The maximum likelihood method for Student’s t-distributed autoregressive model with infinite variance

• Autorzy: Stępniak Paweł , Wyłomańska Agnieszka

Identyfikatory

DOI

10.14708/ma.v48i2.7059

Warianty tytułu

PL

Metoda największej wiarygodności dla modelu autoregresyjnego z rozkładem t-Studenta o nieskończonej wariancji

• Języki publikacji: EN

Abstrakty

EN

Discrete-time series models are popular in different applications. The most classical one is the autoregressive moving average (ARMA) time series that is the stationary model. In its classical version, the ARMA model is based on Gaussian distribution. However, the gaussianity is a too simplistic assumption for many real phenomena description, especially when large observations may appear in the data. Thus, we are departing from the assumption of the normal distribution of the data and propose the infinite-variance AR model based on the Student’s t-distribution. We introduce the maximum likelihood method for the estimation of the model’s parameters. The idea is based on the Maclaurin series expansion of the likelihood function. The effectiveness of the proposed approach is demonstrated using Monte Carlo simulations. Finally, the real financial time series are considered by using the presented methodology.

PL

Dyskretne modele szeregów czasowych są popularne w różnych zastosowaniach. Najbardziej klasycznym z nich jest szereg czasowy autoregresyjny średniej ruchomej (ARMA), który jest modelem stacjonarnym. W swojej klasycznej wersji model ARMA jest oparty na rozkładzie normalnym. Jednak założenie o gaussowskiej strukturze jest zbytnim uproszczeniem w przypadku opisu wielu zjawisk rzeczywistych, zwłaszcza gdy w danych mogą pojawić się duże obserwacje. W związku z tym odchodzimy od założenia o normalnym rozkładzie danych i proponujemy model AR o nieskończonej wariancji oparty na rozkładzie t-Studenta o nieskończonej wariancji. W pracy przedstawiamy metodę największej wiarygodności do estymacji parametrów modelu. Pomysł opiera się na reprezentacji funkcji wiarygodności poprzez szereg Maclaurina. Skuteczność proponowanego podejścia została wykazana za pomocą symulacji Monte Carlo. W końcowej części artykułu przedstawione zostały zastosowania zaproponowanej metodologii do opisu danych rzeczywistych.

Słowa kluczowe

EN

autoregressive model; Student’s t-distribution; estimation; Monte Carlo simulation

PL

model autoregresyjny; rozkład t-studenta; symulacje Monte Carlo

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2020
• Tom: Vol. 48, no. 2
• Strony: 133--156
• Opis fizyczny: Bibliogr. 41 poz., fot., tab., wykr.

Twórcy

autor

Stępniak Paweł

• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, Wrocław 50-370

autor

Wyłomańska Agnieszka

• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, Wrocław 50-370

Bibliografia

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Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).