Adaptive Test for Periodic ARFIMA Models


  • Amine Amimour ivision of Research in Teaching, Didactics of Disciplines and Pedagogical Innovation. National Institute for Research in Education, Zona industrial, Oued romane, El-Achour Algiers, Algeria


Local asymptotic normality property, long memory process, locally asymptotically optimal test, residual autocorrelations, periodically correlated models


The goal of this article is to construct an adaptive test for periodic long memory models, using the main technical tool of Le Cam (1986)’s Local Asymptotic Normality (LAN) property constructed in Amimour and Belaide (2020), and a Correlogram-Based LAN Result. We consider the problem of testing a given ARFIMA model in which the density of the generating white noise is specified
against a periodic ARFIMA model. The perspectives of this work are, first to establish a locally asymptotically most stringent parametric tests, when the density of the innovations and the longmemory parameter are unspecified. Second to define and investigate so-called residual rank tests.


Akharif A, Hallin M. Efficient detection of random coefficients in autoregressive models. Ann Stat.

; 31(2): 2675-704.

Al Zahrani F, Al Sameeh FAR, Musa ACM, Shokeralla AAA. Forecasting diabetes patients

attendance at Al-Baha hospitals using autoregressive fractional integrated moving average

(ARFIMA) models. J Data Anal Inf Process. 2020; 8(3): 183-194.

Amimour A. Modeles de longue m ` emoire ´ a coefficients p ` eriodiques. PhD [dissertation]. Bejaia, ´

Algerie: Facult ´ e des Sciences Exactes, UAMB; 2020. ´

Amimour A, Belaide K. Local asymptotic normality for a periodically time varying long

memory parameter. Commun Stat Theory Methods, 2020; 51(9): 2936-2952. doi:


Amimour A, Belaide K. A long memory time series with a periodic degree of fractional differencing.

ArXiv preprint arXiv : 2008.01939, 2020a.

Amimour A, Belaide K. On the invertibility in periodic arfima models. ArXiv preprint arXiv


Amimour A, Belaide K, Hili O. Minimum Hellinger distance estimates for a periodically time-varying

long memory parameter. Comptes Rendus Math. 2022; 360.G10: 1153-1162. doi: 10.5802/crmath.381.

Benghabrit Y, Hallin M. Rank-based tests for autoregressive against bilinear serial dependence. J

Nonparametr Stat. 1996; 6(2-3): 253-272.

Benghabrit Y, Hallin M. Locally asymptotically optimal tests for autoregressive against bilinear serial

dependence. Stat Sin. 1996; 6(1): 147-169.

Bentarzi M. Modeles de s ` eries chronologiques ´ a coefficients p ` eriodiques. PhD [dissertation]. Alger, ´

Algerie: Institut de Math ´ ematiques, USTHB; 1995. ´

Bentarzi M, Hallin M. Locally optimal tests against periodic autoregression: parametric and nonparametric approaches. Econ Theory. 1996; 12: 88-112.

Bentarzi M, Merzougui M. Adaptive Test for Periodicity in Autoregressive Conditional Heteroskedastic Processes. Commun Stat Simul Comput. 2010; 39: 1735–53.

Bisognin C, Lopes SRC. Properties of seasonal long memory processes. Math Comput Model. 2009;

(9-10): 1837-1851.

Franses PH, Ooms M. A periodic long-memory model for quarterly UK inflation. Int J Forecast.

; 13(1): 117-126.

Gil-Alana LA, Toro J. Estimation and testing of ARFIMA models in the real exchange rate. Int J

Financ Econ. 2002; 7(4): 279–292.

Hajek J, Sidak Z, Sen PK. Theory of Rank Tests. New York: Academic press; 1999.

Hallin M, Taniguchi M, Serroukh A, Choy K. Local Asymptotic Normality for regression models

with long-memory disturbance. Ann Stat. 1999; 27(6): 2054–2080.

Hallin M, Puri ML. Optimal rank-based procedures for time series analysis: Testing an ARMA model

against other ARMA models. Ann Stat. 1988; 16(1): 402-432.

Hosking JRM. Fractional differencing. Biometrika. 1981; 68(1): 165-176.

Hui YV, Li WK. On fractionally differenced periodic processes. Sankhaya. 1995; 57(1): 19-31.

Jibrin SV, Musa Y, Zubair UA, Saidu AS. ARFIMA modelling and investigation of structural break(s)

in West Texas Intermediate and Brent series. CBN J Appl Stat. 2015; 6(2): 59-79.

Le Cam L. Asymptotic methods in statistical decision theory, New York: Springer-Verlag; 1986.

Pipiras V, Taqqu MS. Long-range dependence and self-similarity. UK: Cambridge University Press;

Samorodnitsky G. Long-range dependence. Found trends stoch. syst. 2006; 1(3): 163-257.

Seymour L. An overview of periodic time series with examples. IFAC Proceedings Volumes. 2001;

(12): 61-66.




How to Cite

Amimour, A. . (2023). Adaptive Test for Periodic ARFIMA Models. Thailand Statistician, 21(4), 802–811. Retrieved from