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Fractional lower-order covariance (FLOC)-based estimation for multidimensional PAR(1) model with \(\alpha -\)stable noise

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Abstract

Many real data exhibit periodic behavior. The periodic autoregressive moving average (PARMA) is one of the most common and useful model to describe these data. In the classical case, the PARMA model is considered by the assumption of Gaussian (or finite-variance) distribution of the noise. However, the Gaussian distribution seems to be unsuitable in many real applications, especially when the corresponding data exhibit impulsive-like behavior. Therefore, the extensions of the classical PARMA models are considered and the Gaussian distribution of the noise is replaced by the so-called heavy-tailed distribution. One of the most known distribution that can be used here is the \(\alpha -\) stable one. In this paper, we introduce a new estimation technique for the parameters of the multidimensional periodic autoregressive time series of order 1 (i.e., PAR(1)), which is based on fractional lower-order covariance, the alternative dependence measure adequate for \(\alpha -\)stable distributed models. From theoretical point of view, the use of this technique is justified as in this case the classical measure (i.e., covariance) is not defined. The practical aspect of this technique is discussed. The efficiency of the technique on simulated data is demonstrated using the Monte Carlo approach in different contexts, including the sample size and index of stability \(\alpha \) of the noise’s distribution. Lastly, we present the real data analysis.

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The data are free available.

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Funding

The work of P. Giri and S. Sundar was supported by Indian Institute of Technology Madras, India, under the Project No. SB20210848MAMHRD008558 ”Centre for Computational Mathematics and Data Science, Department of Mathematics, IIT Madras Chennai India.” The work of A. Wyłomańska was supported by the National Center of Science under Opus Grant 2020/37/B/HS4/00120 ”Market risk model identification and validation using novel statistical, probabilistic, and machine learning tools.”

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    Prashant Giri & S. Sundar

  2. Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Agnieszka Wyłomańska

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  1. Prashant Giri
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  2. S. Sundar
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  3. Agnieszka Wyłomańska
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Contributions

[ S.S, A.W, P.G] contributed to conceptualization, [A.W, P.G] contributed to methodology, [ P.G] contributed to formal analysis and investigation, [P.G, A.W] contributed to writing—original draft preparation, [P.G, A.W] contributed to writing—review and editing, and [S.S, A.W] contributed to supervision.

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Correspondence to Prashant Giri.

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A Algorithms and additional tables

A Algorithms and additional tables

In this section, we show that how to solve the consistent system of Eq. (38). Since the FLOC-based matrix can be singular or non-singular, i.e., \(\widehat{\mathbf {R_d}}^{v-1}(0)\) may be singular or non-singular for each \(v = 1,\ldots ,T\). The following procedure (algorithm) can be used to solve the system of Eq. (38) in any case.

figure a

where BICGSTAB is a procedure (algorithm) for solving linear systems of equations \(\mathbf{A} x = b \) (for example) with preconditioning, where \(\mathbf{A} \) is the coefficient matrix, which may be singular or non-singular, \(\mathbf{x} \) is the unknown vector, and \(\mathbf{b} \) is the known vector. It is the most recent and up-to-date process. For more information on BICGSTAB, see [72,73,74,75].

Table 3 The table presents the medians and 90% confidence intervals of FLOC-based estimated PAR(1) model’s coefficient matrices of \(T=2\), where q are ranging from 1.0 to 1.5, and \(\alpha = 1.6\)
Full size table
Table 4 The table presents the medians and 90% confidence intervals of FLOC-based estimated PAR(1) model’s coefficient matrices of \(T=2\), where q are ranging from 1.0 to 1.5, and \(\alpha = 1.6\)
Full size table
Table 5 The table presents the medians and 90% confidence intervals of FLOC-based estimated PAR(1) model’s coefficient matrices of \(T=2\), where q are ranging from 1.0 to 1.5, and \(\alpha = 1.6\)
Full size table

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Giri, P., Sundar, S. & Wyłomańska, A. Fractional lower-order covariance (FLOC)-based estimation for multidimensional PAR(1) model with \(\alpha -\)stable noise. Int J Adv Eng Sci Appl Math 13, 215–235 (2021). https://doi.org/10.1007/s12572-021-00301-0

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