在此研究中，我們建立了基於常態分佈的符號區間值變數的基本敘述統計量，包括平均數，變異數及相關係數。此外，我們還提出了自我區間迴歸 (AIR) 模型來配適區間值時間序列過程。把區間值視為常態分佈的最大和最小順序統計量並且推導出上述模型的概似函數。為了滿足隨機變異，我們將 AIR 模型與自我迴歸異質性變異 (ARCH) 模型結合。模擬結果呈現了估計量的準確性。在實際資料分析中，考慮兩組數據：空氣品質監測資料和 S&P 500 指數。比較每日平均值和每日區間值做主成分分析的負荷量差異。對於 S&P 500 指數，AIR-ARCH 模型的一步預測最高價和最低價比其他方法好，如向量自我迴歸模型和k-最近鄰居方法。

In this study, we establish the fundamental statistical inferences for normality distributed symbolic interval-valued variables. They include the mean, variance and correlation. Further, we propose an auto-interval-regressive (AIR) model to characterize the interval time series process. The likelihood functions of above models are derived by treating the intervals as the largest and smallest order statistics of a normal distribution. To capture the stochastic volatility, we combine the AIR model with auto-regressive conditional heteroscedasticity (ARCH) model. Simulation studies are performed to demonstrate the accuracy of the estimators. In real application, we consider two data sets: the air quality monitoring data and S&P 500 index. We compare the differences of the loadings of principal component analysis based on daily mean and daily interval-valued data. For S&P 500 index, the 1-step predictive high and low prices based on AIR-ARCH model dominate the alternatives, such as vector autoregressive model and k-nearest neighbors method.

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