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

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## Summary

In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE)\(\hat \alpha _{ML} \) based on*T* observations from the first order Gaussian process up to the term of order*T* ^{−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 order*T* ^{−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.

### AMS 1980 subject classifications

Primary 62M10 Secondary 62E20### Key words and phrases

First order autoregressive process maximum likelihood estimate asymptotic expansion probability of concentration third order efficiency## Preview

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### References

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