相依截切資料之非參數推估

NONPARAMETRIC INFERENCE FOR DEPENDENTLY TRUNCATED DATA

楊幼君(You-Jun Yang)

指導教授 ： 數學系

國立臺灣大學/理學院/數學研究所/博士(2012年)

https://doi.org/10.6342/NTU.2012.10629

摘要

存活分析，研究個體從發生起始事件到發生終點事件的時間，在許多科學領域中是相當重要且常見的課題，而截切是樣本收集時時常發生的問題。在存活時間與截切時間獨立的假設下， product-limit 與 Nelson-Aalen 估計量分別為存活時間之存活函數與累積風險函數之非參數最大概似函數估計量。然而，在實際應用裡，存活時間常與發生起點事件的時間相關，導致存活時間與截切時間相關。例如：愛滋病病人從感染到被診斷出有愛滋病的存活時間可能與病人受到感染時的年齡相關。 Cheng et al. (2007) 探討固定截切時間點相依截切資料非參數存活分析之可辨識性及估計問題。本論文推廣 Cheng et al. (2007) 在非參數假設下分析隨機截切時間點相依截切資料，探討條件存活函數與條件累積風險函數的可辨識性，並且利用核估計方法推廣 product-limit 與 Nelson-Aalen 估計量得出條件存活函數與條件累積風險函數之估計量，進而得到條件風險函數估計量。我們並推導其大樣本性質，並且利用模擬研究這些估計量的表現。我們也將這些估計量應用於一筆乳癌資料與一筆愛滋病資料。

關鍵字

條件風險函數 ； 條件存活函數 ； 相依截切 ； 非參數估計 ； 平滑估計 ； 存活資料

並列摘要

Survival analysis, which studies the time period between an initiating event and a terminating event of an individual, is an important and emerging problem in many scientific fields. Truncation often occurs during collection of data. It is well known that the product-limit and Nelson-Aalen estimators are nonparametric maximum likelihood estimators of the survival function and the cumulative hazard function, respectively, when the survival time and the truncation time are independent. However, in practice, this independence assumption may be violated because the survival time often depends on the time of the initiating event. For example, the survival time from infection to diagnosis of AIDS of an AIDS patient may depend on the age of the patient at the infection. In the case that the occurrence time of truncation is fixed, Cheng et al. (2007) investigated identifiability and estimation problems when analyzing dependently truncated data nonparametrically. In this thesis, we further investigate the case that the occurrence time of truncation is random. We use kernel methods to estimate the survival, cumulative hazard, and hazard functions of the conditional distribution and study both asymptotic and numerical behaviors of the estimators. We find that these estimators are asymtotically consistent and normally distributed. We also apply the methods to analyze a breast cancer data set and an AIDS data set.

並列關鍵字

Conditional hazard function ； Conditional survival distribution ； Dependent truncation ； Nonparametric estimation ； Smooth Estimator ； Survival data

參考文獻

Aalen, O. O. (1980), “A model for nonparametric regression analysis of counting processes,” Lecture Notes in Statistics, 2, 1–25.

Alioum, A. and Commenges, D. (1996), “A proportional hazards model for arbitrarily censored and truncated data,” Biometrics, 52, 512–524.

Beaudoin, D. and Lakhal-Chaieb, L. (2008), “Archimedean copula model selection under dependent truncation,” Statistics In Medicine, 27, 4440–4454.

Cao, R. and Gonz ́alez-Manteiga, W. (2008), “Goodness-of-fit tests for conditional models under censoring and truncation,” Journal of Econometrics, 143, 166–190.

Cheng, M.-Y., Hall, P., and Yang, Y.-J. (2007), “Nonparametric inference under dependent truncation,” ACTA Scientiarum Mathematicarum (Szeged), 73, 397– 422, dedicated to S ́andor Cs ̈orgo ̈ on the occasion of his 60th Anniversary.