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

Abstract

In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) &~L based on T observations from the first order Gaussian process up to the term of order T-L The expansion is used to compare &~L with a generalized estimate &c,,c.~ including the least square estimate (LSE) &us, based on the asymptotic probabilities around the true value of the estimates up to the terms of order T-L It is shown that &~L

Higher Order Asymptotic Theory for Time Series Analysis

“…(;12;::: 0, which coincides with the results of Fujikoshi and Ochi(1984). That is, e; is always better than e;.…”

“…For an AR(l) process, Fujikoshi and Ochi(1984) investigated some third-order asymptotic properties of the maximum likelihood estimator.…”

Contributions to multivariate analysis by Professor Yasunori Fujikoshi

“…For some of these studies, see Fujikosih and Isogai [43], Fujikoshi and Ochi [53], Siotani and Fujikoshi [99], Fujikoshi and Nishii [50], Fujikoshi et al [48], Kariya et al [80], Fujikoshi and Watamori [64], Otake et al [86], Fujikoshi [31], Seo et al [96], Naito et al [85], Kanda and Fujikoshi [77], Gupta et al [68], etc. At the time of his retirement of Hiroshima University, the number of published papers written by Professor Fujikoshi was 124, and after retirement, he has written another eight papers.…”

Approximate Distribution of Parameter Estimators for First‐order Autoregressive Models

“…The computation is quite tedious (there is no quick method). Note that the third-order Edgeworth expansion for the estimator can be derived from the first four asymptotic cumulants using the general method described by Bhattacharya and Ghosh (1978), and conversely these cumulants can be deduced from the Edgeworth expansions which have already been obtained by Phillips (1978) for the LS estimator, by Ochi (1983) for the FBLS estimators and by Fujikoshi and Ochi (1984) for the ML estimator. Also, Taniguchi (1986) has provided the leading term of these asymptotic cumulants.…”