An alternative circular smoothing method to nonparametric estimation of periodic functions. (English) Zbl 1514.62946
Summary: This article provides alternative circular smoothing methods in nonparametric estimation of periodic functions. By treating the data as ‘circular’, we solve the “boundary issue” in the nonparametric estimation treating the data as ‘linear’. By redefining the distance metric and signed distance, we modify many estimators used in the situations involving periodic patterns. In the perspective of ‘nonparametric estimation of periodic functions’, we present the examples in nonparametric estimation of (1) a periodic function, (2) multiple periodic functions, (3) an evolving function, (4) a periodically varying-coefficient model and (5) a generalized linear model with periodically varying coefficient. In the perspective of ‘circular statistics’, we provide alternative approaches to calculate the weighted average and evaluate the ‘linear/circular-linear/circular’ association and regression. Simulation studies and an empirical study of electricity price index have been conducted to illustrate and compare our methods with other methods in the literature.
MSC:
| 62-XX | Statistics |
Keywords:
alternative circular smoothing; boundary issue; nonparametric estimation; periodic function; linear/circular-linear/circular association and regressionReferences:
| [1] | Y. Ait-Sahalia and J. Duarte, Nonparametric option pricing under shape restrictions, J. Econ. 116 (2003), pp. 9-47. doi: 10.1016/S0304-4076(03)00102-7 · Zbl 1016.62121 · doi:10.1016/S0304-4076(03)00102-7 |
| [2] | F.E. Benth, J.S. Benth, and S. Koekebakker, Stochastic Modelling of Electricity and Related Markets, World Scientific Publishing Co., Singapore, 2008. · Zbl 1143.91002 · doi:10.1142/6811 |
| [3] | F.E. Benth, R. Kiesel, and A. Nazarova, A critical empirical study of three electricity spot price models, Energy Econ. 34 (2012), pp. 1589-1616. doi: 10.1016/j.eneco.2011.11.012 · doi:10.1016/j.eneco.2011.11.012 |
| [4] | F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 81 (1973), pp. 637-654. doi: 10.1086/260062 · Zbl 1092.91524 · doi:10.1086/260062 |
| [5] | P. Burman, Smoothing sparse contingency tables, Indian J. Stat. 49 (1987), pp. 24-36. · Zbl 0639.62050 |
| [6] | A. Cartea and M. Figueroa, Pricing in electricity markets: A mean reverting jump diffusion model with seasonality, Appl. Math. Financ. 12 (2005), pp. 313-335. doi: 10.1080/13504860500117503 · Zbl 1134.91526 |
| [7] | S.X. Chen and Z. Xu, On smoothing estimation for seasonal time series with long cycles, Stat. Interface 6 (2013), pp. 435-447. doi: 10.4310/SII.2013.v6.n4.a3 · Zbl 1326.62185 · doi:10.4310/SII.2013.v6.n4.a3 |
| [8] | S.X. Chen and Z. Xu, On implied volatility for options some reasons to smile and more to correct, J. Econ. 179 (2014), pp. 1-15. doi: 10.1016/j.jeconom.2013.10.007 · Zbl 1293.91193 · doi:10.1016/j.jeconom.2013.10.007 |
| [9] | C. Erlwein, F.E. Benth, and R. Mamon, HMM filtering and parameter estimation of an electricity spot price model, Energy Econ. 32 (2010), pp. 1034-1043. doi: 10.1016/j.eneco.2010.01.005 · doi:10.1016/j.eneco.2010.01.005 |
| [10] | J. Fan and W. Zhang, Statistical estimation in varying coefficient models, Ann. Stat. 27 (1999), pp. 1491-1518. doi: 10.1214/aos/1017939139 · Zbl 0977.62039 · doi:10.1214/aos/1017939139 |
| [11] | J. Fan and W. Zhang, Statistical methods with varying coefficient models, Stat. Interface 1 (2008), pp. 179-195. doi: 10.4310/SII.2008.v1.n1.a15 · Zbl 1230.62031 · doi:10.4310/SII.2008.v1.n1.a15 |
| [12] | J.J. Fernandez-Duran, Circular distributions based on nonnegative trigonometric sums, Biometrics 60 (2004), pp. 499-503. doi: 10.1111/j.0006-341X.2004.00195.x · Zbl 1274.62352 · doi:10.1111/j.0006-341X.2004.00195.x |
| [13] | J.J. Fernandez-Duran, Models for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums., Biometrics 63 (2007), pp. 579-585. doi: 10.1111/j.1541-0420.2006.00716.x · Zbl 1134.62030 · doi:10.1111/j.1541-0420.2006.00716.x |
| [14] | N.I. Fisher, Statistical Analysis of Circular Data, Cambridge University Press, Cambridge, 1993. · Zbl 0788.62047 · doi:10.1017/CBO9780511564345 |
| [15] | M.G. Genton and P. Hall, Statistical inference for evolving periodic functions, J. R. Stat. Soc.: Ser. B 69 (2007), pp. 643-657. doi: 10.1111/j.1467-9868.2007.00604.x · Zbl 07555369 · doi:10.1111/j.1467-9868.2007.00604.x |
| [16] | P. Hall, Nonparametric methods for estimating periodic functions, with applications in astronomy, in COMPSTAT 2008: Proceedings in Computational Statistics, P. Brito, ed., Physika-Verlag, Heidelberg, 2008, pp. 3-18. · Zbl 1153.85001 |
| [17] | P. Hall, J. Reimann, and J. Rice, Nonparametric estimation of a periodic function, Biometrika 87 (2000), pp. 545-557. doi: 10.1093/biomet/87.3.545 · Zbl 0956.62031 · doi:10.1093/biomet/87.3.545 |
| [18] | P. Hall, G.S. Watson, and J. Cabrera, Kernel density estimation with spherical data, Biometrika 74 (1987), pp. 751-762. doi: 10.1093/biomet/74.4.751 · Zbl 0632.62033 · doi:10.1093/biomet/74.4.751 |
| [19] | P. Hall and J. Yin, Nonparametric methods for deconvolving multiperiodic functions, J. R. Stat. Soc.: Ser. B 65 (2003), pp. 869-886. doi: 10.1046/j.1369-7412.2003.00420.x · Zbl 1059.62031 · doi:10.1046/j.1369-7412.2003.00420.x |
| [20] | J.D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, 1994. · Zbl 0831.62061 |
| [21] | B.E. Hansen, Econometrics, Online Published Book, 2015. Available at http://www.ssc.wisc.edu/bhansen/econometrics/Econometrics.pdfhttp://www.ssc.wisc.edu/bhansen/econometrics/Econometrics.pdf. |
| [22] | W. Hardle, M. Muller, S. Sperlich, and A. Werwatz, Nonparametric and Semiparametric Models, An Introduction. Online Published Book, 2004. Available at http://www.i-xplore.de · Zbl 1059.62032 |
| [23] | T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer-Verlag, New York, 2008. · Zbl 0973.62007 |
| [24] | S.R. Jammalamadaka and A. SenGupta, Topics in Circular Statistics, Series on Multivariate Analysis, Vol. 5, World Scientific Publishing Co., River Edge, NJ, 2001. · Zbl 1006.62050 |
| [25] | J. Janczura, S. Truck, R. Weron, and R.C. Wolff, Identifying spikes and seasonal components in electricity spot price data: A guide to robust modeling, Energy Econ. 38 (2013), pp. 96-110. doi: 10.1016/j.eneco.2013.03.013 · doi:10.1016/j.eneco.2013.03.013 |
| [26] | J. Janczura and R. Weron, An empirical comparison of alternate regime-switching models for electricity spot prices, Energy Econ. 32 (2010), pp. 1059-1073. doi: 10.1016/j.eneco.2010.05.008 · doi:10.1016/j.eneco.2010.05.008 |
| [27] | S.J. Koopman, M. Ooms, and M.A. Carnero, Periodic seasonal reg-ARFIMA-GARCH models for daily electricity spot prices, J. Am. Stat. Assoc. 102 (2007), pp. 16-27. doi: 10.1198/016214506000001022 · Zbl 1284.62786 |
| [28] | K.V. Mardia and P.E. Jupp, Directional Statistics, John Wiley and Sons, West Sussex, 2000. · Zbl 0935.62065 |
| [29] | E.A. Nadaraya, On estimating regression, Theory Probab. Appl. 9 (1964), pp. 141-142. doi: 10.1137/1109020 · doi:10.1137/1109020 |
| [30] | D.S. Osborn, The implications of periodically varying coefficients for seasonal time-series processes, J. Econ. 48 (1991), pp. 373-384. doi: 10.1016/0304-4076(91)90069-P · Zbl 0725.62100 · doi:10.1016/0304-4076(91)90069-P |
| [31] | G. Pennacchi, Theory of Asset Pricing, Prentice Hall, Upper Saddle River, NJ, 2007. |
| [32] | D. Pirino and R. Reno, Electricity prices: A nonparametric approach, Int. J. Theor. Appl. Finan. 13 (2010), 285. doi:10.1142/S0219024910005772. · Zbl 1203.91233 · doi:10.1142/S0219024910005772 |
| [33] | J.S. Simonoff, Smoothing Methods in Statistics, Springer-Verlag, New York, 1996. · Zbl 0859.62035 · doi:10.1007/978-1-4612-4026-6 |
| [34] | Y. Sun, J.D. Hart, and M.G. Genton, Nonparametric inference for periodic sequences, Technometrics 54 (2012), pp. 83-96. doi: 10.1080/00401706.2012.650499 |
| [35] | G.J.G. Upton and B. Fingleton, Spatial Data Analysis by Example Vol. 2 (Categorical and Directional Data), John Wiley and Sons, Hoboken, NJ, 1989. · Zbl 0705.62002 |
| [36] | G.S. Watson, Smooth regression analysis, Sankhya: Indian J. Stat. Ser. A 26 (1964), pp. 359-372. · Zbl 0137.13002 |
| [37] | R. Weron, Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach, John Wiley & Sons, West Sussex, 2006. · doi:10.1002/9781118673362 |
| [38] | R. Weron, Market price of risk implied by Asian-style electricity options and futures, Energy Econ. 30 (2008), pp. 1098-1115. doi: 10.1016/j.eneco.2007.05.004 · doi:10.1016/j.eneco.2007.05.004 |
| [39] | R. Weron, I. Simonsen, and P. Wilman, Modeling highly volatile and seasonal markets: Evidence from the Nord Pool electricity market, in The Application of Econophysics, H. Takayasu, ed., Springer, Tokyo, 2004, pp. 182-191. |
| [40] | Z. XU, Essays on GMO effects on crop yields, the effects of pricing errors on implied volatilities and smoothing for seasonal time series with a long cycle, Graduate theses and diss., Paper 12910, 2012. |
| [41] | F. Ziel, R. Steinert, and S. Husmann, Efficient modeling and forecasting of electricity spot prices, Energy Econ. 47 (2015), pp. 98-111. doi: 10.1016/j.eneco.2014.10.012 · doi:10.1016/j.eneco.2014.10.012 |
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