### Abstract

Zero crossing (ZC) statistic is the number of zero crossings observed in a time series. The expected value of the ZC specifies the first-order autocorrelation of the processes. Hence, we can estimate the autocorrelation by using the ZC estimator. The asymptotic consistency and normality of the ZC estimator for scalar Gaussian processes are already discussed in 1980. In this article, first, we derive the joint asymptotic distribution of the ZC estimator for ellipsoidal processes. Next, we show the variance of the ZC estimator does not attain the Cramer–Rao lower bound (CRLB). However, it is shown that the ZC estimator has robustness when the process is contaminated by an outlier. In contrast with this, we observe that the quasi-maximum likelihood estimator (QMLE) attains the CRLB. However, we can see that QMLE is sensitive for the outlier.

Original language | English |
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Journal | Journal of Time Series Analysis |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

- asymptotic distribution
- outlier
- quasi-maximum likelihood estimator
- spectral density
- Stationary process
- zero crossing

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics