在截切變數具相關性下之半母數推論

Abstract

許多科學研究資料中所記錄的變數是截切後的樣本，例如抽樣對像為六十歲以上的 人即代表壽命為截切變數，壽命高於六十則為截切條件。多數針對截切資料所發展 的統計方法都假設截切變數和所感興趣的變數間彼此為獨立。當兩者可能具有關 聯性時，有學者提出檢定此獨立假設是否合理的統計方法，亦有學者重新定義易 於在截切的條件下分析的相關係數，但是這些方法各有其限制。在計劃中我們假設 兩個變數在未截切前的關聯性服從某個半母數 Archimedean copula 模式，在此假設 下提出推論方法以估計邊際分配函數，截切比例，和關聯性參數。我們將以模擬研究 檢驗所提出方法的可行性和其在有限樣本下的表現，並推導所提出方法的大樣本性 質。

In many useful applications, the variable of interest many be truncated by another random variable. Most existing inference methods are derived under the assumption that the truncation variable is independent of the variable of interest. When these two variables are possibly correlated, some statisticians have proposed methods to test quasi-independence between the two variables or defined a new measure of association conditional on the observable region after truncation. In this project, we assume that the dependence structure before truncation follows a semi-parametric copula model. Then we will study how to estimate the marginal distribution functions; the truncation proportion and the association parameter given truncated data. The whole problem is quite challenging since all of the above three quantities are unknown and estimating each of them under truncation is not an easy task. Simulations will be performed to assess the validity of the estimators and evaluate their finite sample performances. Large sample theory of the proposed method will be developed.