Additive models with autoregressive symmetric errors based on penalized regression splines. (English) Zbl 1505.62303
Summary: In this paper additive models with \(p\)-order autoregressive conditional symmetric errors based on penalized regression splines are proposed for modeling trend and seasonality in time series. The aim with this kind of approach is try to model the autocorrelation and seasonality properly to assess the existence of a significant trend. A backfitting iterative process jointly with a quasi-Newton algorithm are developed for estimating the additive components, the dispersion parameter and the autocorrelation coefficients. The effective degrees of freedom concerning the fitting are derived from an appropriate smoother. Inferential results and selection model procedures are proposed as well as some diagnostic methods, such as residual analysis based on the conditional quantile residual and sensitivity studies based on the local influence approach. Simulations studies are performed to assess the large sample behavior of the maximum penalized likelihood estimators. Finally, the methodology is applied for modeling the daily average temperature of San Francisco city from January 1995 to April 2020.
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
| 62G08 | Nonparametric regression and quantile regression |
| 62J05 | Linear regression; mixed models |
| 62J20 | Diagnostics, and linear inference and regression |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62P12 | Applications of statistics to environmental and related topics |
Keywords:
cubic splines; cyclic splines; daily temperature; model checking; penalized likelihood; Student-\(t\) models; robust estimationReferences:
| [1] | Akaike, H.; Petrov, BN; Csaki, F., Information theory and an extension of the maximum likelihood principle, International symposium on information theory, 267-281 (1973), Hungary: Akademiai Kiado Budapest, Hungary · Zbl 0283.62006 |
| [2] | Barros, M.; Paula, GA, Discussion of Birnbaum-Saunders distributions: a review of models, analysis and applications, Appl Stoch Models Bus Ind, 35, 96-99 (2019) · Zbl 1428.62073 · doi:10.1002/asmb.2408 |
| [3] | Byrd, RH; Lu, P.; Nocedal, J.; Zhu, C., A limited memory algorithm for bound constrained optimization, SIAM J Sci Comput, 16, 1190-1208 (1995) · Zbl 0836.65080 · doi:10.1137/0916069 |
| [4] | Cao, CZ; Lin, JG; Zhu, LX, Heteroscedasticity and/or autocorrelation diagnostics in nonlinear models with AR(1) and symmetrical errors, Stat Pap, 51, 813-836 (2010) · Zbl 1247.62218 · doi:10.1007/s00362-008-0171-y |
| [5] | Cook, RD, Assessment of local influence, J R Stat Soc B, 48, 133-169 (1986) · Zbl 0608.62041 |
| [6] | Cook, RD; Weisberg, S., Residuals and influence in regression (1982), London: Chapman and Hall, London · Zbl 0564.62054 |
| [7] | Cleveland, WS; McRae, JE; Terpenning, I., STL: a seasonal-trend decomposition, J Off Stat, 6, 3-73 (1990) |
| [8] | Cysneiros, FJA; Paula, GA, Restricted methods in symmetrical linear regression models, Comput Stat Data Anal, 49, 689-708 (2005) · Zbl 1429.62285 · doi:10.1016/j.csda.2004.06.001 |
| [9] | Davidon, WC, Variable metric method for minimization, SIAM J Optim, 1, 1-17 (1991) · Zbl 0752.90062 · doi:10.1137/0801001 |
| [10] | Dunn, PK; Smyth, GK, Randomized quantile residuals, J Comput and Graph Stat, 5, 236-244 (1996) |
| [11] | Efron, B.; Hinkley, DV, Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information, Biometrika, 65, 457-487 (1978) · Zbl 0401.62002 · doi:10.1093/biomet/65.3.457 |
| [12] | Eilers, PH; Marx, BD, Flexible smoothing with B-splines and penalties, Stat Sci, 11, 89-102 (1996) · Zbl 0955.62562 · doi:10.1214/ss/1038425655 |
| [13] | Fang, KT; Kotz, S.; Ng, KW, Symmetric multivariate and related distributions (1990), London: Chapman and Hall, London · Zbl 0699.62048 · doi:10.1007/978-1-4899-2937-2 |
| [14] | Fox, J., Applied regression analysis and generalized linear models (2015), London: Sage Publications, London |
| [15] | Green, PJ; Silverman, BW, Nonparametric regression and generalized linear models: a roughness penalty approach (1994), London: Chapman and Hall/CRC, London · Zbl 0832.62032 · doi:10.1007/978-1-4899-4473-3 |
| [16] | Hastie, TJ; Tibshirani, RJ, Generalized additive models (1990), London: Chapman and Hall/CRC, London · Zbl 0747.62061 |
| [17] | Huang, L.; Jiang, H.; Wang, H., A novel partial-linear single-index model for time series data, Comput Stat Data Anal, 134, 110-122 (2019) · Zbl 1507.62077 · doi:10.1016/j.csda.2018.12.012 |
| [18] | Huang, L.; Xia, Y.; Qin, X., Estimation of semivarying coefficient time series models with ARMA errors, Ann Stat, 44, 1618-1660 (2016) · Zbl 1349.62269 · doi:10.1214/15-AOS1392 |
| [19] | Ibacache-Pulgar, G.; Paula, GA; Cysneiros, FJA, Semiparametric additive models under symmetric distributions, TEST, 22, 103-121 (2013) · Zbl 1302.62094 · doi:10.1007/s11749-012-0309-z |
| [20] | Judge, GG; Griffiths, WE; Hill, RC; Lutkepohl, H.; Lee, TC, The theory and practice of econometrics (1985), New York: Wiley, New York · Zbl 0731.62155 |
| [21] | Kissock JK (1999) UD EPA Average Daily Temperature Archive, http://academic.udayton.edu/kissock/http/Weather/default.htm. Accessed 20 Feb 2021 |
| [22] | Lancaster, P.; Salkauskas, K., An introduction curve and surface fitting (1986), London: Academic Press, London · Zbl 0649.65012 |
| [23] | Lee, SY; Xu, L., Influence analyses of nonlinear mixed-effects models, Comput Stat Data Anal, 45, 321-341 (2004) · Zbl 1429.62280 · doi:10.1016/S0167-9473(02)00303-1 |
| [24] | Liu, JM; Chen, R.; Yao, Q., Nonparametric transfer function models, J Econom, 157, 151-164 (2010) · Zbl 1431.62400 · doi:10.1016/j.jeconom.2009.10.029 |
| [25] | Liu, S., On diagnostics in conditionally heteroscedastic time series models under elliptical distributions, J Appl Probab, 41A, 393-405 (2004) · Zbl 1049.62100 · doi:10.1239/jap/1082552214 |
| [26] | Lucas, A., Robustness of the student-t based M-estimator, Commun Stat Theory Methods, 26, 1165-1182 (1997) · Zbl 0920.62041 · doi:10.1080/03610929708831974 |
| [27] | Mittelhammer, RC; Judge, GG; Miller, DJ, Econometric foundations (2000), New York: Cambridge University Press, New York · Zbl 0961.62110 |
| [28] | Paula, GA; Medeiros, MJ; Vilca-Labra, FE, Influence diagnostics for linear models with first-order autoregressive elliptical errors, Stat Probab Lett, 79, 339-346 (2009) · Zbl 1155.62052 · doi:10.1016/j.spl.2008.08.017 |
| [29] | Poon, WY; Poon, YS, Conformal normal curvature and assessment of local influence, J R Stat Soc B, 61, 51-61 (1999) · Zbl 0913.62062 · doi:10.1111/1467-9868.00162 |
| [30] | R Core Team (2020) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.Rproject.org. Accessed 10 Jan 2021 |
| [31] | Relvas, CEM; Paula, GA, Partially linear models with first-order autoregressive symmetric errors, Stat Pap, 57, 795-825 (2016) · Zbl 1373.62391 · doi:10.1007/s00362-015-0680-4 |
| [32] | Schwarz, GE, Estimating the dimension of a model, Ann Stat, 6, 461-464 (1978) · Zbl 0379.62005 · doi:10.1214/aos/1176344136 |
| [33] | Vanegas, LH; Paula, GA, An extension of log-symmetric regression models: R codes and applications, J Stat Comp Simul, 86, 1709-1735 (2016) · Zbl 1510.62191 · doi:10.1080/00949655.2015.1081689 |
| [34] | Wise, J., The autocorrelation function and the spectral density function, Biometrika, 42, 151-159 (1955) · Zbl 0065.12902 · doi:10.1093/biomet/42.1-2.151 |
| [35] | Wood, SN, Generalized additive models: an introduction with R (2017), London: Chapman and Hall/CRC, London · Zbl 1368.62004 · doi:10.1201/9781315370279 |
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