穩健經驗貝氏權重線性回歸模型

Abstract

Ghosh et al. (2022)利用適當的阿爾法概似函數推導出符合指數收斂的穩健貝氏偽後驗分配，以確保參數估計不受數據中離群值的影響。然而，當我們試圖將Ghosh et al. (2022)的結果，應用在階層貝氏線性回歸模型時，我們發現參數估計的計算量過於巨大。因此在階層貝氏模型的框架下，我們推導出穩健經驗貝氏模型，將其應用到權重線性回歸。藉由權重來達到模型的穩健，此權重來自Ghosh and Basu (2013)線性回歸模型的最小密度功率散度估計方法。此穩健模型有效降低參數估計的計算量，同時達到估計的穩健性。最後，本文也展示該穩健方法在數據模擬上的表現。

Basu et al. (1998) proposed a minimum divergence method, based on the density with a single index on its power, to derive a robust estimate and has been widely applied. Ghosh and Basu (2013) further extended the minimum density power divergence method in the linear regression model. Ghosh et al. (2022) proposed a pseudo-posterior Bayesian estimate by equipping the Bayes framework with the density power divergence. However, we observe that the pseudo-posterior Bayesian estimate can hardly be extended to the hierarchical Bayes model due to lack of conjugated priors and thus requires huge computation loading. Here, we introduce a robust empirical Bayesian model by assigning weights on each individual data, where the weights are derived from the minimum density power divergence method. We apply it to the Bayesian linear regression model and use the weights derived in the linear regression model proposed by Ghosh and Basu (2013). The introduction of weights allows us to achieve robustness for the Bayesian estimates. This approach could reduce the computation loading and improve the robustness in estimating the parameters. In the end, we also demonstrate the performance of the robust method in a simulation study.