Sankhya B

pp 1–33 | Cite as

Detection of EXPAR nonlinearity in the Presence of a Nuisance Unidentified Under the Null Hypothesis

  • Nabil AzouaghEmail author
  • Said El Melhaoui


This work considers the problem of detecting the eventual existence of an exponential component in autoregressive models of order p ≥ 1. This problem comes down to test a linear dependence AR(p) against a nonlinear one of exponential autoregressive model EXPAR(p). Following Le Cam’s asymptotic theory, we have established the local asymptotic normality of EXPAR(p) models in the vicinity of AR(p) ones. Then, we have used pseudo-Gaussian methods to extract a pseudo-Gaussian test which is locally asymptotically optimal under Gaussian densities and valid under a large class of non-Gaussian ones. The main problem arising in this context is the fact that the test statistic’s involves nuisance parameter unidentified under the null hypothesis. Contrary to the simple case of order 1 the test statistic’s depends on this nuisance through a complex function. So as to solve this problem, we suggest a method which consists to take the maximum of the test statistic’s over a specific compact set of the nuisance parameter, then, we use the AR-sieve bootstrap procedure to approximate its asymptotic distribution.

Keywords and phrases.

Nonlinearity tests Exponential autoregressive models LAN property Pseudo-Gaussian methods Nuisance parameter AR-sieve bootstrap 

AMS (2000) subject classification.

Primary 62G10 Secondary 62F05 62F40 



  1. Akharif, A. and Hallin, M. (2003). Efficient detection of random coefficients in autoregressive models. Ann. Stat.31, 2, 675–704.MathSciNetCrossRefGoogle Scholar
  2. Allal, J. and El Melhaoui, S. (2006). Optimal detection of exponential component in autoregressive models. J. Time Ser. Anal.27, 6, 793–810.MathSciNetCrossRefGoogle Scholar
  3. Andrews, D. W. and Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica: Journal of the Econometric Society62, 6, 1383–1414.MathSciNetCrossRefGoogle Scholar
  4. Azouagh, N. and El Melhaoui, S. (2019). Detecting exponential component in autoregressive models: comparative study between several tests of nonlinearity. Communications in Statistics-Simulation and Computation, available on
  5. Baragona, R., Battaglia, F. and Cucina, D. (2002). A note on estimating autoregressive exponential models. Quaderni di Statistica4, 1, 71–88.Google Scholar
  6. Berg, A., McMurry, T. and Politis, D.N. (2012). Testing time series linearity: traditional and bootstrap methods. Handbook of Statistics (Ed. S.R. Rao), 27–42, Volume 30. Elsevier.Google Scholar
  7. Berg, A., Paparoditis, E. and Politis, D. N. (2010). A bootstrap test for time series linearity. Journal of Statistical Planning and Inference140, 12, 3841–3857.MathSciNetCrossRefGoogle Scholar
  8. Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli3, 2, 123–148.MathSciNetCrossRefGoogle Scholar
  9. Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput.16, 5, 1190–1208.MathSciNetCrossRefGoogle Scholar
  10. Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika64, 2, 247–254.MathSciNetCrossRefGoogle Scholar
  11. El Melhaoui, S. and Allal, J. (2006). Optimal rank-based detection of exponential component in autoregressive models. Nonparametric Statistics18, 7-8, 431–447.MathSciNetCrossRefGoogle Scholar
  12. Garel, B. and Hallin, M. (1995). Local asymptotic normality of multivariate arma processes with a linear trend. Ann. Inst. Stat. Math.47, 3, 551–579.MathSciNetzbMATHGoogle Scholar
  13. Ghosh, H., Gurung, B. and Gupta, P. (2015). Fitting expar models through the extended kalman filter. Sankhya B77, 1, 27–44.MathSciNetCrossRefGoogle Scholar
  14. Haggan, V. and Ozaki, T. (1981). Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrika68, 1, 189–196.MathSciNetCrossRefGoogle Scholar
  15. Hájek, J. (1972). Local asymptotic minimax and admissibility in estimation, 1, p. 175–194.Google Scholar
  16. Hájek, J. and S~idák, Z. (1967). Theory of rank tests. Academic Press, New York.Google Scholar
  17. Hallin, M., Ingenbleek, J. -F. and Puri, M. L. (1985). Linear serial rank tests for randomness against arma alternatives. Ann. Stat.13, 3, 1156–1181.MathSciNetCrossRefGoogle Scholar
  18. Hallin, M. and Puri, M. L. (1994). Aligned rank tests for linear models with autocorrelated error terms. J. Multivar. Anal.50, 2, 175–237.MathSciNetCrossRefGoogle Scholar
  19. Hallin, M. and Werker, B. J. (1999). Optimal testing for semi-parametric AR models-from gaussian lagrange multipliers to autoregression rank scores and adaptive tests, 158. Dekker, New York, Ghosh, S. (ed.), p. 295–350.Google Scholar
  20. Hansen, B. E. (1996). Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica: Journal of the econometric society64, 2, 413–430.MathSciNetCrossRefGoogle Scholar
  21. Hinich, M. J. (1982). Testing for gaussianity and linearity of a stationary time series. Journal of time series analysis3, 3, 169–176.MathSciNetCrossRefGoogle Scholar
  22. Hyndman, R. J. and Fan, Y. (1996). Sample quantiles in statistical packages. Am. Stat.50, 4, 361–365.Google Scholar
  23. Kreiss, J. -P. (1987). On adaptive estimation in stationary arma processes. Ann. Stat.15, 1, 112–133.MathSciNetCrossRefGoogle Scholar
  24. Lange, K. (2010). Numerical analysis for statisticians. Springer Science & Business Media.Google Scholar
  25. Le Cam, L. (1986). Asymptotic methods in statistical decision theory. Springer-Verlag, New York.CrossRefGoogle Scholar
  26. Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization. The computer journal7, 4, 308–313.MathSciNetCrossRefGoogle Scholar
  27. Ozaki, T. (1993). Non-Gaussian Characteristics of Exponential Autoregressive Processes. Chapman and Hall, London, Ozaki, T (ed.), p. 257–273.Google Scholar
  28. Saikkonen, P. and Luukkonen, R. (1988). Lagrange multiplier tests for testing non-linearities in time series models. Scand. J. Stat.15, 1, 55–68.zbMATHGoogle Scholar
  29. Swensen, A. R. (1985). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. J. Multivar. Anal.16, 1, 54–70.MathSciNetCrossRefGoogle Scholar
  30. Tong, H. (1990). Non-linear time series: a dynamical system approach. Oxford University Press, Oxford.zbMATHGoogle Scholar
  31. Tsay, R. S. (1986). Nonlinearity tests for time series. Biometrika73, 2, 461–466.MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Faculté des Sciences, Département de MathématiquesUniversité Mohammed PremierOujdaMorocco
  2. 2.Faculté de Droit, Département d’ÉconomieUniversité Mohammed PremierOujdaMorocco

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