傳統的統計方法中，在分析具有左截斷(left-truncated)的資料時常常會把感興趣的隨機變數與其左截斷隨機變數具獨立性的假設。但已有許多文章指出在真實資料中獨立性的假設並不合適。此篇論文中我們建模的方式是給定兩隨機變數的邊際分配並使用連結函數(copula function)來建構我們的聯合機率分配。用此全母數模型即可解決獨立性假設不合適的問題。在此種模型下我們對inclusion probability 推導出較為簡化的公式並給出在某些條件下此函數對參數微分的公式。在連結函數為Clayton copula 邊際分配皆為韋伯(Weibull)的模型下，我們推導出對數概似函數對參數的一次及二次微分並使用了隨機牛頓-拉弗森演算法(Randomized Newton-Raphson algorithm)得到參數的最大概似估計量(Maximum likelihood estimator)。在最後我們使用了一組煞車皮的壽命資料作為實例的分析。

Traditional statistical methods for left-truncated lifetime data rely on the independence assumption regarding the truncation variable. However, the dependence between a lifetime variable of interest and its left-truncation variable usually occurs in many real data from reliability and biomedical analysis. In this paper, we propose a copula-based dependence model between and with the marginal distributions specified by parametric models. Then we consider the maximum likelihood estimator (MLE) under the copula-based dependence model between and . To calculate the MLE of the unknown parameters , explicit formulas of the inclusion probability and its partial derivatives are obtained under the Clayton copula and Weibull marginal model, which are new results in this paper. Then we derive explicit expression for the randomized-Newton-Raphson algorithm for maximizing the log-likelihood. We perform simulations to verify the correctness of the proposed method. We illustrate our method by real data from a field reliability study on the lifetimes of brake pads.

References
Akaike H (1973) Information theory and an extension of the maximum likelihood principle, Petrov BN and Csaki F, Proc. 2nd International Symposium on Information Theory, Akademiai Kiado, Budapest, pp.267-281.
Chen YH (2010) Semiparametric marginal regression analysis for dependent competing risks under an assumed copula. Journal of the Royal Statistical Society, Ser. B 72: 235-51.
Ding AA (2012) Copula identifiability conditions for dependent truncated data model. Lifetime Data Analysis 18: 397-407.
Escarela G, Carriere JF (2003) Fitting competing risks with an assumed copula. Statistical Methods in Medical Research 12: 333-349.
Emura T, Wang W (2010) Testing quasi-independence for truncation data. Journal of Multivariate Analysis 101: 223-239.
Emura T, Wang W, Hung HN (2011) Semi-parametric inference for copula models for dependently truncated data. Statistica Sinica 21: 349-367.
Emura T, Konno Y (2012a) Multivariate normal distribution approaches for dependently truncated data. Statistical Papers 53: 133-149.
Emura T, Konno Y (2012b) A goodness-of-fit tests for parametric models based on dependently truncated data. Computational Statistics & Data Analysis 56: 2237-2250.
Emura T, Wang W (2012) Nonparametric maximum likelihood estimation for dependent truncation data based on copulas. Journal of Multivariate Analysis 110: 171-188.
Emura T, Konno Y, Michimae H (2014) Statistical inference based on the nonparametric maximum likelihood estimator under double-truncation. Lifetime Data Analysis 21: 397-418.
Emura T, Konno Y (2014) Erratum to: Multivariate normal distribution approaches for dependently ttruncated data. Statistical Papers 55: 1233-36.
Emura T, Chen YH (2014) Gene selection for survival data under dependent censoring: a
copula-based approach. Statistical Methods in Medical Research DOI: 10.1177/096228021
4533378.
Emura T, Nakatochi M, Murotani K, Rondeau V (2015) A joint frailty-copula model between tumour progression and death for meta-analysis. Statistical Methods in Medical Research.
Emura T (2015) depend.truncation: Statistical Inference for Parametric and Semiparametric Models Based on Dependently Truncated Data. Available online http://cran.r-project.org/web/packages/depend.truncation/depend.truncation.pdf
Emura T, Murotani K (2015) An algorithm for estimating survival under a copula-based dependent truncation model. TEST DOI: 10.1007/s11749-015-0432-8.
Emura T, Wang W (2015) Semiparametric inference for an accelerated failure time model with dependent truncation. Annals of the Institute of Statistical Mathematics DOI 10.1007/s10463-015-0526-9.
Fan TH, Yu CH (2013) Statistical inference on contant stress accelerated life tests under generalized gamma lifetime distributions. Quality and Reliability Engineering International 29: 631-638.
Gardes L, Stupfler G (2014) Estimating extreme quantiles under random truncation. TEST 24(2): 207-227.
Joe H (1993) Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis 46: 262-282.
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation. Computational Statistics DOI: 10.1007/s00180-015-0564-z.
Kalbfleisch JD, Lawless JF (1992) Some useful statistical methods for truncated data. Journal of Quality Technology 24: 145-152.
Knight K (2000) Mathematical Statistics. Chapman and Hall, Boca Raton.
Klein JP, Moeschberger ML (2003) Survival Analysis Techniques for Censored and Truncated Data, (2nd ed.). Springer-Verlag, New York.
Lakhal-Chaieb L, Rivest LP, Abdous B (2006) Estimating survival under a dependent truncation. Biometrika 93: 665–669.
Lawless JF (2003) Statistical Models and Methods for Lifetime Data,(2nd ed.). A John WILEY & SONS, Hoboken, New Jersey.
Lynden-Bell D (1971) A method of allowing for known observational selection in small samples applied to 3RC quasars. Mon Not R Astron Soc 155: 95-118.
Nelsen RB (2006) An Itroduction to copulas, (2nd ed.). Springer Science+Business Media, New York.
R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, R version 3.0.2.
Schwarz G (1978) Estimating the dimension of a model. The Annals of Statistics 6: 461-464.
Schepsmeier U, J, Brechmann (2012) VineCopula: statistical inference of vine copulas.
Available online http://cran.r-project.org/web/packages/VineCopula/
Strzalkowska-Kominiak E, Stute W (2013) Empirical copulas for consecutive survival data copulas in survival analysis. TEST 22: 688-714.
Schepsmeier U, J (2014) Derivatives and fisher information of bivariate copulas. Statistical Papers 55: 525-542.