## Rank-based estimation for all-pass time series models.(English)Zbl 1117.62089

Summary: An autoregressive-moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models are useful for identifying and modeling noncausal and noninvertible autoregressive-moving average processes. We establish asymptotic normality and consistency for rank-based estimators of all-pass model parameters. The estimators are obtained by minimizing the rank-based residual dispersion function given by L. A. Jaeckel [Ann. Math. Stat. 43, 1449–1458 (1972; Zbl 0277.62049)]. These estimators can have the same asymptotic efficiency as maximum likelihood estimators and are robust. The behavior of the estimators for finite samples is studied via simulation and rank estimation is used in the deconvolution of a simulated water gun seismogram.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators 62F35 Robustness and adaptive procedures (parametric inference) 62E20 Asymptotic distribution theory in statistics 86A15 Seismology (including tsunami modeling), earthquakes

Zbl 0277.62049
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### References:

 [1] Allal, J., Kaaouachi, A. and Paindaveine, D. (2001). $$R$$-estimation for ARMA models. J. Nonparametr. Statist. 13 815–831. · Zbl 0994.62087 [2] Andrews, B., Davis, R. A. and Breidt, F. J. (2006). Maximum likelihood estimation for all-pass time series models. J. Multivariate Anal. 97 1638–1659. · Zbl 1102.62091 [3] Andrews, M. E. (2003). Parameter estimation for all-pass time series models. Ph.D. dissertation, Dept. Statistics, Colorado State Univ. [4] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, New York. · Zbl 0944.60003 [5] Breidt, F. J. and Davis, R. A. (1992). Time-reversibility, identifiability and independence of innovations for stationary time series. J. Time Ser. Anal. 13 377–390. · Zbl 0753.62058 [6] Breidt, F. J., Davis, R. A. and Trindade, A. A. (2001). Least absolute deviation estimation for all-pass time series models. Ann. Statist. 29 919–946. · Zbl 1012.62094 [7] Brockwell, P. J. and Davis, R. A. (1991). Time Series : Theory and Methods , 2nd ed. Springer, New York. · Zbl 0709.62080 [8] Chi, C.-Y. and Kung, J.-Y. (1995). A new identification algorithm for allpass systems by higher-order statistics. Signal Processing 41 239–256. · Zbl 0969.94002 [9] Chien, H.-M., Yang, H.-L. and Chi, C.-Y. (1997). Parametric cumulant based phase estimation of 1-D and 2-D nonminimum phase systems by allpass filtering. IEEE Trans. Signal Processing 45 1742–1762. · Zbl 0883.93058 [10] Davis, R. A., Knight, K. and Liu, J. (1992). $$M$$-estimation for autoregressions with infinite variance. Stochastic Process. Appl. 40 145–180. · Zbl 0801.62081 [11] Giannakis, G. B. and Swami, A. (1990). On estimating noncausal nonminimum phase ARMA models of non-Gaussian processes. IEEE Trans. Acoust. Speech Signal Process. 38 478–495. · Zbl 0706.93064 [12] Hallin, M. (1994). On the Pitman non-admissibility of correlogram-based methods. J. Time Ser. Anal. 15 607–611. · Zbl 0807.62068 [13] Huang, J. and Pawitan, Y. (2000). Quasi-likelihood estimation of non-invertible moving average processes. Scand. J. Statist. 27 689–702. · Zbl 0964.62091 [14] Jaeckel, L. A. (1972). Estimating regression coefficients by minimizing the dispersion of the residuals. Ann. Math. Statist. 43 1449–1458. · Zbl 0277.62049 [15] Jurečková, J. and Sen, P. K. (1996). Robust Statistical Procedures : Asymptotics and Interrelations. Wiley, New York. · Zbl 0862.62032 [16] Koul, H. L. and Ossiander, M. (1994). Weak convergence of randomly weighted dependent residual empiricals with applications to autoregression. Ann. Statist. 22 540–562. · Zbl 0836.62063 [17] Koul, H. L. and Saleh, A. K. Md. E. (1993). $$R$$-estimation of the parameters of autoregressive [$$\mathrmAr(p)$$] models. Ann. Statist. 21 534–551. · Zbl 0795.62076 [18] Koul, H. L., Sievers, G. L. and McKean, J. W. (1987). An estimator of the scale parameter for the rank analysis of linear models under general score functions. Scand. J. Statist. 14 131–141. · Zbl 0628.62035 [19] Lii, K.-S. and Rosenblatt, M. (1988). Nonminimum phase non-Gaussian deconvolution. J. Multivariate Anal. 27 359–374. · Zbl 0658.60069 [20] Mukherjee, K. and Bai, Z. D. (2002). $$R$$-estimation in autoregression with square-integrable score function. J. Multivariate Anal. 81 167–186. · Zbl 1011.62093 [21] Robinson, P. M. (1987). Time series residuals with application to probability density estimation. J. Time Ser. Anal. 8 329–344. · Zbl 0625.62071 [22] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London. · Zbl 0617.62042 [23] Terpstra, J. T., McKean, J. W. and Naranjo, J. D. (2001). Weighted Wilcoxon estimates for autoregression. Aust. N. Z. J. Stat. 43 399–419. · Zbl 0992.62083
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