摘要(英) |
Truncation often occurs in lifetime data analysis, where samples are collected under certain time constraints. This thesis considers parametric inference when random samples are subject to double-truncation, i.e., both left- and right-truncations. Efron and Petrosian (1999) proposed to fit the special exponential family (SEF)
where and ,
for doubly-truncated data, but did not study it’s computational and theoretical properties. This thesis fills this gap.
We develop computational algorithms for Newton-Raphson and fixed point iteration techniques to obtain maximum likelihood estimator (MLE) of the parameters, and then compare the performance of these two methods by simulations. To stabilize the convergence under the three-parameter SEF, we propose a randomized Newton-Raphson method. Also, we study the asymptotic properties of the MLE based on the theory of independent but not identically distributed (i.n.i.d) random variables that accommodate the heterogeneity of truncation intervals. Lifetime data from the Channing House study are used for illustration.
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參考文獻 |
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.
Bakoyannis G, Touloumi G. (2012) Practical methods for competing risks data: a review. Statistical Method in Medical Research : 21: 257-272.
Balakrishnan N, Asit Basu P (1996) The Exponential Distribution: Theory, Methods and Applications. Taylor & Francis Ltd, United States.
Bradley RA, Gart JJ (1962) The asymptotic properties of ML estimators when sampling from associated population. Biometrika 49: 205-214.
Burden RL, Faires JD (2011) Numerical Analysis. Cengage Learning, Boston.
Chen YH (2009) Weighted Breslow-type and maximum likelihood estimation in semiparametric transformation models. Biometrika 96: 235-251.
Chen YH (2012) Maximum likelihood analysis of semi-competing risks data with semiparametric regression models. Lifetime Data Analysis 18: 36-57.
Chang SM, Genton MG (2007) Extreme value distributions for the skew-symmetric family of distributions. Communications in statistics-Theory and methods 36:1705-1717.
Cohen AC (1991) Truncated and Censored Samples. Marcel Dekker, New York.
Casella G, Berger RL (2002) Statistical Inference. Duxbury Thomson Learning, Australia.
Castillo JD (1994) The singly truncated normal distribution: A non-steep exponential family. Annals of the Institute of Statistical Mathematics, 46: 57-66.
Cheng YJ (2014) Personal communication. Date: 2014/06/23. Place: National Central University.
Commenges D (2002) Inference for multi-state models from interval-censored data. Statistical Methods in Medical Research 11: 167-182.
Efron B, Tibshirani R (1996) Using specially designed exponential families for density estimation. The Annals of Statistics 24: 2431-2461.
Efron B, Petrosian R (1999) Nonparametric methods for doubly truncated data. Journal of the American Statistical Association 94: 824-834.
Emura T, Konno Y (2012) Multivariate normal distribution approaches for dependently truncated data. Statistical Papers 53:133-149.
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. DOI: 10.1007/s10985-014-9297-5.
Emura T, Chen YH (2014) Gene selection for survival data under dependent censoring: A copula-based approach. Statistical Methods in Medical Research. DOI: 10.1177/0962280214533378.
Klein JP, Moeschberger ML (2003) Survival Analysis Techniques for Censored and Truncated Data. Springer, New York.
Knight K (2000) Mathematical Statistics. Chapman and Hall, Boca Raton.
Lehmann EL, Casella G (1998) Theory of Point Estimation. Springer, New York.
Lehmann EL, Romano JP (2005) Testing Statistical Hypotheses. Springer, New York.
Miller RG, Efron B, Brown BW, Moses LE (1980) Survival analysis with incomplete observations, Hyde J, Biostatistics Casebook, Wiley, New York, pp. 31-46.
Moreira C, de Uña-Álvarez J (2010) Bootstrapping the NPMLE for doubly truncated data. Journal of Nonparametric Statistics 22: 567-583.
Moreira C, de Uña-Álvarez J, Keilegom IV (2014) Goodness-of-fit tests for a semiparametric model under random double truncation. Computational Statistics. DOI: 10.1007/s00180-014-0496-z.
Moreira C, de Uña-Álvarez J (2012) Kernel density estimation with doubly-truncated data. Electronic Journal of Statistics 6: 501-521.
Moreira C, Keilegom IV (2013) Bandwidth selection for kernel density estimation with doubly truncated data. Computational Statistics & Data analysis 61: 107-123.
Nelsen RB (2006) An Introduction To Copulas. Springer, New York.
Robertson HT, Allison DB (2012) A novel generalized normal distribution for human longevity and other negatively skewed data. PLoS ONE 7: e37025.
R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, R version 3.0.2.
Shao J (2003) Mathematical Statistics. Springer, New York.
Sankaran PG, Sunoj SM (2004) Identification of models using failure rate and mean residual life of doubly truncated random variables. Statistical Papers 45: 97-109.
Shen PS (2010) Nonparametric analysis of doubly truncated data. Annals of the Institute of Statistical Mathematics 62: 835-853.
Stovring H, Wang MC (2007) A new approach of nonparametric estimation of incidence and lifetime risk based on birth rates and incidence events. BMC Medical Research Methodology 7: 53.
van der Vaart AW (1998) Asymptotic Statistics. Cambridge University Press, Cambridge.
Zhu H, Wang MC (2012) Analysing bivariate survival data with interval sampling and application to cancer epidemiology. Biometrika 99: 345-361
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