Testing the Number of Components in Normal Mixture Regression Models

REFERENCES

• Andrews, D. W.K. (1999), “Estimation When a Parameter is on a Boundary,” Econometrica, 67, 1341–1383.
   Web of Science ®Google Scholar
• Azaïs, J.-M., Gassiat, E., and Mercadier, C. (2009), “The Likelihood Ratio Test for General Mixture Models With or Without Structural Parameter,” ESAIM: Probability and Statistics, 13, 301–327.
   Web of Science ®Google Scholar
• Boiteau, G., Singh, M., Singh, R., Tai, G., and Turner, T. (1998), “Rate of Spread of PVYn by Alate Myzus Persicae (Sulzer) From Infected to Healthy Plants Under Laboratory Conditions,” Potato Research, 41, 335–344.
   Web of Science ®Google Scholar
• Chen, H., and Chen, J. (2001), “The Likelihood Ratio Test for Homogeneity in Finite Mixture Models,” Canadian Journal of Statistics, 29, 201–215.
   Web of Science ®Google Scholar
• ——— (2003), “Tests for Homogeneity in Normal Mixtures in the Presence of a Structural Parameter,” Statistica Sinica, 13, 351–365.
   Web of Science ®Google Scholar
• Chen, H., Chen, J., and Kalbfleisch, J.D. (2004), “Testing for a Finite Mixture Model With Two Components,” Journal of the Royal Statistical Society, Series B, 66, 95–115.
   Google Scholar
• Chen, J. (1995), “Optimal Rate of Convergence for Finite Mixture Models,” Annals of Statistics, 23, 221–233.
   Web of Science ®Google Scholar
• Chen, J., and Kalbfleisch, J.D. (1996), “Penalized Minimum-Distance Estimates in Finite Mixture Models,” Canadian Journal of Statistics, 24, 167–175.
   Web of Science ®Google Scholar
• Chen, J., and Khalili, A. (2008), “Order Selection in Finite Mixture Models With a Non-Smooth Penalty,” Journal of the American Statistical Association, 103, 1674–1683.
   Web of Science ®Google Scholar
• Chen, J., and Li, P. (2009), “Hypothesis Test for Normal Mixture Models: The EM Approach,” Annals of Statistics, 37, 2523–2542.
   Web of Science ®Google Scholar
• Chen, J., Li, P., and Fu, Y. (2012), “Inference on the Order of a Normal Mixture,” Journal of the American Statistical Association, 107, 1096–1105.
   Web of Science ®Google Scholar
• Chen, J., Tan, X., and Zhang, R. (2008), “Inference for Normal Mixtures in Mean and Variance,” Statistica Sinica, 18, 443–465.
   Web of Science ®Google Scholar
• Chernoff, H., and Lander, E. (1995), “Asymptotic Distribution of the Likelihood Ratio Test that a Mixture of two Binomials is a Single Binomial,” Journal of Statistical Planning and Inference, 43, 19–40.
   Web of Science ®Google Scholar
• Conway, K., and Deb, P. (2005), “Is Parental Care Really Ineffective? Or, is the ‘Deveil’ in the Distribution?,” Journal of Health Economics, 24, 489–513.
   PubMed Web of Science ®Google Scholar
• Dacunha-Castelle, D., and Gassiat, E. (1999), “Testing the Order of a Model Using Locally Conic Parametrization: Population Mixtures and Stationary ARMA Processes,” Annals of Statistics, 27, 1178–1209.
   Web of Science ®Google Scholar
• Dempster, A.P., Laird, N.M., and Rubin, D.B. (1977), “Maximum Likelihood From Incomplete Data via EM Algorithm” (with discussion), Journal of the Royal Statistical Society, Series B, 39, 1–38.
   Google Scholar
• Efron, B. (2004), “Large-Scale Simultaneous Hypothesis Testing: The Choice of a Null Hypothesis,” Journal of the American Statistical Association, 99, 96–104.
   Web of Science ®Google Scholar
• Efron, B., Tibshirani, R., Storey, J.D., and Tusher, V. (2001), “Empirical Bayes Analysis of a Microarray Experiment,” Journal of the American Statistical Association, 96, 1151–1160.
   Web of Science ®Google Scholar
• Garel, B. (2001), “Likelihood Ratio Test for Univariate Gaussian Mixture,” Journal of Statistical Planning and Inference, 96, 325–350.
   Web of Science ®Google Scholar
• ——— (2005), “Asymptotic Theory of the Likelihood Ratio Test for the Identification of a Mixture,” Journal of Statistical Planning and Inference, 131, 271–296.
   Web of Science ®Google Scholar
• Ghosh, J.K., and Sen, P.K. (1985), “On the Asymptotic Performance of the Log-likelihood Ratio Statistic for the Mixture Model and Related Results,” in Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (vol. 2), eds. L. Le Cam, and R. Olshen, Belmont, CA: Wadsworth, pp. 789–806.
   Google Scholar
• Golub, T.R., Slonim, D.K., Tamayo, P., Huard, C., Gaasenbeek, M., Mesirov, J.P., Coller, H., Loh, M.L., Downing, J.R., Caligiuri, M.A., Bloomfield, C.D., and Lander, E.S. (1999), “Molecular Classification of Cancer: Class Discovery and Class Prediction by Gene Expression Monitoring,” Science, 286, 531–537.
   PubMed Web of Science ®Google Scholar
• Hartigan, J. (1985), “Failure of Log-likelihood Ratio Test,” in Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (vol. 2), eds. L. Le Cam, and R. Olshen, Berkeley, CA: University of California Press, pp. 807–810.
   Google Scholar
• Hathaway, R.J. (1985), “A Constrained Formulation of Maximum-Likelihood Estimation for Normal Mixture Distributions,” Annals of Statistics, 13, 795–800.
   Web of Science ®Google Scholar
• Henna, J. (1985), “On Estimating of the Number of Constituents of a Finite Mixture of Continuous Distributions,” The Annals of the Institute of Statistical Mathematics, 37, 235–240.
   Web of Science ®Google Scholar
• Jacobs, R.A., Jordan, M.I., Nowlan, S.J., and Hinton, G.E. (1991), “Adaptive Mixtures of Local Experts,” Neural Computation, 3, 79–87.
   PubMed Web of Science ®Google Scholar
• James, L.F., Priebe, C.E., and Marchette, D.J. (2001), “Consistent Estimation of Mixture Complexity,” The Annals of Statistics, 29, 1281–1296.
   Web of Science ®Google Scholar
• Keribin, C. (2000), “Consistent Estimation of the Order of Mixture Models,” Sankhya, Series A, 62, 49–62.
   Google Scholar
• Kiefer, J., and Wolfowitz, J. (1956), “Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters,” Annals of Mathematical Statistics, 27, 887–906.
   Google Scholar
• Kon, S. (1984), “Models of Stock Returns—A Comparison,” Journal of Finance, 39, 147–165.
   Web of Science ®Google Scholar
• Kon, S., and Jen, F. (1978), “Estimation of Time-Varying Systematic Risk and Performance for Mutual Fund Portfolios: An Application of Switching Regression,” Journal of Finance, 33, 457–475.
   Web of Science ®Google Scholar
• Lee, M.-L. T., Kuo, F.C., Whitmore, G.A., and Sklar, J. (2000), “Importance of Replication in Microarray Gene Expression Studies: Statistical Methods and Evidence From Repetitive cDNA Hybridizations,” Proceedings of the National Academy of Sciences of the United State of America, 9834–9838.
   PubMed Web of Science ®Google Scholar
• Lemdani, M., and Pons, O. (1997), “Likelihood Ratio Tests for Genetic Linkage,” Statistics and Probability Letters, 33, 15–22.
   Web of Science ®Google Scholar
• Li, P., and Chen, J. (2010), “Testing the Order of a Finite Mixture,” Journal of the American Statistical Association, 105, 1084–1092.
   Web of Science ®Google Scholar
• Li, P., Chen, J., and Marriott, P. (2009), “Non-Finite Fisher Information and Homogeneity: An EM Approach,” Biometrika, 96, 411–426.
   Web of Science ®Google Scholar
• Lindsay, B.G. (1995), Mixture Models: Theory, Geometry, and Applications, BethesdaMD: Institute of Mathematical Statistics.
   Google Scholar
• Lindsay, B.G., and Roeder, K. (1992), “Residual Diagnostics for Mixture Models,” Journal of the American Statistical Association, 87, 785–794.
   Web of Science ®Google Scholar
• Liu, X., and Shao, Y. (2003), “Asymptotics for Likelihood Ratio Tests Under Loss of Identifiability,” Annals of Statistics, 31, 807–832.
   Web of Science ®Google Scholar
• Maitra, R., and Melnykov, V. (2010), “Simulating Data to Study Performance of Finite Mixture Modeling and Model-Based Clustering Algorithms,” Journal of Computational and Graphical Statistics, 19, 354–376.
   Web of Science ®Google Scholar
• Mankiw, N.G., Romer, D., and Weil, D.N. (1992), “A Contribution to the Empirics of Economic Growth,” Quarterly Journal of Economics, 107, 407–437.
   Web of Science ®Google Scholar
• McLachlan, G., and Peel, D. (2000), Finite Mixture Models, New York: Wiley.
   Google Scholar
• Miloslavsky, M., and van der Laan, M.J. (2003), “Fitting of Mix- Tures With Unspecified Number of Components Using Cross-Validation Distance Estimate,” Computational Statistics and Data Analysis, 41, 413–428.
   Web of Science ®Google Scholar
• Quandt, R.E., and Ramsey, J.B. (1978), “Estimating Mixtures of Normal Distributions and Switching Regressions,” Journal of the American Statistical Association, 73, 730–738.
   Web of Science ®Google Scholar
• R Core Team (2014), R: A Language and Environment for Statistical Computing, Vienna, Austria: R Foundation for Statistical Computing.
   Google Scholar
• Roeder, K. (1994), “A Graphical Technique for Detecting the Number of Components in a Mixture of Normals,” Journal of the American Statistical Association, 89, 487–495.
   Web of Science ®Google Scholar
• Rotnitzky, A., Cox, D.R., Bottai, M., and Robins, J. (2000), “Likelihood-Based Inference With Singular Information Matrix,” Bernoulli, 6, 243–284.
   Web of Science ®Google Scholar
• Shen, J., and He, X. (2015), “Inference for Subgroup Analysis With a Structured Logistic-Normal Mixture Model,” Journal of the American Statistical Association, 110, 303–312.
   Web of Science ®Google Scholar
• Titterington, D.M., Smith, A. F.M., and Makov, U.E. (1985), Statistical Analysis of Finite Mixture Distributions, New York: Wiley.
   Google Scholar
• Tucker, A.L. (1992), “A Reexamination of Finite- and Infinite-Variance Distributions as Models of Daily Stock Returns,” Journal of Business and Economic Statistics, 10, 73–81.
   Web of Science ®Google Scholar
• Turner, T.R. (2000), “Estimating the Propagation Rate of a Viral Infection of Potato Plants via Mixtures of Regressions,” Journal of the Royal Statistical Society, Series C, 49, 371–384.
   Web of Science ®Google Scholar
• Venkataraman, S. (1997), “Value at Risk for a Mixture of Normal Distributions: The Use of Quasi-Bayesian Estimation Techniques,,” Economic Perspectives, Federal Reserve Bank of Chicago, 21, 2–13.
   Google Scholar
• White, H. (1982), “Maximum Likelihood Estimation of Misspecified Models,” Econometrica, 50, 1–25.
   Web of Science ®Google Scholar
• Windham, M.P., and Cutler, A. (1992), “Information Ratios for Validating Mixture Analysis,” Journal of the American Statistical Association, 87, 1188–1192.
   Web of Science ®Google Scholar
• Woo, M.-J., and Sriram, T.N. (2006), “Robust Estimation of Mixture Complexity,” Journal of the American Statistical Association, 101, 1475–1486.
   Web of Science ®Google Scholar
• Zhu, H., and Zhang, H. (2006), “Asymptotics for Estimation and Testing Procedures under Loss of Identifiability,” Journal of Multivariate Analysis, 97, 19–45.
   Web of Science ®Google Scholar
• Zhu, H.-T., and Zhang, H. (2004), “Hypothesis Testing in Mixture Regression Models,” Journal of the Royal Statistical Society, Series B, 66, 3–16.
   Google Scholar