Finite Mixture of Heteroscedastic Single-Index Models

Author(s)
Peng Zeng

ABSTRACT

In many applications a heterogeneous population consists of several subpopulations. When each subpopulation can be adequately modeled by a heteroscedastic single-index model, the whole population is characterized by a finite mixture of heteroscedastic single-index models. In this article, we propose an estimation algorithm for fitting this model, and discuss the implementation in detail. Simulation studies are used to demonstrate the performance of the algorithm, and a real example is used to illustrate the application of the model.

In many applications a heterogeneous population consists of several subpopulations. When each subpopulation can be adequately modeled by a heteroscedastic single-index model, the whole population is characterized by a finite mixture of heteroscedastic single-index models. In this article, we propose an estimation algorithm for fitting this model, and discuss the implementation in detail. Simulation studies are used to demonstrate the performance of the algorithm, and a real example is used to illustrate the application of the model.

Cite this paper

P. Zeng, "Finite Mixture of Heteroscedastic Single-Index Models,"*Open Journal of Statistics*, Vol. 2 No. 1, 2012, pp. 12-20. doi: 10.4236/ojs.2012.21002.

P. Zeng, "Finite Mixture of Heteroscedastic Single-Index Models,"

References

[1] W. H?rdle and T. M. Stoker, “Investigating Smooth Multiple Regression by the Method of Average Derivatives,” Journal of the American Statistical Association, Vol. 84, No. 408, 1989, pp. 986-995. doi:10.2307/2290074

[2] W. K. Newey and T. M. Stoker, “Efficiency of Weighted Average Derivative Estimators and Index Models,” Econometrica, Vol. 61, No. 5, 1993, pp. 1199-1223. doi:10.2307/2951498

[3] Y. Xia, “Asymptotic Distributions for Two Estimators of the Single-Index Model,” Econometric Theory, Vol. 22, No. 6, 2006, pp. 1112-1137. doi:10.1017/S0266466606060531

[4] Y. Xia, H. Tong, and W. K. Li, “Single-Index Volatility Models and Estimation,” Statistica Sinica, Vol. 12, 2002, pp. 785-799.

[5] R. E. Quandt and J. B. Ramsey, “Estimating Mixtures of Normal Distributions and Swithcing Regressions (with Discussions),” Journal of the American Statistical Association, Vol. 73, No. 364, 1978, pp. 730-752. doi:10.2307/2286266

[6] W. S. DeSarbo and W. L. Cron, “A Maximum Likelihood Methodology for Clusterwise Linear Regression,” Journal of Classification, Vol. 5, No. 2, 1988, pp. 248-282. doi:10.1007/BF01897167

[7] R. A. Jacobs, M. I. Jordan, S. J. Nowland and G. E. Hinton, “Adaptive Mixtures of Local Experts,” Neural Computation, Vol. 3, No. 1, 1991, pp. 79-87. doi:10.1162/neco.1991.3.1.79

[8] P. Wang, M. L. Puterman, I. Cockburn and N. Le, “Mixed Poisson Regression Models with Covariate Dependent Rates,” Biometrics, Vol. 52, No. 2, 1996, pp. 381-400. doi:10.2307/2532881

[9] G. J. McLachlan and D. Peel, “Finite Mixture Models,” John Wiley & Sons, New York, 2000. doi:10.1002/0471721182

[10] A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximum Likelihood from Incomplete Data via the EM Algorithm (with Discussion),” Journal of the Royal Statistical Society, Series B, Vol. 39, 1977, pp. 1-38.

[11] C. F. J. Wu, “On the Convergence Properties of the EM Algorithm,” The Annals of Statistics, Vol. 11, No. 1, 1983, pp. 95-103. doi:10.1214/aos/1176346060

[12] J. Fan and I. Gijbels, “Local Polynomial Modelling and its Applications,” Chapman & Hall Ltd, London, 1996.

[13] P. Hall and R. J. Carroll, “Variance Function Estimation in Regression: The Effect of Estimating the Mean,” Journal of the Royal Statistical Society, Series B, Vol. 51, 1989, pp. 3-14.

[14] D. Ruppert, M. P. Wand, U. Holst, and O. H¨ossjer, “Local Polynomial Variance Function Estimation,” Technometrics, Vol. 39, No. 3, 1997, pp. 262-273. doi:10.2307/1271131

[15] J. Fan and Q. Yao, “Efficient Estimation of Conditional Variance Functions in Stochastic Regression,” Biometrika, Vol. 85, No. 3, 1998, pp. 645-660. doi:10.1093/biomet/85.3.645

[16] J. H. Friedman and W. Stuetzle, “Projection Pursuit Regression,” Journal of the American Statistical Association, Vol. 76, No. 376, 1981, pp. 817-823. doi:10.2307/2287576

[17] R. D. Cook, “Regression Graphics: Ideas For Studying Regressions Through Graphics,” John Wiley and Sons, New York, 1998.

[18] K.-C. Li, “Sliced Inverse Regression for Dimension Reduction (with Discussion),” Journal of the American Statistical Association, Vol. 86, No. 414, 1991, pp. 316-342. doi:10.2307/2290563

[19] Y. Zhu and P. Zeng, “Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression,” Journal of the American Statistical Association, Vol. 101, No. 476, 2006, pp. 1638-1651. doi:10.1198/016214506000000140

[20] D. Ruppert, S. J. Sheather and M. P. Wand, “An Effective Bandwidth Selector for Local Least Squares Regression,” Journal of the American Statistical Association, Vol. 90, No. 432, 1995, pp. 1257-1270. doi:10.2307/2291516

[21] B. W. Silverman, “Density Estimation for Statistics and Data Analysis,” Chapman & Hall Ltd, London, 1996.

[1] W. H?rdle and T. M. Stoker, “Investigating Smooth Multiple Regression by the Method of Average Derivatives,” Journal of the American Statistical Association, Vol. 84, No. 408, 1989, pp. 986-995. doi:10.2307/2290074

[2] W. K. Newey and T. M. Stoker, “Efficiency of Weighted Average Derivative Estimators and Index Models,” Econometrica, Vol. 61, No. 5, 1993, pp. 1199-1223. doi:10.2307/2951498

[3] Y. Xia, “Asymptotic Distributions for Two Estimators of the Single-Index Model,” Econometric Theory, Vol. 22, No. 6, 2006, pp. 1112-1137. doi:10.1017/S0266466606060531

[4] Y. Xia, H. Tong, and W. K. Li, “Single-Index Volatility Models and Estimation,” Statistica Sinica, Vol. 12, 2002, pp. 785-799.

[5] R. E. Quandt and J. B. Ramsey, “Estimating Mixtures of Normal Distributions and Swithcing Regressions (with Discussions),” Journal of the American Statistical Association, Vol. 73, No. 364, 1978, pp. 730-752. doi:10.2307/2286266

[6] W. S. DeSarbo and W. L. Cron, “A Maximum Likelihood Methodology for Clusterwise Linear Regression,” Journal of Classification, Vol. 5, No. 2, 1988, pp. 248-282. doi:10.1007/BF01897167

[7] R. A. Jacobs, M. I. Jordan, S. J. Nowland and G. E. Hinton, “Adaptive Mixtures of Local Experts,” Neural Computation, Vol. 3, No. 1, 1991, pp. 79-87. doi:10.1162/neco.1991.3.1.79

[8] P. Wang, M. L. Puterman, I. Cockburn and N. Le, “Mixed Poisson Regression Models with Covariate Dependent Rates,” Biometrics, Vol. 52, No. 2, 1996, pp. 381-400. doi:10.2307/2532881

[9] G. J. McLachlan and D. Peel, “Finite Mixture Models,” John Wiley & Sons, New York, 2000. doi:10.1002/0471721182

[10] A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximum Likelihood from Incomplete Data via the EM Algorithm (with Discussion),” Journal of the Royal Statistical Society, Series B, Vol. 39, 1977, pp. 1-38.

[11] C. F. J. Wu, “On the Convergence Properties of the EM Algorithm,” The Annals of Statistics, Vol. 11, No. 1, 1983, pp. 95-103. doi:10.1214/aos/1176346060

[12] J. Fan and I. Gijbels, “Local Polynomial Modelling and its Applications,” Chapman & Hall Ltd, London, 1996.

[13] P. Hall and R. J. Carroll, “Variance Function Estimation in Regression: The Effect of Estimating the Mean,” Journal of the Royal Statistical Society, Series B, Vol. 51, 1989, pp. 3-14.

[14] D. Ruppert, M. P. Wand, U. Holst, and O. H¨ossjer, “Local Polynomial Variance Function Estimation,” Technometrics, Vol. 39, No. 3, 1997, pp. 262-273. doi:10.2307/1271131

[15] J. Fan and Q. Yao, “Efficient Estimation of Conditional Variance Functions in Stochastic Regression,” Biometrika, Vol. 85, No. 3, 1998, pp. 645-660. doi:10.1093/biomet/85.3.645

[16] J. H. Friedman and W. Stuetzle, “Projection Pursuit Regression,” Journal of the American Statistical Association, Vol. 76, No. 376, 1981, pp. 817-823. doi:10.2307/2287576

[17] R. D. Cook, “Regression Graphics: Ideas For Studying Regressions Through Graphics,” John Wiley and Sons, New York, 1998.

[18] K.-C. Li, “Sliced Inverse Regression for Dimension Reduction (with Discussion),” Journal of the American Statistical Association, Vol. 86, No. 414, 1991, pp. 316-342. doi:10.2307/2290563

[19] Y. Zhu and P. Zeng, “Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression,” Journal of the American Statistical Association, Vol. 101, No. 476, 2006, pp. 1638-1651. doi:10.1198/016214506000000140

[20] D. Ruppert, S. J. Sheather and M. P. Wand, “An Effective Bandwidth Selector for Local Least Squares Regression,” Journal of the American Statistical Association, Vol. 90, No. 432, 1995, pp. 1257-1270. doi:10.2307/2291516

[21] B. W. Silverman, “Density Estimation for Statistics and Data Analysis,” Chapman & Hall Ltd, London, 1996.