A Khmaladze-transformed test of fit with ML estimation in the presence of recurrent events

References

• Aalen, O. and Husebye, E. (1991). Statistical Analysis of Repeated Event Forming Renewal Processes, Statistics in Medicine 10: 1227–1240.
   PubMed Web of Science ®Google Scholar
• Adekpedjou, A., DeMel, W. A., and Zamba, K. D. (2015). Data Dependent Cells Chi-Square Test with Recurrent Events, Scandinavian Journal of Statistics 42: 1045–1065.
   Web of Science ®Google Scholar
• Adekpedjou, A., Peña, E., and Quiton, J. (2010). Estimation and Efficiency with Recurrent Event Data under Informative Monitoring, Journal of Statistical Planning and Inference 140: 597–615.
   PubMed Web of Science ®Google Scholar
• Adekpedjou, A. and Zamba, K. D. (2012). A Chi-Squared Goodness of Fit Test for Recurrent Event Data, Journal of Statistical Theory and Applications 11: 97–119.
   Google Scholar
• Akritas, M. G. (1988). Pearson-Type Goodness of Fit Tests: The Univariate Case, Journal of American Statistical Association 83: 222–230.
   Web of Science ®Google Scholar
• Andersen, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. (1993). Statistical Models Based on Counting Processes, New York: Springer.
   Google Scholar
• Bai, J. (2003). Testing Parametric Conditional Distributions of Dynamic Models, Review of Economics and Statistics 85: 531–549.
   Web of Science ®Google Scholar
• Billingsley, P. (1999). Convergence of Probability Measures, New York: Wiley.
   Google Scholar
• Borgan, Ø. (1991). Maximum Likelihood Estimation in Parametric Counting Process Models, with Application to Censored Failure Time Data, Scandinavian Journal of Statistics 11: 1–16.
   Google Scholar
• Breslow, N. and Crowley, J. (1974). A Large Sample Study of The Life Table and Product Limit Estimates under Random Censorship, Annals of Statistics 2: 437–453.
   Web of Science ®Google Scholar
• Chang, I. S. and Hsiung, C. A. (1988). Counting Process Approach to Time Sequential Estimation with Censored Data, Sequential Analysis 7: 127–148.
   Google Scholar
• Chernoff, H. and Lehmann, E. L. (1954). The Use of Maximum Likelihood Estimates in Chi-Square Tests for Goodness of Fit, Annals of Statistics 25: 579–586.
   Web of Science ®Google Scholar
• Durbin, J. (1973). Weak Convergence of Sample Distribution Functions When Parameters Are Estimated, Annals of Statistics 1: 279–290.
   Web of Science ®Google Scholar
• Fisher, R. A. (1922). On the Interpretation of Chi-Square from Contingency Tables, and the Calculation of P, Journal of Royal Statistical Society 85: 87–94.
   Google Scholar
• Fisher, R. A. (1924). The Condition under Which Chi-Square Measures the Discrepancy between Observation and Hypothesis, Journal of Royal Statistical Society 87: 442–450.
   Google Scholar
• Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis, New York: Wiley.
   Google Scholar
• Gaudoin, O., Yang, B., and Xie, M. (2006). Confidence Intervals for the Scale Parameter of the Power-Law Process, Communications in Statistics - Theory & Methods 35: 1525–1538.
   Web of Science ®Google Scholar
• Ghosh, B. K. and Sen, P. K. (1991). Handbook of Sequential Analysis, New York: Dekker.
   Google Scholar
• Habib, M. G. and Thomas, D. R. (1986). Chi-Square Goodness of Fit Tests for Randomly Censored Data, Annals of Statistics 14: 759–765.
   Web of Science ®Google Scholar
• Hjort, N. L. (1990). Goodness of Fit Tests in Models for Life History Data Based on Cumulative Hazard Rates, Annals of Statistics 18: 1221–1258.
   Web of Science ®Google Scholar
• Hollander, M. and Peña, E. A. (1992). A Chi-Squared Goodness of Fit Test for Randomly Censored Data, Journal of American Statistical Association 87: 458–463.
   Web of Science ®Google Scholar
• Jacod, J. (1974/1975). Multivariate Point Processes: Predictable Projection, Radon-Nikodym Derivatives, Representation of Martingales, Zeitschrift f$\ddot{u}$r Wahrscheinlichkeitstheorie Verwandte Gebiete 31: 235–253.
   Google Scholar
• Kaplan, E. L. and Meier, P. (1958). Nonparametric Estimation from Incomplete Observations, Journal of American Statistical Association 53: 457–481.
   Web of Science ®Google Scholar
• Khmaladze, È. V. (1981). Martingale Approach in the Theory of Goodness of Fit Test, Theory of Probability and Its Applications 26: 240–257.
   Web of Science ®Google Scholar
• Khmaladze, È. V. and Koul, H. L. (2004). Martingale Transforms Goodness of Fit Tests in Regression Models, Annals of Statistics 32: 995–1034.
   Web of Science ®Google Scholar
• Kolmogorov, A. N. (1933). Sulla Determinazione Empirica di Una Legge di Distribuzione, Giornal dell’ Istituto Italiano degli Attuari 4: 83–91.
   Google Scholar
• Koul, H. L. and Sakhanenko, L. (2005). Goodness of Fit Testing in Regression: A Finite Sample Comparison of Bootstrap Methodology and Khmaladze Transformation, Statistics and Probability Letters 74: 290–302.
   Web of Science ®Google Scholar
• Koziol, J. A. and Green, S. B. (1976). A Cramér-von Mises Statistic for Randomly Censored Data, Biometrika 63: 465–474.
   Web of Science ®Google Scholar
• Li, G. and Doss, H. (1993). Generalized Pearson-Fisher Chi-Square Goodness of Fit Tests, with Applications to Models with Life History Data, Annals of Statistics 21: 772–797.
   Web of Science ®Google Scholar
• Limnios, N. and Nikulin, M. (2000). Recent Advances in Reliability Theory, Boston: Birkha¨user.
   Google Scholar
• Lovric, M. (2011). International Encyclopedia of Statistical Science, Berlin: Springer.
   Google Scholar
• Miller, R. G, Jr.(1983). What Price Kaplan-Meier?, Biometrics 39: 1077–1081.
   PubMed Web of Science ®Google Scholar
• Park, W. J. and Kim, Y. G. (1992). Goodness of Fit Tests for the Power Law Process, IEEE Transactions on Reliability 41: 107–111.
   Web of Science ®Google Scholar
• Peña, E. A., Strawderman, R. L., and Hollander, M. (2000). A Weak Convergence Result Relevant in Recurrent and Renewal Models, in Recent Advances in Reliability, N. Limnios and M. Nikulin, eds., pp. 489–514, Boston: Birkhäuser.
   Google Scholar
• Peña, E. A., Strawderman, R. L., and Hollander, M. (2001a). Nonparametric Estimation with Recurrent Event Data, Journal of American Statistical Association 96: 1299–1315.
   Web of Science ®Google Scholar
• Peña, E. A., Strawderman, R. L., and Hollander, M. (2001b). Nonparametric Estimation with Recurrent Event Data, Cornell University Biometrics Department, Technical Report BU-1568-M.
   Google Scholar
• Pierce, D. A. (1982). The Asymptotic Effect of Substituting Estimators for Parameters in Certain Types of Statistics, Annals of Statistics 10: 475–478.
   Web of Science ®Google Scholar
• Proschan, F. (1963). Theoretical Explanation of Observing Decreasing Failure Rate, Technometrics 5: 375–383.
   Web of Science ®Google Scholar
• Rigdon, S. E. and Basu, A. P. (2000). Statistical Methods for the Reliability of Repairable Systems, Toronto: Wiley.
   Google Scholar
• Seigmund, D. (1985). Sequential Analysis, New York: Springer.
   Google Scholar
• Selke, T. (1988). Weak Convergence of the Aalen Estimator for a Censored Renewal Process, in Statistical Decision Theory and Related Topics, IV, S. S. Gupta and J. O. Berger, eds., pp. 183–194, New York: Springer.
   Google Scholar
• Smirnov, N. V. (1933). Estimate of Deviation between Empirical Distribution Functions in Two Independent Samples, Bulletin Moscow University 2: 3–16.
   Google Scholar
• Spiekerman, C. F. and Lin, D. Y. (1998). Marginal Regression Models for Multivariate Failure Time Data, Journal of American Statistical Association 93: 1164–1175.
   Web of Science ®Google Scholar
• White, H. (1982). Maximum Likelihood Estimation of Misspecified Models, Econometrica 50: 1–25.
   Web of Science ®Google Scholar
• Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis, Philadelphia: SIAM.
   Google Scholar
• Woodroofe, M. (1991). The Role of Renewal Theory in Sequential Analysis, in Handbook of Sequential Analysis, B. K. Ghosh and P. K. Sen, eds., pp. 145–167, New York: Dekker.
   Google Scholar
• Zamba, K. D. and Adekpedjou, A. (2011). Parameter Estimation for Correlated Recurrent Events under Informative Monitoring, Statistical Methodology 8: 273–290.
   Google Scholar