Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

References

• Auestad , B. , Tjostheim , D. ( 1990 ). Identification of nonlinear time series: First order characterisation and order determination . Biometrica 77 : 669 – 687 .
   Web of Science ®Google Scholar
• Berkes , I. , Philipp , W. ( 1977 ). An almost sure invariance principle for the empirical distribution function of mixing random variables . Z. Wahrscheinlichkeitstheorie verw. Gebiete 41 : 115 – 137 .
   Web of Science ®Google Scholar
• Burke , M. D. , Csörgő , S. , Horváth , L. (1981). Strong approximations of some biometric estimates under random censorship. Z. Wahrsch. Verw. Gebiete 56:87–112.
   Web of Science ®Google Scholar
• Burke , M. D. , Csörgő , S. , Horváth , L. ( 1988 ). A correction to and improvment of “Strong Approximations of Some Biometric Estimates under Random Censorship.” Probab. Theor. Relat. Field. 79 : 51 – 57 .
   Web of Science ®Google Scholar
• Cai , Z. ( 1998 ). Asymptotic properties of Kaplan–Meier estimator for censored dependent data . Stat. Probab. Lett. 37 : 381 – 389 .
   Web of Science ®Google Scholar
• Cheng , K. F. ( 1984 ). On almost sure representation for quantiles of the product limit estimator with applications . Sankhya 46 : 426 – 443 .
   Google Scholar
• Csörgő , M. ( 1983 ). Quantile Processes with Statistical Applications . Philadelphia : SIAM .
   Google Scholar
• Dhompongsa , S. ( 1984 ). A note on the almost sure approximation of the empirical process of weakly dependent random variables . Yokohama Mathematical Journal 32 : 113 – 121 .
   Google Scholar
• Diebolt , J. , Guégan , D. ( 1993 ). Tail behaviour of the stationary density of general nonlinear autoregressive processes of order 1 . J. Appl. Probab. 30 : 315 – 329 .
   Web of Science ®Google Scholar
• Doukhan , P. ( 1996 ). Mixing: Properties and Examples . Lecture Notes in Statistics , Vol. 61. New York : Springer Verlag .
   Google Scholar
• Engle , R. F. ( 1982 ). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation . Econometrics 50 ( 4 ): 987 – 1007 .
   Web of Science ®Google Scholar
• Fotopoulos , S. , Ahn , S. K. , Cho , S. ( 1994 ). Strong approximation of the quantile processes and its applications under strong mixing properties . J. Multivariate Anal. 51 : 17 – 47 .
   Google Scholar
• Gastwirth , J. L. ( 1971 ). A general definition of the Lorenz curve . Econometrica 39 : 1037 – 1039 .
   Web of Science ®Google Scholar
• Gill , R. D. ( 1980 ). Censoring and stochastic integrals . Mathematical Centre Tracs . Vol. 124 . Amsterdam : Mathematisch Centrum .
   Google Scholar
• Masry , E. , Tjostheim , D. ( 1995 ). Nonparametric estimation and identification of nonlinear ARCH time series: Strong convergence and asymptotic normality . Econ. Theor. 11 : 258 – 289 .
   Web of Science ®Google Scholar
• Masry , E. , Tjostheim , D. ( 1997 ). Additive nonlinear ARX time series and projection estimates . Econ. Theor. 13 : 214 – 256 .
   Web of Science ®Google Scholar
• Ould-Saïd , E. , Sadki , O. ( 2005 ). Strong approximation of quantile function for strong mixing and censored processes . Comm. Stat. Theor. Meth. 34 : 1449 – 1459 .
   Web of Science ®Google Scholar
• Ozaki , T. ( 1979 ). Nonlinear time series models for nonlinear random vibrations. Technical Report. Manchester, UK: University of Manchester .
   Google Scholar
• Padgett , W. T. , Thombs , L. A. ( 1989 ). A smooth nonparametric quantile estimation from right censored data . Stat. Probab. Lett. 7 : 113 – 121 .
   Google Scholar
• Parzen , E. ( 1979 ). Nonparametric statistical data modeling . J. Am. Stat. Assoc. 74 : 105 – 131 .
   Web of Science ®Google Scholar
• Sander , J. ( 1975 ). The weak convergence of quantiles of the product limit estimator. Technical Report Number 5. Palo Alto, CA: Stanford University, Department of Statistics .
   Google Scholar
• Sen , P. K. ( 1972 ). On the Bahadur representation of samples quantiles for sequences of α-mixing random variables . J. Multivariate Anal. 2 : 77 – 95 .
   Google Scholar
• Shorack , G. R. , Wellner , J. A. ( 1986 ). Empirical Processes with Applications to Statistics . New York : Wiley-Interscience .
   Google Scholar
• Tse , S. M. ( 2003 ). Strong Gaussian approximations in the left truncated and right censored model . Statistica Sinica 13 : 275 – 282 .
   Web of Science ®Google Scholar
• Tse , S. M. ( 2005 ). Quantile processes for left truncated and right censored data . Ann. Inst. Stat. Math. 57 : 61 – 69 .
   Web of Science ®Google Scholar
• Yoshihara , K. I. ( 1995 ). The Bahadur representation of sample quantiles for sequences of strongly mixing random variables . Stat. Probab. Lett. 24 : 299 – 304 .
   Web of Science ®Google Scholar
• Yu , H. ( 1996 ). A note on strong approximation for quantile processes of strong mixing sequences . Stat. Probab. Lett. 30 , 1 – 7 .
   Google Scholar
• Zhou , Y. , Yip , S. F. ( 1999 ). A strong representation of the product-limit estimator for left truncated and right censored data . J. Multivariate Anal. 69 : 261 – 280 .
   Web of Science ®Google Scholar