An F-type test for detecting departure from monotonicity in a functional linear model

References

• Ait-Saïdi, A., Ferraty, F., Kassa, R., and Vieu, P. (2008), ‘Cross-validated Estimations in the Single-functional Index Model’, Statistics, 42, 475–494. doi: 10.1080/02331880801980377
   Web of Science ®Google Scholar
• Akakpo, N., Balabdaoui, F., and Durot, C. (2014), ‘Testing Monotonicity via Local Least Concave Majorants’, Bernoulli, 20, 514–544. doi: 10.3150/12-BEJ496
   Web of Science ®Google Scholar
• Aneiros-Pérez, G., and Vieu, P. (2006), ‘Semi-functional Partial Linear Regression’, Statistics & Probability Letters, 76, 1102–1110. doi: 10.1016/j.spl.2005.12.007
   Web of Science ®Google Scholar
• Aubin, J.P. (2000), Applied Functional Analysis (2nd ed.), New York: Wiley.
   Google Scholar
• Bigot, M., and Gadat, S. (2010), ‘Smoothing under Diffeomorphic Constraints with Homeomorphic Splines’, SIAM Journal on Numerical Analysis, 48, 224–243. doi: 10.1137/080727555
   Web of Science ®Google Scholar
• Birke, M., and Neumeyer, N. (2013), ‘Testing Monotonicity of Regression Functions – an Empirical Process Approach’, Scandinavian Journal of Statistics, 40, 438–454. doi: 10.1111/j.1467-9469.2012.00820.x
   Web of Science ®Google Scholar
• Bongiorno, E.G., Salinelli, E., Goia, A., and Vieu, P. (2014), Contributions in Infinite-Dimensional Statistics and Related Topics, Bologna: Società Editrice Esculapio.
   Google Scholar
• Bowman, A., Jones, M., and Gijbels, I. (1998), ‘Testing Monotonicity of Regression’, Journal of Computational and Graphical Statistics, 7, 489–500.
   Web of Science ®Google Scholar
• Buja, A., Hastie, T., and Tibshirani, R. (1989), ‘Linear Smoothers and Additive Models (with Discussion)’, Annals of Statistics, 17, 453–510. doi: 10.1214/aos/1176347115
   Web of Science ®Google Scholar
• Burba, F., Ferraty, F., and Vieu, P. (2009), ‘k-Nearest Neighbour Method in Functional Nonparametric Regression’, Journal of Nonparametric Statistics, 21, 453–469. doi: 10.1080/10485250802668909
   Web of Science ®Google Scholar
• Canty, A., and Ripley, B. (2015), boot: Bootstrap R (S-Plus) Functions, R package version 1.3-17.
   Google Scholar
• Cardot, H., Ferraty, F., and Sarda, P. (2003), ‘Spline Estimators for the Functional Linear Model’, Statistica Sinica, 13, 571–591.
   Web of Science ®Google Scholar
• Carroll, R.J., Delaigle, A., and Hall, P. (2011), ‘Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error’, Journal of the American Statistical Association, 106, 191–202. doi: 10.1198/jasa.2011.tm10355
   PubMed Web of Science ®Google Scholar
• Chen, D., Hall, P., and Müller, H.G. (2011), ‘Single and Multiple Index Functional Regression Models with Nonparametric Link’, The Annals of Statistics, 39, 1720–1747. doi: 10.1214/11-AOS882
   Web of Science ®Google Scholar
• Crambes, C., Kneip, A., and Sarda, P. (2009), ‘Smoothing Splines Estimators for Functional Linear Regression’, The Annals of Statistics, 37, 35–72. doi: 10.1214/07-AOS563
   Web of Science ®Google Scholar
• Craven, P., and Wahba, W. (1979), ‘Smoothing Noisy Data with Spline Functions: Estimating the Correct Degree of Smoothing by the Method of Generalized Cross-Validation’, Numerische Mathematik, 31, 377–403. doi: 10.1007/BF01404567
   Web of Science ®Google Scholar
• Cuevas, A. (2014), ‘A Partial Overview of the Theory of Statistics with Functional Data’, Journal of Statistical Planning and Inference, 147, 1–23. doi: 10.1016/j.jspi.2013.04.002
   Web of Science ®Google Scholar
• de Leeuw, J., Hornik, K., and Mair, P. (2009), ‘Isotone Optimization in R: Pool-Adjacent-Violators Algorithm (PAVA) and Active Set Methods’, Journal of Statistical Software, 32, 1–24. doi: 10.18637/jss.v032.i05
   Web of Science ®Google Scholar
• Delaigle, A., and Hall, P. (2012), ‘Methodology and Theory for Partial Least Squares Applied to Functional Data’, The Annals of Statistics, 40, 322–352. doi: 10.1214/11-AOS958
   Web of Science ®Google Scholar
• Ferraty, F. (2011), Recent Advances in Functional Data Analysis and Related Topics, Heidelberg: Springer.
   Google Scholar
• Ferraty, F., Goia, A., Salinelli, E., and Vieu, P. (2013), ‘Functional Projection Pursuit Regression’, Test, 22, 293–320. doi: 10.1007/s11749-012-0306-2
   Web of Science ®Google Scholar
• Ferraty, F., González-Manteiga, W., Martínez-Calvo, A., and Vieu, P. (2012), ‘Presmoothing in Functional Linear Regression’, Statistica Sinica, 22, 69–94. doi: 10.5705/ss.2010.085
   Web of Science ®Google Scholar
• Ferraty, F., Keilegom, I.V., and Vieu, P. (2010), ‘On the Validity of the Bootstrap in Nonparametric Functional Regression’, Scandinavian Journal of Statistics, 37, 286–306. doi: 10.1111/j.1467-9469.2009.00662.x
   Web of Science ®Google Scholar
• Ferraty, F., Park, J., and Vieu, P. (2011), ‘Estimation of a Functional Single Index Model’, in Recent Advances in Functional Data Analysis and Related Topics, F. Ferraty, ed., Springer: Heidelberg, pp. 111–116.
   Google Scholar
• Ferraty, F., and Vieu, P. (2006), Nonparametric Functional Data Analysis: Theory and Practice (1st ed.), New York: Springer.
   Google Scholar
• García-Portugués, E., González-Manteiga, W., and Febrero-Bande, M. (2014), ‘A Goodness-of-Fit Test for the Functional Linear Model with Scalar Response’, Journal of Computational and Graphical Statistics, 23, 761–778. doi: 10.1080/10618600.2013.812519
   Web of Science ®Google Scholar
• Goia, A., and Vieu, P. (2015), ‘A Partitioned Single Functional Index Model’, Computational Statistics, 30, 673–692. doi: 10.1007/s00180-014-0530-1
   Web of Science ®Google Scholar
• González-Manteiga, W., and Martínez-Calvo, A. (2011), ‘Bootstrap in Functional Linear Regression’, Journal of Statistical Planning and Inference, 141, 453–461. doi: 10.1016/j.jspi.2010.06.027
   Web of Science ®Google Scholar
• Hall, P., and Heckman, N.E. (2000), ‘Testing for Monotonicity of a Regression Mean by Calibrating for Linear Functions’, The Annals of Statistics, 28, 20–39. doi: 10.1214/aos/1016120363
   Web of Science ®Google Scholar
• Hansen, P.C. (1998), Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (Vol. 4), Philadelphia: SIAM.
   Google Scholar
• Hardle, W., and Marron, J. (1991), ‘Bootstrap Simultaneous Error Bars for Nonparametric Regression’, The Annals of Statistics, 19, 778–796. doi: 10.1214/aos/1176348120
   Web of Science ®Google Scholar
• D.A. Harville (1977), ‘Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems’, Journal of the American Statistical Association, 72, 320–338. doi: 10.1080/01621459.1977.10480998
   Web of Science ®Google Scholar
• Horváth, L., and Kokoszka, P. (2012), Inference for Functional Data with Applications (1st ed.), Berlin: Springer.
   Google Scholar
• Hsing, T., and Eubank, R. (2015), Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators, Chichester: John Wiley & Sons.
   Google Scholar
• Kauermann, G., Claeskens, G., and Opsomer, J.D. (2009), ‘Bootstrapping for Penalized Spline Regression’, Journal of Computational and Graphical Statistics, 18, 126–146. doi: 10.1198/jcgs.2009.0008
   Web of Science ®Google Scholar
• Kelly, C., and Rice, J. (1990), ‘Monotone Smoothing with Application to Dose-response Curves and the Assessment of Synergism’, Biometrics, 46, 1071–1085. doi: 10.2307/2532449
   PubMed Web of Science ®Google Scholar
• Kudraszow, N.L., and Vieu, P. (2013), ‘Uniform Consistency of kNN Regressors for Functional Variables’, Statistics & Probability Letters, 83, 1863–1870. doi: 10.1016/j.spl.2013.04.017
   Web of Science ®Google Scholar
• Lian, H. (2011), ‘Functional Partial Linear Model’, Journal of Nonparametric Statistics, 23, 115–128. doi: 10.1080/10485252.2010.500385
   Web of Science ®Google Scholar
• Mair, P., de Leeuw, J.D., and Hornik, K. (2015), isotone: Active Set and Generalized PAVA for Isotone Optimization, R package version 1.1-0, http://CRAN.R-project.org/package=isotone.
   Google Scholar
• Mammen, E., and Thomas-Agnan, C. (1999), ‘Smoothing Splines and Shape Restrictions’, Scandinavian Journal of Statistics, 26, 239–252. doi: 10.1111/1467-9469.00147
   Web of Science ®Google Scholar
• Mammen, E., Marron, J.S., Turlach, B.A., and Wand, M.P. (2001), ‘A General Projection Framework for Constrained Smoothing’, Statistical Science, 16, 232–248. doi: 10.1214/ss/1009213727
   Web of Science ®Google Scholar
• Martínez Calvo, A. (2013), ‘Estimates and Bootstrap Calibration for Functional Regression with Scalar Response’, Ph.D. dissertation, Universidade de Santiago de Compostela.
   Google Scholar
• Montoya, E.L., and Meiring, W. (2013), ‘On the Relative Efficiency of a Monotone Parameter Curve Estimator in a Functional Nonlinear Model’, Statistics and Computing, 23, 425–436. doi: 10.1007/s11222-012-9320-1
   Web of Science ®Google Scholar
• Pinheiro, J.C., Bates, D.M. (2002), Mixed Effects Models in S and S-Plus (1st ed.), New York: Springer.
   Google Scholar
• Pinheiro, J.C., Bates, D., DebRoy, S., Sarkar, D., and R Core Team. (2015), nlme: Linear and Nonlinear Mixed Effects Models, R package version 3.1-120 http://CRAN.R-project.org/package=nlme.
   Google Scholar
• R Core Team (2014), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, http://www.R-project.org/.
   Google Scholar
• Rachdi, M., Laksaci, A., Demongeot, J., Abdali, A., and Madani, F. (2014), ‘Theoretical and Practical Aspects of the Quadratic Error in the Local Linear Estimation of the Conditional Density for Functional Data’, Computational Statistics & Data Analysis, 73, 53–68. doi: 10.1016/j.csda.2013.11.011
   Web of Science ®Google Scholar
• Ramsay, J.O. (1988), ‘Monotone Regression Splines in Action’, Statistical Science, 3, 425–441. doi: 10.1214/ss/1177012761
   Google Scholar
• Ramsay, J.O. (1998), ‘Estimating Smooth Monotone Functions’, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60, 365–375. doi: 10.1111/1467-9868.00130
   Web of Science ®Google Scholar
• Ramsay, J.O., and Silverman, B.W. (2002), Applied Functional Data Analysis (1st ed.), New York: Springer.
   Google Scholar
• Ramsay, J.O., and Silverman, B.W. (2005), Functional Data Analysis (2nd ed.), New York: Springer.
   Google Scholar
• Ramsay, J.O., Wickham, H., Graves, S., and Hooker, G. (2014), fda: Functional Data Analysis, R package version 2.4.3, http://CRAN.R-project.org/package=fda.
   Google Scholar
• Reiss, P.T., and Ogden, R.T. (2007), ‘Functional Principal Component Regression and Functional Partial Least Squares’, Journal of the American Statistical Association, 102, 984–996. doi: 10.1198/016214507000000527
   Web of Science ®Google Scholar
• Robinson, G.K. (1991), ‘That BLUP Is a Good Thing: The Estimation of Random Effects’, Statistical Science, 6, 15–32. doi: 10.1214/ss/1177011926
   Google Scholar
• Ruppert, D., Wand, M.P., and Carroll, R.J. (2003), Semiparametric Regression (1st ed.), New York: Cambridge University Press.
   Google Scholar
• Schipper, M., Taylor, J.M.G., and Lin, X. (2007), ‘Bayesian Generalized Monotonic Functional Mixed Models for the Effects of Radiation Dose Histograms on Normal Tissue Complications’, Statistics in Medicine, 26, 4643–4656. doi: 10.1002/sim.2887
   PubMed Web of Science ®Google Scholar
• Schipper, M., Taylor, J.M.G., and Lin, X. (2008), ‘Generalized Monotonic Functional Mixed Models with Application to Modelling Normal Tissue Complications’, Journal of the Royal Statistical Society: Series C (Applied Statistics), 57, 149–163. doi: 10.1111/j.1467-9876.2007.00606.x
   Web of Science ®Google Scholar
• Wahba, G. (1982), ‘Constrained Regularization for Ill Posed Linear Operator Equations, with Applications in Meteorology and Medicine’, in Statistical Decision Theory and Related Topics, III (Vol. 2), eds. S.S. Gupta and J.O. Berger, New York: Academic Press, pp. 383–418.
   Google Scholar
• Wand, M.P., and Ormerod, J.T. (2008), ‘On Semiparametric Regression with O'Sullivan Penalised Splines’, Australian & New Zealand Journal of Statistics, 50, 179–198. doi: 10.1111/j.1467-842X.2008.00507.x
   Web of Science ®Google Scholar
• Wang, G., Feng, X., and Chen, M. (2015), ‘Functional Partial Linear Single-index Model’, Scandinavian Journal of Statistics, 43, 261–274. doi: 10.1111/sjos.12178
   Web of Science ®Google Scholar
• Zhang, J. (2004), ‘A Simple and Efficient Monotone Smoother Using Smoothing Splines’, Journal of Nonparametric Statistics, 16, 779–796. doi: 10.1080/10485250410001681167
   Web of Science ®Google Scholar