Constrained Estimators and Consistency of a Regression Model on a Lexis Diagram

REFERENCES

• Ameriks, J., and Zeldes, S.P. (2004), “How Do Household Portfolio Shares Vary With Age?” Technical Report, Graduate School of Business, Columbia University. Available at http://www1.gsb.columbia. edu/mygsb/faculty/research/pubfiles/16/Ameriks_Zeldes_age_Sept_2004d.pdf.
   Google Scholar
• Aswani, A., Bickel, P., and Tomlin, C. (2011), “Regression on Manifolds: Estimation of the Exterior Derivative,” Annals of Statistics, 39, 48–81.
   Web of Science ®Google Scholar
• Barndorff-Nielsen, O.E., and Cox, D.R. (1994), Inference and Asymptotics, New York: Chapman and Hall.
   Google Scholar
• Bretscher, O. (2004), Linear Algebra with Applications (3rd. ed.), Upper Saddle River, NJ: Prentice Hall.
   Google Scholar
• Carstensen, B. (2007), “Age-Period-Cohort Models for the Lexis Diagram,” Statistics in Medicine, 26, 3018–3045.
   PubMed Web of Science ®Google Scholar
• Clayton, D., and Schifflers, E. (1987), “Models for Temporal Variation in Cancer Rates. II: Age-Period-Cohort Models,” Statistics in Medicine, 6, 469–481.
   PubMed Web of Science ®Google Scholar
• Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties,” Journal of American Statistical Association, 96, 1348–1360.
   Web of Science ®Google Scholar
• Fan, J., and Lv, J. (2011), “Nonconcave Penalized Likelihood with NP-Dimensionality,” IEEE Transactions on Information Theory, 57, 5467–5484.
   PubMed Web of Science ®Google Scholar
• Fan, J., and Peng, H. (2004), “Nonconcave Penalized Likelihood With a Diver-ging Number of Parameters,” Annals of Statistics, 32, 928–961.
   Web of Science ®Google Scholar
• Fienberg, S.E., and Mason, W. M. (eds.) (1985), “Specification and Implementation of Age, Period and Cohort Models,” in Cohort Analysis in Social Research: Beyond Identifiability Problem, New York: Springer, pp. 45–88.
   Google Scholar
• Fu, W. J. (1998), “Penalized Regressions: The Bridge Versus the Lasso,” Journal of Computational and Graphical Statistics, 7, 397–416.
   Web of Science ®Google Scholar
• ——— (2000), “Ridge Estimator in Singular Design With Application to Age-Period-Cohort Analysis of Disease Rates,” Communications in Statistics—Theory and Method, 29, 263–278.
   Web of Science ®Google Scholar
• ——— (2008), “A Smoothing Cohort Model in Age-Period-Cohort Analysis With Applications to Homicide Arrest Rates and Lung Cancer Mortality Rates,” Sociological Methods and Research, 36, 327–361.
   Web of Science ®Google Scholar
• Fu, W.J., and Hall, P. (2006), “Asymptotic Properties of Estimators in Age-Period-Cohort Analysis,” Statistics and Probability Letters, 76, 1925–1929.
   Web of Science ®Google Scholar
• Fu, W.J., Land, K., and Yang, Y. (2011), “On the Intrinsic Estimator and Constrained Estimators in Age-Period-Cohort Models,” Sociological Methods and Research, 40, 453–466.
   Web of Science ®Google Scholar
• Fukuda, K. (2006), “Age-Period-Cohort Decomposition of Aggregate Data: An Application to US and Japanese Household Saving Rates,” Journal of Applied Econometrics, 21, 981–998.
   Web of Science ®Google Scholar
• Heuer, C. (1997), “Modeling of Time Trends and Interactions in Vital Rates Using Restricted Regression Splines,” Biometrics, 53, 161–177.
   PubMed Web of Science ®Google Scholar
• Holford, T.R. (1991), “Understanding the Effects of Age, Period and Cohort on Incidence and Mortality Rates,” Annual Review of Public Health, 12, 425–457.
   PubMed Web of Science ®Google Scholar
• Jiang, J.M. (1996), “REML Estimation: Asymptotic Behavior and Related Topics,” Annals of Statistics, 24, 255–286.
   Web of Science ®Google Scholar
• ——— (2007), Linear and Generalized Linear Mixed Models and Their Applications, New York: Springer.
   Google Scholar
• ——— (2010), Large Sample Techniques for Statistics, New York: Springer.
   Google Scholar
• Keilman, N., and Pham, D.Q. (2000), “Predictive Intervals for Age-Specific Fertility,” European Journal of Population, 16, 41–66.
   PubMed Web of Science ®Google Scholar
• Kiefer, J., and Wolfowitz, J. (1956), “Consistency of the Maximum Likelihood Estimation in the Presence of Infinitely Many Incidental Parameters,” Annals of Mathematical Statistics, 27, 887–906.
   Google Scholar
• Kuang, D., Nielsen, B., and Nielsen, J.P. (2008), “Identification of the Age-Period-Cohort Model and the Extended Chain-Ladder Model,” Biometrika, 95, 979–986.
   Web of Science ®Google Scholar
• Kupper, L.L., Janis, J.M., Karmous, A., and Greenberg, B.G. (1985), “Statistical Age-Period-Cohort Analysis: A Review and Critique,” Journal of Chronic Disease, 38, 811–830.
   PubMedGoogle Scholar
• Kupper, L.L., Janis, J.M., Salama, I.A., Yoshizawa, C.N., Greenberg, B.G., Winsborough, H.H. (1983), “Age-Period-Cohort Analysis: An Illustration of the Problems in Assessing Interaction in One Observation Per Cell Data,” Communications in Statistics—Theory and Methods, 12, 201–217.
   Web of Science ®Google Scholar
• Lam, C., and Fan, J. (2008), “Profile-Kernel Likelihood Inference With Diverging Number of Parameters,” Annals of Statistics, 36, 2232–2260.
   PubMed Web of Science ®Google Scholar
• Lv, J., and Fan, Y. (2009), “A Unified Approach to Model Selection and Sparse Recovery Using Regularized Least Squares,” Annals of Statistics, 37, 3498–3528.
   Web of Science ®Google Scholar
• Mason, W.M., and Smith, H.L. (1985), “Age–Period–Cohort Analysis and the Study of Deaths From Pulmonary Tuberculosis,” in Cohort Analysis In Social Research: Beyond the Identification Problem, eds. W. M. Mason and S. E. Fienberg, New York: Springer-Verlag, pp. 151–227.
   Google Scholar
• McCullagh, P., and Nelder, J.A. (1989), Generalized Linear Models (2nd ed.), New York: Chapman and Hall.
   Google Scholar
• Murphy, S.A., and van der Vaart, A.W. (2000), “On Profile Likelihood,” Journal of American Statistical Association, 95, 449–465.
   Web of Science ®Google Scholar
• Neyman, J., and Scott, E. (1948), “Consistent Estimates Based on Partially Consistent Observations,” Econometrica, 16, 1–32.
   Web of Science ®Google Scholar
• O’Brien, R.M. (2011), “Intrinsic Estimators as Constrained Estimators in Age-Period-Cohort Accounting Models,” Sociological Methods and Research, 40, 467–470.
   Web of Science ®Google Scholar
• Robertson, C., and Boyle, P. (1998), “Age-Period-Cohort Models of Chronic Disease Rates: I Modeling Approach,” Statistics in Medicine, 30, 1305–1323.
   Web of Science ®Google Scholar
• Rodgers, W. L. (1982), “Estimable Functions of Age, Period, and Cohort Effects,” American Sociological Review, 47, 774–787.
   Web of Science ®Google Scholar
• Rosenberg, P.S., and Anderson, W.F. (2011), “Age-Period-Cohort Models in Cancer Surveillance Research: Ready for Prime Time?” Cancer Epidemiology, Biomarkers & Prevention, 20, 1263–1268.
   PubMed Web of Science ®Google Scholar
• Scheffé, H. (1959), The Analysis of Variance, New York: Wiley.
   Google Scholar
• Schiffman, M.H., Brinton, L.A., Devesa, S., and Fraumeni,Jr, J.F. (1996), “Cervical Cancer,” in Cancer Epidemiology and Prevention (2nd ed.), eds. Schottenfeld, D., Fraumeni, J. F. Jr, New York: Oxford University Press, pp. 1090–1116.
   Google Scholar
• Searle, S.R. (1971), Linear Models, New York: Wiley.
   Google Scholar
• Severini, T.A. (2000), Likelihood Methods in Statistics, London: Oxford University Press.
   Google Scholar
• Smith, H. (2008), “Advances in Age-Period-Cohort Analysis,” Sociological Methods and Research, 36, 287–296.
   Web of Science ®Google Scholar
• Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288.
   Google Scholar
• Tu, Y.-K., Kramer, N., and Lee, W.C. (2012), “Addressing the Identification Problem in Age-Period-Cohort Analysis: A Tutorial on the Use of Partial Least Squares and Principal Component Analysis,” Epidemiology, 23, 583–593.
   PubMed Web of Science ®Google Scholar
• Tu, Y.-K., Smith, G.D., and Gilthorpe, M.S. (2011), “A New Approach to Age-Period-Cohort Analysis Using Partial Least Squares Regression: The Trend in Blood Pressure in Glasgow Alumni Cohort,” PLoS One, 6, e19401.
   PubMed Web of Science ®Google Scholar
• Williams, D. (1991), Probability With Martingales, Cambridge, UK: Cambridge Mathematical University Press.
   Google Scholar
• Winship, C., and Harding, D.J. (2008), “A Mechanism-Based Approach to the Identification of Age-Period-Cohort Models,” Sociological Methods and Research, 36, 362–401.
   Web of Science ®Google Scholar
• Yang, Y., Fu, W.J., and Land, K.C. (2004), “A Methodological Comparison for Age-Period-Cohort Models: The Intrinsic Estimator Method Versus Conventional Generalized Linear Models,” Sociological Methodology, 34, 75–110.
   Web of Science ®Google Scholar