CORRIGENDUM: TIME SERIES MODELS WITH ASYMMETRIC INNOVATIONS

In the AR[1] time series model (0 ≤ ϵ < 1)

where ε _i are iid and have a gamma distribution G(k, σ): {1/σ ^k Γ(k)}e ^−ε/σε ^k−1 (0 < ε < ∞), k > 1, Tiku et al. [1] worked out estimators of ϵ and σ, respectively. Their estimator is essentially the ML (maximum likelihood) estimator and is linear in nature. Unfortunately, it can assume negative values in certain situations, e.g., when outliers enter the time series at very early stages. We give a quadratic estimator of σ which is always real and positive.

If L denotes the likelihood function of y ₁, y ₂,…,y_n (conditional to , Model 2 of Vinod and Shenton [2]) and z _(i) = (y _[i] − ϵy _[i]−1)/σ(1 ≤ i ≤ n) are the ordered variates (for a given ϵ), then the likelihood equations are (Tiku et al. [1])

w_i = y _[i] and w _i−1 = y _[i]−1. Simplified version of [3] gives the linear estimator of σ mentioned above (Tiku et al. [1]).

Following Islam et al. [3], we do not simplify [3] but linearize z _(i) ⁻¹, namely,

where t _(i) = E{z _(i)}. The values of t _(i) can easily be calculated from the equation given in Tiku et al. ([1], p. 1334).

Modified likelihood equations are obtained by incorporating [4] in 2-3. The solutions are the following MML estimators:

where

Since β _i > 0 (1 ≤ i ≤ n), is always real and positive. From the results given in Akkaya and Tiku [4], it follows that are asymptotically fully efficient, i.e., they are unbiased and their variance–covariance matrix is I ⁻¹, where I is the Fisher information matrix which exists for all k > 2.

Efficiency: To compare the efficiencies of with those of , in situations where the linear estimator is always positive, e.g., when ε_i have one of the following distributions

we simulated their means and variances from [100000/n] Monte Carlo runs. The random numbers generated were standardized to have variance σ². The values are given in Table 1. It can be seen that are as efficient as . In general, the former have little less bias but the same mean square errors as the later.

Table 1. Means and Variances of the Estimators; ϵ = 0.5, σ = 1

	Mean				n (Variance)
n
Distribution (1)
30	0.541	0.539	0.920	0.953	0.027	0.027	0.674	0.723
100	0.515	0.515	0.974	0.978	0.015	0.016	0.689	0.654
Distribution (2)
30	0.528	0.521	0.946	0.970	0.057	0.053	0.516	0.535
100	0.511	0.508	0.978	0.988	0.044	0.045	0.497	0.486
Distribution (3)
30	0.521	0.516	0.960	0.977	0.072	0.072	0.458	0.471
100	0.509	0.506	0.985	0.992	0.067	0.069	0.460	0.455
Distribution (4)
30	0.467	0.460	0.938	0.966	0.056	0.056	0.473	0.487
100	0.495	0.498	0.970	0.981	0.047	0.047	0.495	0.498

It may be noted that the MML estimators above are enormously more efficient than the least square estimators [1], 3-4.

Unknown location: In certain situations one may work with the model

The MML estimators are obtained exactly along the same lines as above. For example,

and, similarly, the quadratic estimator which is of the same form as in 5 and is always real and positive. Here,

and so on. These estimators are essentially as efficient as given in Tiku et al. [1]; can, however, assume negative values in certain situations. It may be noted that the estimator needs a little correction arising from the fact that (1/n)∑ _i=1 ⁿ α _i ≅ 2/(k − 1) for large n, and not 1/(k−1) as stated in Tiku et al. ([1], p. 1345).

Computations: The estimators above are computed exactly along the same lines as in Akkaya and Tiku [4], i.e., are computed first in two iterations; is then computed from 9 in a single iteration.

Acknowledgments