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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 $\left\{{X}_{t}\right\}$,

${X}_{t}={e}_{t}+b{e}_{t-1}{X}_{t-1},$

is considered, where $\left\{{e}_{t}\right\}$ 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.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 62E20 Asymptotic distribution theory in statistics 62F10 Point estimation