Sign Test for Hypothesis about the Order of Equation in Moving Average Model

The article deals with constructing the sign test for the hypothesis about the order of equation in moving average. We found the asymptotic distribution of the test statistics which appeared to be the central χ²-distribution under the null hypothesis and the noncentral χ²-distribution under the alternative one. Knowing the asymptotic distribution makes it possible to calculate the asymptotic relative efficiency of the constructed sign test criterion with respect to the known criteria. In our research we give an example of calculating the asymptotic relative efficiency of the constructed sign test criterion in relation to the classical criterion, based on a sample covariance ratio. Moreover, we determine the values of the asymptotic relative efficiency for a normal distribution, the double exponential distribution (Laplace distribution) and contaminated normal distribution (Tukey distribution). It is shown that if the innovation process in the moving average model is contaminated with Gaussian outliers, the asymptotic relative efficiency of this test can be arbitrarily large compared to the traditional criterion, based on a sample correlation coefficient.

References

[1] Schelter B., Winterhalder M., Timmer J. Handbook of time series analysis: recent theoretical developments and applications. Weinheim, Wiley, 2006. 508 p.

[2] Montgomery D.C., Jennings C.L., Kulahci M. Introduction to time series analysis and forecasting. Hoboken, Wiley, 2015. 655 p.

[3] Tsay R.S. Analysis of time series. Hoboken, Wiley, 2010. 667 p.

[4] Rao T.S., Rao S.S., Rao C.R. Handbook of statistics. Vol. 30. Time series analysis: methods and applications. Amsterdam, Elsevier, 2012. 755 p.

[5] Wilson G.T., Reale M., Haywood J. Models for dependent time series. Boca Raton, CRC Press, 2015. 334 p.

[6] Daraio C., Simar L. Advanced robust and nonparametric methods in efficiency analysis. N.Y., Springer, 2007. 260 p.

[7] Huber P., Ronchetti E.M. Robust statistics. Hoboken, Wiley, 2009. 360 p.

[8] Hettmansperger T.P., McKean J.W. Robust nonparametric statistical methods. Boca Raton, CRC Press, 2011. 535 p.

[9] Wilcox R.R. Introduction to robust estimation and hypothesis testing. Amsterdam, Elsevier, 2012. 689 p.

[10] Andrews B. Rank-based estimation for autoregressive moving average time series models. J. Time Ser. Anal., 2008, vol. 29, no. 1, pp. 51-73.

[11] Goryainov V.B. Identification of a spatial autoregression by rank methods. Automation and Remote Control, 2011, vol. 72, no. 5, pp. 975-988.

[12] Goryainov V.B. Least-modules estimates for spatial autoregression coefficients. Journal of Computer and Systems Sciences International, 2011, vol. 50, no. 4, pp. 565-572.

[13] Goryainova E.R., Goryainov V.B. Sign tests in moving-average model. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2008, no. 1, pp. 76-86 (in Russ.).

[14] Goryainov V.B., Goryainova E.R. Nonparametric identification of the spatial autoregression model under a priori stochastic uncertainty. Automation and Remote Control, 2010, no. 2, pp. 198-208.

[15] Truquet L., Yao J. On the quasi-likelihood estimation for random coefficient autoregressions. Statistics, 2012, vol. 46, no. 4, pp. 505-521.

[16] McLeod A.I. On the distribution and applications of residual autocorrelations in Box-Jenkins models. J. R. Statist. Soc. B, 1978, vol. 40, pp. 296-302.

[17] Hallin M., Puri M.L. Optimal rank-based procedures for time series analysis: Testing an ARMA model against other ARMA models. The Annals of Statistics, 1988, vol. 16, pp. 402-432.

[18] Maronna R.A., Martin D., Yohai V. Robust statistics: Theory and methods. Chichester, Wiley, 2006. 403 p.

[19] Bustos O., Fraiman R., Yohai V.J. Asymptotic behaviour of the estimates based on residual autocovariances for ARMA models. Lecture Notes in Statist, 1984, vol. 26, pp. 26-49.

[20] Mukantseva L.A. Testing normality in one-dimensional and multi-dimensional linear regression. Theory of Probability and its Applications, 1978, vol. 22, pp. 591-602.