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Prediction and calibration for multiple correlated variables. (English) Zbl 1422.62247
Summary: The standard approach for prediction of multiple correlated outcome measures overpredicts the unknown observation in the linear model setup if associated covariate measures follow a certain distribution. It is desired to have a nonempty confidence region when some covariate measures are missing and required to be estimated. This article develops a methodology for prediction and proposes a shrinkage predictor with a smaller risk compared to the one based on the maximum likelihood estimate. It also provides an algorithm for constructing a nonempty confidence region for unknown covariates. Proposed methodology is shown to perform well in terms of maintaining a smaller risk in prediction and the coverage probability in calibration. Results are illustrated with a recent behavioral science dataset.
MSC:
62J07 Ridge regression; shrinkage estimators (Lasso)
62M20 Inference from stochastic processes and prediction
Software:
pls; ridge; shrink
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[1] Allegrini, F.; Wentzell, P. D.; Olivieri, A. C., Generalized error-dependent prediction uncertainty in multivariate calibration, Anal. Chim. Acta, 903, 51-60, (2016)
[2] Atkins, M. S.; Shernoff, E. S.; Frazier, S. L.; Schoenwald, S. K.; Cappella, E.; Marinez-Lora, A.; Mehta, T. G.; Lakind, D.; Cua, G.; Bhaumik, R.; Bhaumik, D., Redesigning community mental health services for urban children: Supporting schooling to promote mental health, J. Consult. Clin. Psychol., 83, 839-852, (2015)
[3] Benton, D.; Krishnamoorthy, K.; Mathew, T., Inferences in multivariate-univariate calibration problems, J. R. Stat. Soc. Ser. D, 52, 15-39, (2003)
[4] Brandwein, A. C.; Strawderman, W. E., Stein estimation for spherically symmetric distributions: Recent developments, Statist. Sci., 27, 11-23, (2012) · Zbl 1330.62285
[5] Breiman, L.; Friedman, J. H., Predicting multivariate responses in multiple linear regression, J. R. Stat. Soc. Ser. B Stat. Methodol., 59, 3-54, (1997) · Zbl 0897.62068
[6] Brown, P. J., Multivariate calibration, J. R. Stat. Soc. Ser. B Stat. Methodol., 44, 287-321, (1982) · Zbl 0511.62083
[7] Brown, L. D.; Zhao, L. H., A geometrical explanation of Stein shrinkage, Statist. Sci., 27, 24-30, (2012) · Zbl 1330.62282
[8] Cai, T. T., Minimax and adaptive inference in nonparametric function estimation, Statist. Sci., 27, 31-50, (2012) · Zbl 1330.62059
[9] Casella, G.; Hwang, J. T.G., Shrinkage confidence procedures, Statist. Sci., 27, 51-60, (2012) · Zbl 1330.62283
[10] Chételat, D.; Wells, M. T., Improved multivariate normal mean estimation with unknown covariance when \(p\) is greater than \(n,\) Ann. Statist., 40, 3137-3160, (2012) · Zbl 1296.62048
[11] Constantine, A. G., The distribution of Hotelling’s generalised \(T_0^2,\) Ann. Math. Stat., 37, 215-225, (1966) · Zbl 0138.13902
[12] Copas, J., Regression, prediction and shrinkage, J. R. Stat. Soc. Ser. B Stat. Methodol., 45, 311-354, (1983) · Zbl 0532.62048
[13] Cule, E.; Moritz, S., ridge: Ridge Regression with Automatic Selection of the Penalty Parameter, (2018), https://CRANR-project.org/package=ridge, R package version 23
[14] Datta, G.; Ghosh, M., Small area shrinkage estimation, Statist. Sci., 27, 95-114, (2012) · Zbl 1330.62286
[15] Davis, A.; Hayakawa, T., Some distribution theory relating to confidence regions in multivariate calibration, Ann. Inst. Statist. Math., 39, 141-152, (1987) · Zbl 0656.62058
[16] Dunkler, D.; Sauerbrei, W.; Heinze, G., Global, parameterwise and joint shrinkage factor estimation, J. Stat. Softw., 69, 1-19, (2016)
[17] Fourdrinier, D.; Wells, M. T., On improved loss estimation for shrinkage estimators, Statist. Sci., 27, 61-81, (2012) · Zbl 1330.62287
[18] Fujikoshi, Y.; Nishii, R., On the distribution of a statistic in multivariate inverse regression analysis, Hiroshima Math. J., 14, 215-225, (1984) · Zbl 0561.62050
[19] George, E. I.; Liang, F.; Xu, X., From minimax shrinkage estimation to minimax shrinkage prediction, Statist. Sci., 27, 82-94, (2012) · Zbl 1330.62288
[20] George, E. I.; Strawderman, W. E., A tribute to Charles Stein, Statist. Sci., 27, 1-2, (2012) · Zbl 1332.00110
[21] Hotelling, H., Multivariate quality control, illustrated by the air testing of sample bombsights, (Eisenhart, C.; Hastay, M. W.; Wallis, W. A., Techniques of Statistical Analysis, (1947), McGraw-Hill: McGraw-Hill New York), 111-184
[22] James, W.; Stein, C., Estimation with quadratic loss, (Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Contributions to the Theory of Statistics, vol. 1, (1961), University of California Press: University of California Press Berkeley, CA), 361-379 · Zbl 1281.62026
[23] LaMotte, L. R.; Wells, J. D., Inverse prediction for multivariate mixed models with standard software, Statist. Papers, 57, 929-938, (2016) · Zbl 1351.62115
[24] Martens, H.; Naes, T., Multivariate Calibration, (1989), Wiley: Wiley Chichester · Zbl 0732.62109
[25] Mathew, T.; Kasala, S., An exact confidence region in multivariate calibration, Ann. Statist., 22, 94-105, (1994) · Zbl 0795.62023
[26] Mathew, T.; Sharma, M. K., Joint confidence regions in the multivariate calibration problem, J. Statist. Plann. Inference, 100, 427-441, (2002) · Zbl 0987.62042
[27] Mathew, T.; Sharma, M. K.; Nordstrom, K., Tolerance regions and multiple-use confidence regions in multivariate calibration, Ann. Statist., 26, 1989-2013, (1998) · Zbl 0929.62071
[28] Mathew, T.; Zha, W., Conservative confidence regions in multivariate calibration, Ann. Statist., 24, 707-725, (1996) · Zbl 0859.62062
[29] Mathew, T.; Zha, W., Multiple use confidence regions in multivariate calibration, J. Amer. Statist. Assoc., 92, 1141-1150, (1997) · Zbl 1067.62522
[30] Mevik, B.-H.; Wehrens, R.; Liland, K. H., pls: Partial Least Squares and Principal Component Regression, (2018), https://CRANR-project.org/package=pls, R Package version 27-0
[31] Morris, C. N.; Lysy, M., Shrinkage estimation in multilevel normal models, Statist. Sci., 27, 115-134, (2012) · Zbl 1330.62290
[32] Muirhead, R. J., Aspects of Multivariate Statistical Theory, (1982), Wiley: Wiley New York · Zbl 0556.62028
[33] Oman, S. D., Confidence regions in multivariate calibration, Ann. Statist., 16, 174-187, (1988) · Zbl 0637.62033
[34] Peng, J.; Guo, L.; Hu, Y.; Rao, K.; Xie, Q., Maximum correntropy criterion based regression for multivariate calibration, Chemometr. Intell. Lab., 161, 27-33, (2017)
[35] Sundberg, R.; Brown, P. J., Multivariate calibration with more variables than observations, Technometrics, 31, 365-371, (1989)
[36] Tan, X., Improved minimax estimation of a multivariate normal mean under heteroscedasticity, Bernoulli, 21, 574-603, (2015) · Zbl 1311.62086
[37] Tibshirani, R. J., Regression shrinkage and selection via the Lasso, J. R. Stat. Soc. Ser. B Stat. Methodol., 58, 267-288, (1994) · Zbl 0850.62538
[38] Uhlig, H., On singular Wishart and singular multivariate beta distributions, Ann. Statist., 22, 1, 395-405, (1994) · Zbl 0795.62052
[39] Wasserman, L. A., All of Nonparametric Statistics, (2006), Springer: Springer New York · Zbl 1099.62029
[40] Wei, W.; Balabdaoui, F.; Held, L., Calibration tests for multivariate Gaussian forecasts, J. Multivariate Anal., 154, 216-233, (2017) · Zbl 1352.62087
[41] Wu, X.; Liu, Z.; Li, H., A novel algorithm for linear multivariate calibration based on the mixed model of samples, Anal. Chim. Acta, 801, 43-47, (2013)
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