Measures of central tendency have been widely used for summarising statistical data, with the mean being the most popular summary statistic. However, in reallife applications it is not always the most representative measure of central location, especially when dealing with data which is skewed or contains outliers. Alternative statistics with less bias are the median and the mode. Median and quantile regression has been used in different fields to examine the effect of factors at different points of the distribution. Mode estimation, on the other hand, has found many applications in cases where the analysis focuses on obtaining information about the most typical value or pattern. This thesis demonstrates that mode also plays an important role in the analysis of big data, which is becoming increasingly important in many sectors of the global economy. However, mode regression has not been widely applied, even though there is a clear conceptual benefit, due to the computational and theoretical limitations of the existing estimators. Similarly, despite the popularity of the binary quantile regression model, computational straight forward estimation techniques do not exist. Driven by the demand for simple, well-found and easy to implement inference tools, this thesis develops a series of new regression methods for mode and binary quantile regression. Chapter 2 deals with mode regression methods from the Bayesian perspective and presents one parametric and two non-parametric methods of inference. Chapter 3 demonstrates a mode-based, fast pattern-identification method for big data and proposes the first fully parametric mode regression method, which effectively uncovers the dependency of typical patterns on a number of covariates. The proposed approach is demonstrated through the analysis of a decade-long dataset on the Body Mass Index and associated factors, taken from the Health Survey for England. Finally, Chapter 4 presents an alternative binary quantile regression approach, based on the nonlinear least asymmetric weighted squares, which can be implemented using standard statistical packages and guarantees a unique solution.

我國的信用卡市場在民國八十二年全面開放以來，發展至今不過10餘年，已成為全球成長最快速的信用卡市場之一。但近年來也隨著信用卡業務已有相當顯著的成長，然而信用卡不僅只是一種支付工具，也屬於免擔保的信用融資，對發卡銀行而言，風險很高。故本文對於銀行要如何快速且正確的掌握客戶信用與還款能力，以防範呆帳發生，也變得日趨重要。 故本文利用Binary Quantile Regression可用於探討解釋變數對於被解釋變數在給定「特定分位數之下的邊際效果」，提供不同分位數的估計結果，可用於觀察被解釋變數的整個分配狀況。在實證上，二元分量迴歸模型不只可用來解釋平均的狀況，更常用來觀察分配尾端的情況。在以ROC與CAP的信用風險模型來驗證其Binary Quantile Regression的效力。

本研究主要是應用二元分量迴歸BQR（Binary Quantile Regression）模型的方法估計銀行對中小企業授信之信用評等，以期提早偵測出可能會有違約還款的企業，達到授信時的預警效果。信用評等目的為協助金融機構在貸放前更明確的瞭解企業的信用風險，並具以衡量是否核准貸款的重要依據。在過去的研究中最廣為應用的計量方法主要為有母數（parametric）區別迴歸模型，包括Logit Model和Probit Model等區別迴歸模型，這二種模型在正確的條件設定之下，模型的預測結果可以說相當的好，但若是估計資料的分配並未符合所設定的條件，或者是資料具有無法觀察到的異質變異（heteroskedastic），則估計結果會有顯著的偏誤。傳統區別模型的一般設定如下，假設發生違約的機率給定為： ，此處 表示實際上是否真的發生違約逾期還款的情形。 為了在估計時更能控制風險，最近許多有關信用評等的研究方法傾向使用半無母數（semiparametric）單一指數模型以及無母數（nonparametric）的估計方法，如類神經網路與歸納樹（classification trees）分析方法。 而本文主要是將半無母數的分量迴歸區別模型和過去以有母數為主的Probit及Logit區別迴歸模型做比較。Koenker和Bassett（1978）提出分量迴歸估計方法（Quantile Regression Methods），分量迴歸可以更完整的反應出共變異效果對被解釋變數的影響，除此之外，分量迴歸模式提供使用上較多的彈性，在估計時無需對母體的分配做假設，另外，和傳統的最小平方（OLS）估計法不同在於OLS給予估計參數的分量為50%，因此OLS估計出的迴歸線只有一條，因此分析解釋變數對被解釋變數的影響是平均效果；分量迴歸區別模型則給予估計參數不同百分比的分量，從而可在相同樣本下得到不同的分量迴歸線，觀察解釋變數對於被解釋變數影響程度的變化，因此藉由不同分量估計出不同的迴歸係數 ，可以更加瞭解整體分配的全貌。