## Principal components analysis of regularly varying functions.(English)Zbl 1428.62258

Summary: The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not require this assumption. Instead, we assume that the observed functions are regularly varying. We derive the asymptotic distribution of the sample covariance operator and of the sample functional principal components. We obtain a number of results on the convergence of moments and almost sure convergence. We apply the new theory to establish the consistency of the regression operator in a functional linear model.

### MSC:

 62H25 Factor analysis and principal components; correspondence analysis 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference 62R10 Functional data analysis

### Keywords:

functional data; principal components; regular variation

fda (R)
Full Text:

### References:

 [1] Anderson, P.L. and Meerschaert, M.M. (1997). Periodic moving averages of random variables with regularly varying tails. Ann. Statist.25 771-785. · Zbl 0900.62488 [2] Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics149. New York: Springer. · Zbl 0962.60004 [3] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal.12 136-154. · Zbl 0539.62064 [4] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab.13 179-195. · Zbl 0562.60026 [5] Davis, R. and Resnick, S. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist.14 533-558. · Zbl 0605.62092 [6] de Acosta, A. (1981). Inequalities for $$B$$ -valued random vectors with applications to the strong law of large numbers. Ann. Probab.9 157-161. · Zbl 0449.60002 [7] de Acosta, A. and Giné, E. (1979). Convergence of moments and related functionals in the general central limit theorem in Banach spaces. Z. Wahrsch. Verw. Gebiete48 213-231. [8] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance. Applications of Mathematics (New York) 33. Berlin: Springer. · Zbl 0873.62116 [9] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol.68 109-126. · Zbl 1141.62048 [10] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist.38 1845-1884. · Zbl 1189.62141 [11] Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer Series in Statistics. New York: Springer. [12] Horváth, L., Kokoszka, P. and Rice, G. (2014). Testing stationarity of functional time series. J. Econometrics179 66-82. · Zbl 1293.62186 [13] Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley Series in Probability and Statistics. Chichester: Wiley. · Zbl 1338.62009 [14] Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94) 121-140. · Zbl 1164.28005 [15] Klüppelberg, C. and Mikosch, T. (1994). Some limit theory for the self-normalised periodogram of stable processes. Scand. J. Stat.21 485-491. · Zbl 0809.62081 [16] Kokoszka, P. and Reimherr, M. (2013). Asymptotic normality of the principal components of functional time series. Stochastic Process. Appl.123 1546-1562. · Zbl 1275.62066 [17] Kokoszka, P. and Reimherr, M. (2017). Introduction to Functional Data Analysis. Texts in Statistical Science Series. Boca Raton, FL: CRC Press. · Zbl 1411.62004 [18] Kokoszka, P.S. and Taqqu, M.S. (1996). Parameter estimation for infinite variance fractional ARIMA. Ann. Statist.24 1880-1913. · Zbl 0896.62092 [19] Kokoszka, P., Stoev, S. and Xiong, Q. (2019). Supplement to “Principal components analysis of regularly varying functions.” DOI:10.3150/19-BEJ1113SUPP. [20] Kuelbs, J. and Mandrekar, V. (1974). Domains of attraction of stable measures on a Hilbert space. Studia Math.50 149-162. · Zbl 0304.60002 [21] Linde, W. (1986). Probability in Banach Spaces - Stable and Infinitely Divisible Distributions, 2nd ed. A Wiley-Interscience Publication. Chichester: Wiley. · Zbl 0665.60005 [22] Lucca, D.O. and Moench, E. (2015). The pre-FOMC announcement drift. J. Finance70 329-371. [23] Meerschaert, M.M. (1984). Multivariable Domains of Attraction and Regular Variation. Ann Arbor, MI: ProQuest LLC. Thesis (Ph.D.) - University of Michigan. [24] Meerschaert, M.M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley Series in Probability and Statistics: Probability and Statistics. New York: Wiley. · Zbl 0990.60003 [25] Mikosch, T., Gadrich, T., Klüppelberg, C. and Adler, R.J. (1995). Parameter estimation for ARMA models with infinite variance innovations. Ann. Statist.23 305-326. · Zbl 0822.62076 [26] Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. New York: Springer. · Zbl 1079.62006 [27] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust4. New York: Springer. · Zbl 0633.60001 [28] Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer. [29] Rvačeva, E.L. (1962). On domains of attraction of multi-dimensional distributions. In Select. Transl. Math. Statist. and Probability, Vol. 2 183-205. Providence, R.I.: Amer. Math. Soc. [30] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling. New York: CRC Press. · Zbl 0925.60027 [31] Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional linear regression analysis for longitudinal data. Ann. Statist.33 2873-2903. · Zbl 1084.62096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.