Goodness of Fit Test of an Autocorrelated Time Series Cubic Smoothing Spline Model

Authors

  • Samuel Olorunfemi Adams Department of Statistics, University of Abuja, Abuja, Nigeria https://orcid.org/0000-0001-9881-1925
  • Davies Abiodun Obaromi Department of Statistics, Confluence University of Science and Technology, Osara, Kogi State, Nigeria
  • Alumbugu Auta Irinews Department of Mathematics and Statistics, Federal Polytechnic, Nasarawa, Nasarawa State, Nigeria

Keywords:

Cubic spline; goodness-of-fit test; Generalized Maximum Likelihood (GML); Generalized Cross-Validation (GCV); and Mallow CP criterion (MCP)

Abstract

We investigated the finite properties as well as the goodness of fit test for the cubic smoothing spline selection methods like the Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV) and Mallow CP criterion (MCP) estimators for time-series observation when there is the presence of Autocorrelation in the error term of the model. The Monte-Carlo study considered 1,000 replication with six sample sizes: 30; 60; 120; 240; 480 and 960, four degree of autocorrelations; 0.1; 0.3; 0.5; and 0.9 and three smoothing parameters; lambdaGML= 0.07271685, lambdaGCV= 0.005146929, lambdaMCP= 0.7095105. The cubic smoothing spline selection methods were also applied to a real-life dataset. The Predictive mean square error, R-square, and adjusted R-square criteria for assessing finite properties and goodness of fit among competing models discovered that the performance of the estimators is affected by changes in the sample sizes and autocorrelation levels of the simulated and real-life data set. The study concluded that the Generalized Cross-Validation estimator provides a better fit for Autocorrelated time series observation. It is recommended that the GCV works well at the four autocorrelation levels and provides the best fit for time-series observations at all sample sizes considered. This study can be applied to; non –parametric regression, non –parametric forecasting, spatial, survival and econometric observations.

Dimensions

Q.Kong, T. Siauw &A.M.Bayen, Python Programming and Numerical Methods: A Guide for Engineers and Scientist, Elsevier, ISBN: 978-0-12819549-9. (2020) https://doi.org/10.1016/C2018-0-04165-1

R. G. McClarren, Computational nuclear engineering and radiological science using python, https://doi.org/10.1016/C2016-0-03507-16 Elsevier (2018) 439. [3] J. R. Buchanan, “Cubic Spline Interpolation: MATH 375, Numerical Analysis”, Banach. Millersville.edu (2010).

J. Chen, “Testing goodness of fit of polynomial models via spline smoothing techniques”, Statistics and Probability Letters 19 (1994) 65. https://doi.org/10.1016/0167-7152(94)90070-1

N. Caouder & S. Huet, “Testing goodness-of-fit for nonlinear regression models with heterogeneous variances”, Computational Statistics and Data Analysis 23 (1998) 491. https://doi.org/10.1016/S0167-9473(96)00049-1

C.M. Crainiceanua & D. Ruppert, “Likelihood ratio tests for goodnessof-fit of a nonlinear regression model”, Journal of Multivariate Analysis 91 (2004) 35.

M. Tang, Y. Pei, W. Wong, & J. Li, “Goodness-of-fit tests for correlated paired binary data”, Statistical Method in Medical Research 21 (2012) 331. https://doi.org/10.1177/0962280210381176

S. T. Chen, L. Xiao, & M. Staicu, “A smoothing-based goodness-off it test of covariance for functional data”, Biometrics 75 (2019) 562. https://doi.org/10.1111/biom.13005

S. J. T. Hidalgo, C.W. Michael, M. E. Stephanie & M.R. Kosorok, “Goodness-of-fit test for nonparametric regression models: smoothing spline ANOVA models as example”, Computational Statistics and Data Analysis 122 (2018) 135. https://doi.org/10.1016/j.csda.2018.01.004

S. O. Adams & H.U. Yahaya, “Comparative study of GCV-MCP hybrid smoothing methods for predicting time series observations”, =American Journal of Theoretical and Applied Statistics 9 (2020) 219. https://doi:10.11648/j.ajtas.20200905.15

G.Wahba, Applications of Statistics in P. Krishnaiah Edition, A survey of somesmoothing problems and the method of generalized cross-validation for solving them, Northern Holland, Amsterdam (1977).

P. Craven & G. Wahba (1979), “Smoothing noisy data with spline functions”, Numerical Mathematics 31 (1979) 377.

G. Wahba, “A comparison of GCV and GML for choosing the smoothing parameters in the generalized spline smoothing problem”, The Annals of Statistics 4 (1985) 1378.

G. Wahba, “Optimal convergence properties of variable knot kernel and orthogonal series methods for density estimation”, Annals of Statistics 3 (1975) 15.

R. D. Cook &S.Weisberg, “Residuals and influence in regression”, Journal of the American Statistical Association 86 (1982) 328.

G. Wahba, ”Automatic smoothing of the log periodogram”, Journal of the American Statistical Association 75 (1980) 122.

B.W. Silverman, “Spline smoothing: the equivalent variable kernel method”, Annal of Statistics 12 (1984) 898.

D. Xiang & G. Wahba, “Approximate Smoothing Spline Methods for Large Data Sets in the Binary Case. Proceedings of the 1997”, ASA Joint Statistical Meetings, Biometrics Section (1998) 94.

G. Wahba, “Spline models for observational data”, CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia: SIAM 59 (1990) 1.

P. J. Diggle & M.F. Hutchinson, “On spline smoothing with autocorrelated errors”, Australian Journal of Statistics 31 (1998) 166.

C. L. Mallows, “Some comments on Cp”, Technometrics 15 (1973) 661.

R Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/ (2020).

C. Daniel, “One at a time plans” Journal of American Statistical Association, 68 (1973) 353.

S.O. Adams & R. A. Ipinyomi, “A new smoothing method for time series data in the presence of autocorrelated error”, Asian Journal of Probability and Statistics (AJPAS), 04 (2019) 1. https://doi.org/10.9734/ajpas/2019/v4i430121

Central Bank of Nigeria Statistical Bulleting, (2019) Edition.

C. Yanrong, Z.W. Tracy, L. Haiqun, & Y. Yan, “Penalized spline estimation for functional coefficient regression models”, Computational Statistics and Data Analysis 54 (2010) 891.

M. A. Lukas, F .R. De Hoog&R.S.Anderssen, “Practical Use of Robust GCV and Modified GCV for Spline Smoothing”, Computational Statistics 31 (2016) 269.

A. R. Devi, I .N. Budiantara & V. Vita-Ratnasari, “ Unbiased risk and cross-validation method for selecting optimal knots in multivariable nonparametric regression spline truncated (case study: the unemployment rate in central java, Indonesia, 2015)”, AIP Conference Proceedings, (2018).

Published

2021-08-29

How to Cite

Goodness of Fit Test of an Autocorrelated Time Series Cubic Smoothing Spline Model. (2021). Journal of the Nigerian Society of Physical Sciences, 3(3), 191-200. https://doi.org/10.46481/jnsps.2021.265

Issue

Section

Original Research

How to Cite

Goodness of Fit Test of an Autocorrelated Time Series Cubic Smoothing Spline Model. (2021). Journal of the Nigerian Society of Physical Sciences, 3(3), 191-200. https://doi.org/10.46481/jnsps.2021.265