Estimation for the first-order diagonal bilinear time series model. (English) Zbl 0711.62078

The problem of estimation of the parameter b in the simple diagonal bilinear model \(\{X_ t\}\), \[ X_ t=e_ t+be_{t-1}X_{t-1}, \] is considered, where \(\{e_ t\}\) is Gaussian white noise with zero mean and possibly unknown variance \(\sigma^ 2\). The asymptotic normality of the moment estimator of b is established for the two cases when \(\sigma^ 2\) is known and \(\sigma^ 2\) is unknown. It is noted that the limit distribution of the least-squares cannot easily be derived analytically. A bootstrap comparison of the sampling distributions of the least-squares and moment estimates shows that both are asymptotically normal with the least-squares estimate being the more efficient.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20 Asymptotic distribution theory in statistics
62F10 Point estimation
Full Text: DOI


[1] Akamanam S. I., J. Time Ser. Anal. 7 pp 157– (1986)
[2] Anderson T. W., The Statistical Analysis of Time Series (1971) · Zbl 0225.62108
[3] Basawa I. V., Statistical Inference for Stochastic Processes (1980) · Zbl 0448.62070
[4] Beran R. J., Ann. Statist. 10 pp 14– (1982)
[5] Beran R. J., Jahresber. Dtsch. Math. Verin. 86 pp 14– (1984)
[6] DOI: 10.1214/aos/1176349847 · Zbl 0622.62051
[7] Bhaskara Rao M., J. Time Ser. Anal. 4 pp 95– (1983)
[8] DOI: 10.1214/aos/1176345637 · Zbl 0449.62034
[9] DOI: 10.1109/TAC.1974.1100617 · Zbl 0285.93015
[10] Chung K. L., A Course in Probability Theory, 2. ed. (1974) · Zbl 0345.60003
[11] Diananda P. H., Proc. Camb. Philos. Soc. 49 pp 239– (1953)
[12] DOI: 10.1214/ss/1177013815
[13] DOI: 10.1214/aos/1176345638 · Zbl 0449.62046
[14] DOI: 10.1214/aos/1176346705 · Zbl 0542.62051
[15] Fuller W. A., Introduction to Statistical Time Series (1976) · Zbl 0353.62050
[16] Granger C. W. J., An Introduction to Bilinear Time Series Models. (1978) · Zbl 0379.62074
[17] Granger C. W. J., Applied Time Series Analysis pp 25– (1978)
[18] Granger C. W. J., Stock. Proc. Appl. 8 pp 83– (1978)
[19] Haggan, Ozaki, and O. D. Anderson (1980 ) Amplitude-dependent exponential AR model fitting for non-linear random vibrations. Proc. Int. Time Series Meet., Nottingham (). · Zbl 0447.62093
[20] DOI: 10.1215/S0012-7094-48-01568-3 · Zbl 0031.36701
[21] Jones D. A., Proc. R. Soc. London, Ser. A 360 pp 71– (1978)
[22] W. K. Kim(1987 ) Estimation and asymptotic distribution results for the simple bilinear time series model. Ph.D. Dissertation, University of Georgia, Athens, GA.
[23] Lehmann E. L., Theory of Point Estimation (1983) · Zbl 0522.62020
[24] Mohler R. R., Bilinear Control Processes (1973) · Zbl 0343.93001
[25] DOI: 10.2307/3213316 · Zbl 0466.62082
[26] Priestley M. B., J. Time Ser. Anal. 1 pp 47– (1980)
[27] DOI: 10.1016/0304-4149(82)90045-X · Zbl 0485.62100
[28] Rao C. R., Linear Statistical Inference and its Applications, 2. ed. (1965) · Zbl 0137.36203
[29] Rohatgi V. K., An Introduction to Probability Theory and Mathematical Statistics (1976) · Zbl 0354.62001
[30] S. A. O. Sesay(1982 ) Sampling properties of the estimates of parameters of the bilinear model BL(1,0.1,1). M. Sc. Dissertation, University of Manchester Institute of Science and Technology.
[31] S. A. O. Sesay, and Rao. T. Subba(1989 ) Frequency domain estimation of bilinear time series models. J. Time Ser. Anal., to be published. · Zbl 0755.62067
[32] T. SubbaRao(1978 ) On the estimation of parameters of bilinear time series models. Technical Report 79, Department of Mathematics, University of Manchester Institute of Science and Technology.
[33] T. SubbaRao(1978 ) On the theory of bilinear time series models. Technical Report 87, Department of Mathematics, University of Manchester Institute of Science and Technology.
[34] T. SubbaRao(1979 ) On the theory of bilinear time series models II. Technical Report 121, Department of Mathematics, University of Manchester Institute of Science and Technology.
[35] Subba Rao T., J. R. Statist. Soc., Ser. B 43 pp 244– (1981)
[36] Subba Rao T., An Introduction to Bispectral Analysis and Bilinear Time Series Models (1984) · Zbl 0543.62074
[37] Tong H., J. R. Statist. Soc., Ser. B 42 pp 245– (1980)
[38] Varadarajan V. S., Sankhya 20 pp 221– (1958)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.