New methods to estimate the observed noise variance for an ARMA model

For an ARMA model with an observed noise, the observed noise variance estimation is not only a part of the model identification, but also its estimation accuracy affects the following MA parameter estimation accuracy directly. However, it is difficult to improve the estimation accuracy of the observed noise variance, especially when the observed noise variance is small. In this paper, two new methods are proposed to estimate the observed noise variance accurately. In the first method, the lower lags of the auto-covariance function are used to estimate the observed noise variance with high estimation accuracy, but it is valid only when the AR order is greater than the MA order. In the second method, the ARMA model is approximated as a high-order AR model so that it is effective even though the AR order is equal to or less than the MA order. If the observed noise variance is too small, its estimation error may be too large to valid the estimate. An empirical criterion is proposed to judge the necessity of estimating the observed noise variance. The proposed methods are verified by simulations and applied to the random noise modeling for gyroscopes tentatively.