Modified Cp Criterion for Optimizing Ridge and Smooth Parameters in the MGR Estimator for the Nonparametric GMANOVA Model

Author(s)
Isamu Nagai

ABSTRACT

Longitudinal trends of observations can be estimated using the generalized multivariate analysis of variance (GMANOVA) model proposed by [10]. In the present paper, we consider estimating the trends nonparametrically using known basis functions. Then, as in nonparametric regression, an overfitting problem occurs. [13] showed that the GMANOVA model is equivalent to the varying coefficient model with non-longitudinal covariates. Hence, as in the case of the ordinary linear regression model, when the number of covariates becomes large, the estimator of the varying coefficient becomes unstable. In the present paper, we avoid the overfitting problem and the instability problem by applying the concept behind penalized smoothing spline regression and multivariate generalized ridge regression. In addition, we propose two criteria to optimize hyper parameters, namely, a smoothing parameter and ridge parameters. Finally, we compare the ordinary least square estimator and the new estimator.

Longitudinal trends of observations can be estimated using the generalized multivariate analysis of variance (GMANOVA) model proposed by [10]. In the present paper, we consider estimating the trends nonparametrically using known basis functions. Then, as in nonparametric regression, an overfitting problem occurs. [13] showed that the GMANOVA model is equivalent to the varying coefficient model with non-longitudinal covariates. Hence, as in the case of the ordinary linear regression model, when the number of covariates becomes large, the estimator of the varying coefficient becomes unstable. In the present paper, we avoid the overfitting problem and the instability problem by applying the concept behind penalized smoothing spline regression and multivariate generalized ridge regression. In addition, we propose two criteria to optimize hyper parameters, namely, a smoothing parameter and ridge parameters. Finally, we compare the ordinary least square estimator and the new estimator.

KEYWORDS

Generalized ridge regression, GMANOVA model, Mallows' statistic, Non-iterative estimator, Shrinkage estimator, Varying coefficient model

Generalized ridge regression, GMANOVA model, Mallows' statistic, Non-iterative estimator, Shrinkage estimator, Varying coefficient model

Cite this paper

I. Nagai, "Modified Cp Criterion for Optimizing Ridge and Smooth Parameters in the MGR Estimator for the Nonparametric GMANOVA Model,"*Open Journal of Statistics*, Vol. 1 No. 1, 2011, pp. 1-14. doi: 10.4236/ojs.2011.11001.

I. Nagai, "Modified Cp Criterion for Optimizing Ridge and Smooth Parameters in the MGR Estimator for the Nonparametric GMANOVA Model,"

References

[1] A. C. Atkinson, “A note on the generalized information criterion for choice of a model,” Biometrika, vol. 67, no. 2, March 1980, pp. 413-418., pp. 291-293.

[2] P. J. Green and B. W. Silverman, “Nonparametric Regression and Generalized Linear Models,” Chapman & Hall/CRC, 1994.

[3] D. A. Harville, “Matrix Algebra from a Statistician’s Perspective,” New York Springer, 1997.

[4] A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics, vol. 12, No. 1, February 1970, pp. 55-67.

[5] A. M. Kshirsagar and W. B. Smith, “Growth Curves,” Marcel Dekker, 1995.

[6] J. F. Lawless, “Mean squared error properties of generalized ridge regression,” Journal of the American Statistical Association, vol. 76, no. 374, 1981, pp. 462-466.

[7] C. L. Mallows, “Some comments on Cp,” Technometrics, vol. 15, no. 1, November 1973, pp. 661-675.

[8] C. L. Mallows, “More comments on Cp,” Technometrics, vol. 37, no. 4, November 1995, pp. 362-372.

[9] I. Nagai, H. Yangihara and K. Satoh, “Optimization of Ridge Parameters in Multivariate Generalized Ridge Regression by Plug-in Methods,” TR 10-03, Statistical Research Group, Hiroshima University, 2010.

[10] R. F. Potthoff and S. N. Roy, “A generalized multivariate analysis of variance model useful especially for growth curve problems,” Biometrika, vol. 51, no. 3–4, December 1964, pp. 313-326.

[11] K. S. Riedel and K. Imre, “Smoothing spline growth curves with covariates,” Communications in Statistics – Theory and Methods, vol. 22, no. 7, 1993, pp. 1795-1818.

[12] F. J. Richard, “A flexible growth function for empirical use,” Journal of Experimental Botany, vol. 10, no. 2, 1959, pp. 290–301.

[13] K. Satoh and H. Yanagihara, “Estimation of varying coefficients for a growth curve model,” American Journal of Mathematical and Management Sciences, 2010 (in press).

[14] M. Siotani, T. Hayakawa and Y. Fujikoshi, “Modern Multivariate Statistical Analysis: A Graduate Course and Handbook,” American Sciences Press, Columbus, Ohio, 1985.

[15] R. S. Sparks, D. Coutsourides and L. Troskie, “The multivariate ,” Communications in Statistics - Theory and Methods, vol. 12, no. 15, 1983, pp. 1775-1793.

[16] Y. Takane, K. Jung and H. Hwang, “Regularized reduced rank growth curve models,” Computational Statistics and Data Analysis, vol. 55, no. 2, February 2011, pp. 1041-1052.

[17] H. Yanagihara and K. Satoh, “An unbiased Cp criterion for multivariate ridge regression,” Journal of Multivariate Analysis, vol. 101, no. 5, May 2010, pp. 1226-1238.

[18] H. Yanagihara, I. Nagai and K. Satoh, “A bias-corrected Cp criterion for optimizing ridge parameters in multivariate generalized ridge regression,” Japanese Journal of Applied Statistics, vol. 38, no. 3, October 2009, pp. 151-172 (in Japanese).

[1] A. C. Atkinson, “A note on the generalized information criterion for choice of a model,” Biometrika, vol. 67, no. 2, March 1980, pp. 413-418., pp. 291-293.

[2] P. J. Green and B. W. Silverman, “Nonparametric Regression and Generalized Linear Models,” Chapman & Hall/CRC, 1994.

[3] D. A. Harville, “Matrix Algebra from a Statistician’s Perspective,” New York Springer, 1997.

[4] A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics, vol. 12, No. 1, February 1970, pp. 55-67.

[5] A. M. Kshirsagar and W. B. Smith, “Growth Curves,” Marcel Dekker, 1995.

[6] J. F. Lawless, “Mean squared error properties of generalized ridge regression,” Journal of the American Statistical Association, vol. 76, no. 374, 1981, pp. 462-466.

[7] C. L. Mallows, “Some comments on Cp,” Technometrics, vol. 15, no. 1, November 1973, pp. 661-675.

[8] C. L. Mallows, “More comments on Cp,” Technometrics, vol. 37, no. 4, November 1995, pp. 362-372.

[9] I. Nagai, H. Yangihara and K. Satoh, “Optimization of Ridge Parameters in Multivariate Generalized Ridge Regression by Plug-in Methods,” TR 10-03, Statistical Research Group, Hiroshima University, 2010.

[10] R. F. Potthoff and S. N. Roy, “A generalized multivariate analysis of variance model useful especially for growth curve problems,” Biometrika, vol. 51, no. 3–4, December 1964, pp. 313-326.

[11] K. S. Riedel and K. Imre, “Smoothing spline growth curves with covariates,” Communications in Statistics – Theory and Methods, vol. 22, no. 7, 1993, pp. 1795-1818.

[12] F. J. Richard, “A flexible growth function for empirical use,” Journal of Experimental Botany, vol. 10, no. 2, 1959, pp. 290–301.

[13] K. Satoh and H. Yanagihara, “Estimation of varying coefficients for a growth curve model,” American Journal of Mathematical and Management Sciences, 2010 (in press).

[14] M. Siotani, T. Hayakawa and Y. Fujikoshi, “Modern Multivariate Statistical Analysis: A Graduate Course and Handbook,” American Sciences Press, Columbus, Ohio, 1985.

[15] R. S. Sparks, D. Coutsourides and L. Troskie, “The multivariate ,” Communications in Statistics - Theory and Methods, vol. 12, no. 15, 1983, pp. 1775-1793.

[16] Y. Takane, K. Jung and H. Hwang, “Regularized reduced rank growth curve models,” Computational Statistics and Data Analysis, vol. 55, no. 2, February 2011, pp. 1041-1052.

[17] H. Yanagihara and K. Satoh, “An unbiased Cp criterion for multivariate ridge regression,” Journal of Multivariate Analysis, vol. 101, no. 5, May 2010, pp. 1226-1238.

[18] H. Yanagihara, I. Nagai and K. Satoh, “A bias-corrected Cp criterion for optimizing ridge parameters in multivariate generalized ridge regression,” Japanese Journal of Applied Statistics, vol. 38, no. 3, October 2009, pp. 151-172 (in Japanese).