Fig 1.
The profile log-likelihood of data simulated from four distinct MA(1) models, demonstrating a few examples of multimodal likelihood surfaces.
The solid, black line indicates the true value of θ1; the dotted line is the CSS-initialization. The dashed lines correspond to the estimate using stats:arima (red) and our proposed algorithm (implemented in arima2::arima, blue).
Fig 2.
Proportion of simulated data with improved likelihood from using multiple restarts (Algorithm 1).
Table 1.
Table summarizing the computing times of each simulated dataset, in seconds.
For large n, the CSS initialization is generally close to a maximum, causing the default initialization to converge quicker than random initializations.
Fig 3.
Proportion of models that achieved nominal coverage of Bonferroni adjusted 95% confidence intervals.
The dashed line denotes the target coverage level. Parameter estimates were obtained using Algorithm 1. (A) Confidence intervals created using Fisher’s information matrix. (B) Confidence intervals created using profile likelihoods.
Fig 4.
Data is generated from ARMA (p, q) models with (p, q) , and the corresponding AIC table is created.
The Y-axis shows the percentage of tables that were consistent. M is the number of times a maxima is observed before the algorithm terminates, so M = 1 corresponds to the standard maximization procedure.
Fig 5.
Average depth of Lake Michigan-Huron from 1860-2014.
Table 2.
AIC values for an ARMA(p, q) model fit to Lake Michigan-Huron depths.
Table 2a was computing using only a single parameter initialization. Table 2b was computed using Algorithm 1. Highlighted cells show where the likelihood was improved (AIC reduced) using our algorithm.
Table 3.
Parameter values of ARMA(p, q) model fit to Lake Michigan-Huron depth data.
The same parameters are returned with a single initialization, or when using Algorithms 1 or 2.
Fig 6.
Evidence for an AR(1) model for the Lake Michigan-Huron data.
(A) Profile likelihood confidence interval (PLCI) for θ1 which includes the value . The vertical dotted line represents the lower end of the approximate confidence interval; all points on the solid black line lie within the confidence interval, and points on the dashed red line are outside the interval. (B) Histogram of re-estimated θ1 values using simulated data simulated from the
model that was calibrated to the Lake Michigan-Huron data. (C) Histogram of re-estimated θ1 values using data simulated from the AR(1) model that was calibrated to the Lake Michigan-Huron data.
Fig 7.
Inverted AR and MA polynomial roots to the fitted AR(1) and ARMA(2, 1) models to the Lake Michigan-Huron data using a single parameter initialization.
Fig 8.
Residual plots for models fit to the Lake Michigan-Huron data.
(A) Residuals of fitted AR(1) model. (B) Residuals of fitted ARMA(2,1) model.