AbstractRecently, some modern data architecture contains rich resources since data can be recorded in short time intervals. However, it also faces several problems, such as missing data, repeated observations, and the recording of non-equidistant time points. To overcome these issues, one can consider to preserve more information by reorganizing the data in the form of daily maximum and minimum values. To characterize such data, we propose an auto-interval-regressive moving averaging (AIRMA) model by using the order statistics from normal distributions. Furthermore, to better capture the heteroscedasticity in volatility, we design a generalized heteroscedastic volatility (GHV) model, which can be combined with the AIRMA model to be a GHVAIRMA model. We derive the likelihood functions of the aforementioned models to obtain the maximum likelihood estimators. Monte Carlo simulations are then conducted to evaluate our methods of estimation and confirm their validity. Real data examples from the S&P 500 Index and PM2.5 levels are used to demonstrate our method.
Keywords: auto-interval-regressive moving averaging model, heteroscedastic volatility, interval-valued data, order statistics, symbolic data analysis.