Goodness of fit for generalized shrinkage estimation
Authors:
C.-L. Cheng, Shalabh and A. Chaturvedi
Journal:
Theor. Probability and Math. Statist. 100 (2020), 191-214
MSC (2010):
Primary 62J07, 62J05
DOI:
https://doi.org/10.1090/tpms/1106
Published electronically:
August 5, 2020
Full-text PDF
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References |
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Additional Information
Abstract: The present paper develops a goodness-of-fit statistic for the linear regression models fitted by the shrinkage type estimators. A family of double $k$-class estimators is considered as a shrinkage estimator which encompasses several estimators as its particular case. The covariance matrix of error term is assumed to be a non-identity matrix under two situations, known and unknown. The goodness-of-fit statistics based on the idea of coefficient of determination in a multiple linear regression model is proposed for the family of double $k$-class estimators. Its first and second order moments up to the first order of approximation are derived, and finite sample properties are studied using the Monte-Carlo simulation.
References
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References
- A. K. Srivastava, V. K. Srivastava, and A. Ullah, The coefficient of determination and its adjusted version in linear regression models, Econometric Reviews 14 (1995), 229–240. MR 1341247
- A. T. K. Wan and A. Chaturvedi, Double $k$-class estimators in regression models with non-spherical disturbances, J. Multivar. Anal. 79 (2001), no. 2, 226–250. MR 1868290
- A. Ullah and S. Ullah, Double $k$-class estimators of coefficients in linear regression, Econometrica 46 (1978), no. 3, 705–722. MR 491910
- A. Ullah and S. Ullah, Errata: Double $k$-class estimators of coefficients in linear regression, Econometrica 49 (1978), no. 2, 554. MR 640530
- A. Ullah and V. K. Srivastava, Moments of the ratio of quadratic forms in non-normal variables with econometric examples, J. Econometrics 52 (1994), 129–142. MR 1279727
- A. Chaturvedi and Shalabh, Risk and Pitman closeness properties of feasible generalized double $k$-class estimators in linear regression models with non-spherical disturbances under balanced loss function, J. Multivar. Anal. 90 (2004), no. 2, 229–256. MR 2081777
- C.-L. Cheng, Shalabh, and G. Garg, Coefficient of determination for multiple measurement error models, J. Multivar. Anal. 126 (2014), 137–152. MR 3173087
- C.-L. Cheng, Shalabh, and G. Garg, Goodness of fit in restricted measurement error models, J. Multivar. Anal. 145 (2016), 101–116. MR 3459941
- J. S. Crämer, Mean and variance of $R^2$ in small and moderate samples, J. Econometrics 35 (1987), 253–266. MR 903186
- K. Ohtani, Minimum mean squared error estimation of each individual coefficients in a linear regression model, J. Statist. Plann. Inference 62 (1997), 301–316. MR 1468168
- K. Ohtani, Exact small sample properties of an operational variant of the minimum mean squared error estimator, Communications in Statistics (Theory and Methods) 25 (1996), 1223–1231. MR 1394280
- K. Ohtani and D. E. A. Giles, The absolute error risks of regression “goodness of fit” measures, J. Quantitative Economics 12 (1996), 17–26.
- K. Ohtani and H. Hasegawa, On small scale properties of $R^2$ in a linear regression model with multivariate $t$ errors and proxy variables, Econometric Theory 9 (1993), 504–515. MR 1241988
- M. D. Smith, Comparing approximations to the expectation of quadratic forms in normal variables, Econometric Reviews 15 (1996), 81–95. MR 1379463
- R. A. L. Carter, V. K. Srivastava, and A. Chaturvedi, Selecting a double k-class estimator for regression coefficients, Stat. Probab. Lett. 18 (1993), no. 5, 363–371 MR 1247447
- R. W. Farebrother, The minimum mean square error linear estimator and ridge regression, Technometrics 17 (1975), 127–128. MR 375646
- Shalabh, G. Garg, and C. Heumann, Performance of double $k$-class estimators for coefficients in linear regression models with non-spherical disturbances under asymmetric losses, J. Multivar. Anal. 112 (2012), 35–47. MR 2957284
- T. J. Rothenberg, Hypothesis Testing in Linear Models when the Error Covariance Matrix is Nonscalar, Econometrica 52 (1984), no. 4, 827–842. MR 750362
- T. W. Anderson, An Introduction to Multivariate Analysis, John Wiley, New Jersey. 2003. MR 1990662
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Additional Information
C.-L. Cheng
Affiliation:
Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, R.O.C.
Email:
clcheng@stat.sinica.edu.tw
Shalabh
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur - 208 016, India
Email:
shalab@iitk.ac.in
A. Chaturvedi
Affiliation:
Department of Statistics, Allahabad University, Allahabad - 211 002, India
Email:
anoopchaturv@gmail.com
Keywords:
Linear regression,
non-spherical disturbances,
coefficient of determination ($R^2$),
shrinkage estimation,
generalized least squares estimator,
feasible double $k$-class estimators,
feasible generalized least squares estimator,
double $k$-class estimators,
goodness of fit
Received by editor(s):
December 20, 2018
Published electronically:
August 5, 2020
Article copyright:
© Copyright 2020
American Mathematical Society