Estimating Parameters in Autoregressive Models in Non-Normal Situations: Asymmetric Innovations

Estimation of autoregressive models with epsilon-skew-normal innovations

“…They established general results on maximum likelihood estimates (MLE) and as real examples, they fitted ARMA models with log-normal and gamma innovations to the sunspot and the Canadian lynx data respectively, demonstrating that linear time series model with nonGaussian innovations can be a useful tool in time series modelling. Tiku, Wong and Bian [7] and Tiku, Wong, Vaughan and Bian [8] considered the estimation of AR models with symmetric innovations that follow a shift-scaled Student's t distribution, and Tiku, Wong and Bian [9], Akkaya and Tiku [10] and Wong and Bian [11] studied AR models with asymmetric innovations distributed according to gamma and generalised logistic distributions. These authors derived modified MLE of the parameters that are easy to compute.…”

Autoregressive models with short-tailed symmetric distributions

“…Modified likelihood equations are obtained exactly along the same lines as in [1]. That is, the likelihood equations ∂ ln L/∂μ = 0, ∂ ln L/∂δ = 0, ∂ ln L/∂φ = 0 and ∂ ln L/∂σ = 0 are first expressed in terms of the ordered variates z (i) = {w (i) − μ}/σ , where (for given δ and φ)…”

“…They showed that their estimators have good robustness properties with respect to numerous long-tailed distributions (kurtosis >3) and to outliers in a sample. Tiku et al [28] and Akkaya and Tiku [1] considered situations where the innovations have longtailed symmetric (LTS) or skew distributions. The maximum likelihood (ML) estimators being intractable, they derived the modified maximum likelihood (MML) estimators of the parameters in Equation (1) from complete samples, thus avoiding censoring of observations.…”

“…In recent years, however, there has been a realization that non-normal distributions occur so frequently in practice. In that context, Tan and Lin [20] used the robust likelihood function (based on Type II censored samples) of Tiku [23] to obtain robust estimators of the parameters in Equation (1). They showed that their estimators have good robustness properties with respect to numerous long-tailed distributions (kurtosis >3) and to outliers in a sample.…”

“…has many applications in agricultural, biological and biomedical sciences besides business and economics; see the references in [1]. For δ = 0, it reduces to a time series AR(1) model considered in [2] and [29].…”