Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary

In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE)$\hat \alpha _{ML}$ based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare$\hat \alpha _{ML}$ with a generalized estimate$\hat \alpha _{c_1 ,c_2 }$ including the least square estimate (LSE)$\hat \alpha _{LS}$, based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that$\hat \alpha _{ML}$ (or the modified MLE$\hat \alpha _{ML}^*$) is better than$\hat \alpha _{c_1 ,c_2 }$ (or the modified estimate$\hat \alpha _{c_1 ,c_2 }^*$). Further, we note that$\hat \alpha _{ML}^*$ does not attain the bound for third order asymptotic median unbiased estimates.