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Theory and Methods

Generalized Fiducial Inference for Ultrahigh-Dimensional Regression

Pages 760-772 | Received 01 Apr 2013, Published online: 06 Jul 2015

REFERENCES

  • Barbieri, M.M., and Berger, J.O. (2004), “Optimal Predictive Model Selection,” The Annals of Statistics, 32, 870–897.
  • Berger, J.O., Bernardo, J.M., and Sun, D. (2009), “The Formal Definition of Reference Priors,” The Annals of Statistics, 37, 905–938.
  • Berger, J.O., and Pericchi, L.R. (2001), “Objective Bayesian Methods for Model Selection: Introduction and Comparison,” in Model Selection of IMS Lecture Notes Monograph Series (Vol. 110), ed. P. Lahiri, Beachwood, OH: Institute of Mathematical Statistics, pp. 135–207.
  • Bühlmann, P., Kalisch, M., and Maathuis, M.H. (2010), “Variable Selection in High-Dimensional Linear Models: Partially Faithful Distributions and the PC-Simple Algorithm,” Biometrika, 97, 261–278.
  • Chen, J., and Chen, Z. (2008), “Extended Bayesian Information Criteria for Model Selection With Large Model Spaces,” Biometrika, 95, 759–771.
  • Cho, H., and Fryzlewicz, P. (2011), “High Dimensional Variable Selection via Tilting,” Journal of the Royal Statistical Society, Series B, 74, 593–622.
  • Cisewski, J., and Hannig, J. (2012), “Generalized Fiducial Inference for Normal Linear Mixed Models,” The Annals of Statistics, 40, 2102–2127.
  • Dempster, A.P. (2008), “The Dempster-Shafer Calculus for Statisticians,” International Journal of Approximate Reasoning, 48, 365–377.
  • Dong, Y. (2007), “Inference After Model Selection,” Ph.D. dissertation, University of Minnesota.
  • Edlefsen, P.T., Liu, C., and Dempster, A.P. (2009), “Estimating Limits From Poisson Counting Data Using Dempster–Shafer Analysis,” Annals of Applied Statistics, 3, 764–790.
  • Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004), “Least Angle Regression,” The Annals of Statistics, 32, 407–499.
  • Fan, J., Feng, Y., and Song, R. (2011), “Nonparametric Independence Screening in Sparse Ultra-High-Dimensional Additive Models,” Journal of the American Statistical Association, 106, 544–557.
  • Fan, J., Guo, S., and Hao, N. (2012), “Variance Estimation Using Refitted Cross-Validation in Ultrahigh Dimensional Regression,” Journal of the Royal Statistical Society, Series B, 74, 37–65.
  • Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties,” Journal of the American Statistical Association, 96, 1348–1360.
  • Fan, J., and Lv, J. (2008), “Sure Independence Screening for Ultrahigh Dimensional Feature Space,” Journal of the Royal Statistical Society, Series B, 70, 849–911.
  • ——— (2010), “A Selective Overview of Variable Selection in High Dimensional Feature Space,” Statistica Sinica, 20, 101–148.
  • ——— (2011), “Non-Concave Penalized Likelihood With Np-Dimensionality,” IEEE Transactions on Information Theory, 57, 5467–5484.
  • Fan, J., and Peng, H. (2004), “Nonconcave Penalized Likelihood With a Diverging Number of Parameters,” The Annals of Statistics, 32, 928–961.
  • Fisher, R.A.. (1930), “Inverse Probability,” Proceedings of the Cambridge Philosophical Society, xxvi, 528–535.
  • Hannig, J. (2009), “On Generalized Fiducial Inference,” Statistica Sinica, 19, 491–544.
  • ——— (2013), “Generalized Fiducial Inference via Discretization,” Statistica Sinica, 23, 489–514.
  • Hannig, J., Iyer, H.K., and Patterson, P. (2006), “Fiducial Generalized Confidence Intervals,” Journal of American Statistical Association, 101, 254–269.
  • Hannig, J., and Lee, T. C.M. (2009), “Generalized Fiducial Inference for Wavelet Regression,” Biometrika, 96, 847–860.
  • Huang, J., Ma, S., and Zhang, C. (2008), “Adaptive Lasso for Sparse High-Dimensional Regression Models,” Statistica Sinica, 18, 1603–1618.
  • Leeb, H., and Pötscher, B.M. (2008), “Can One Estimate the Unconditional Distribution of Post-Model-Selection Estimators?,” Econometric Theory, 24, 338–376.
  • Lidong, E., Hannig, J., and Iyer, H.K. (2008), “Fiducial Intervals for Variance Components in an Unbalanced Two-Component Normal Mixed Linear Model,” Journal of American Statistical Association, 103, 854–865.
  • Luo, S., and Chen, Z. (2013), “Extended BIC for Linear Regression Models With Diverging Number of Relevant Features and High or Ultra-High Feature Spaces,” Journal of Statistical Planning and Inference, 143, 494–504.
  • Martin, R., and Liu, C. (2013), “Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference,” Journal of the American Statistical Association, 108, 301–313.
  • Martin, R., Zhang, J., and Liu, C. (2010), “Dempster-Shafer Theory and Statistical Inference With Weak Beliefs,” Statistical Science, 25, 72–87.
  • McNally, R.J., Iyer, H.K., and Mathew, T. (2003), “Tests for Individual and Population Bioequivalence Based on Generalized P-Values,” Statistics in Medicine, 22, 31–53.
  • Meier, L., Geer, S., and Bühlmann, P. (2009), “High-Dimensional Additive Modeling,” The Annals of Statistics, 37, 3779–3821.
  • Meinshausen, N., and Buhlmann, P. (2006), “High-Dimensional Graphs and Variable Selection With the Lasso,” The Annals of Statistics, 34, 1436–1462.
  • Pötscher, B.M., and Leeb, H. (2009), “On the Distribution of Penalized Maximum Likelihood Estimators: The LASSO, SCAD, and Thresholding,” Journal of Multivariate Analysis, 100, 2065–2082.
  • Ravikumar, P., Lafferty, J., Liu, H., and Wasserman, L. (2009), “Sparse Additive Models,” Journal of the Royal Statistical Society, Series B, 71, 1009–1030.
  • Rissanen, J. (1989), Stochastic Complexity in Statistical Inquiry, Singapore: World Scientific.
  • ——— (2007), Information and Complexity in Statistical Modeling, New York: Springer.
  • Salome, D.. (1998), “Statistical Inference via Fiducial Methods,” Ph.D. dissertation, University of Groningen.
  • Shen, X., Huang, H.-C., and Ye, J. (2004), “Inference After Model Selection,” Journal of the American Statistical Association, 99, 751–762.
  • Singh, K., Xie, M., and Strawderman, W.E. (2005), “Combining Information From Independent Sources Through Confidence Distributions,” The Annals of Statistics, 33, 159–183.
  • Sonderegger, D., and Hannig, J. (2014), “Fiducial Theory for Free-Knot Splines,” in Contemporary Developments in Statistical Theory, a Festschrift in honor of Professor Hira L. Koul, eds. S. Lahiri, A. Schick, A. SenGupta, and T. N. Sriram, New York: Springer, pp. 155–189.
  • Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288.
  • Wang, H. (2009), “Forward Regression for Ultra-High Dimensional Variable Screening,” Journal of the American Statistical Association, 104, 1512–1524.
  • Wang, J. C.-M., and Iyer, H.K. (2005), “Propagation of Uncertainties in Measurements Using Generalized Inference,” Metrologia, 42, 145–153.
  • Weerahandi, S. (1993), “Generalized Confidence Intervals,” Journal of the American Statistical Association, 88, 899–905.
  • ——— (1995), Exact Statistical Methods for Data Analysis, Springer Series in Statistics, New York: Springer-Verlag.
  • Xie, M., and Singh, K. (2013), “Confidence Distribution, the Frequentist Distribution Estimator of a Parameter: A Review,” International Statistical Review, 81, 3–39.
  • Xie, M., Singh, K., and Strawderman, W.E. (2011), “Confidence Distributions and a Unified Framework for Meta-Analysis,” Journal of the American Statistical Association, 106, 320–333.
  • Zhang, C., and Huang, J. (2008), “The Sparsity and Bias of the Lasso Selection in High-Dimensional Linear Regression,” The Annals of Statistics, 36, 1567–1594.
  • Zhang, J., and Liu, C. (2011), “Dempster-Shafer Inference With Weak Beliefs,” Statistica Sinica, 21, 475–494.
  • Zhao, P., and Yu, B. (2006), “On Model Selection Consistency of Lasso,” The Journal of Machine Learning Research, 7, 2541–2563.

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