical Statistics
Print ISBN: 978-3-319-67385-1

Electronic ISBN: 978-3-319-67386-8

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-319-67386-8_20

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner