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THIRD‐ORDER ASYMPTOTIC PROPERTIES OF ESTIMATORS IN GAUSSIAN ARMA PROCESSES WITH UNKNOWN MEAN
Author(s) -
Kakizawa Yoshihide
Publication year - 1996
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1996.tb00283.x
Subject(s) - mathematics , estimator , autoregressive model , autoregressive–moving average model , asymptotic analysis , statistics , gaussian , mean squared error , delta method , physics , quantum mechanics
. This paper deals with the third‐order asymptotic theory for Gaussian autoregressive moving‐average (ARMA) processes with unknown mean μ. We are interested in the estimation of ρ = ( α 1 …, α p , β 1 …, β q ), where α 1 …, α ρ and β 1 …, β q are the coefficients of the autoregressive part and the moving‐average part, respectively. First, we investigate the third‐order asymptotic optimality of the bias adjusted maximum likelihood estimator (MLE) of ρ in the presence of the nuisance parameters μ and s̀ 2 (innovation variance). Next, for a Gaussian AR(1μ μ, s̀ 2 ), we propose a mean corrected estimator α c1c2 of the autoregressive coefficient. We make a comparison between the bias adjusted estimator α c1c2 * and the bias adjusted MLE, in terms of their probabilities of concentration around the true value, or equivalently, in terms of their mean squared errors. Finally some numerical studies are provided in order to verify the third‐order asymptotic theory.