Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
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CONFLICT AMONG THE SHRINKAGE ESTIMATORS INDUCED BY W, LR AND LM TESTS UNDER A STUDENT'S t REGRESSION MODEL

  • Kibria, B.M.-Golam (Department of Statistics, Florida International University)
  • Published : 2004.12.01
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Abstract

The shrinkage preliminary test ridge regression estimators (SPTRRE) based on Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests for estimating the regression parameters of the multiple linear regression model with multivariate Student's t error distribution are considered in this paper. The quadratic biases and risks of the proposed estimators are compared under both null and alternative hypotheses. It is observed that there is conflict among the three estimators with respect to their risks because of certain inequalities that exist among the test statistics. In the neighborhood of the restriction, the SPTRRE based on LM test has the smallest risk followed by the estimators based on LR and W tests. However, the SPTRRE based on W test performs the best followed by the LR and LM based estimators when the parameters move away from the subspace of the restrictions. Some tables for the maximum and minimum guaranteed efficiency of the proposed estimators have been given, which allow us to determine the optimum level of significance corresponding to the optimum estimator among proposed estimators. It is evident that in the choice of the smallest significance level to yield the best estimator the SPTRRE based on Wald test dominates the other two estimators.

Keywords

References

  1. AHMED, S. E. (2002). 'Simultaneous estimation of coefficients of variation', Journal of Statistical Planning and Inference, 104, 31-51 https://doi.org/10.1016/S0378-3758(01)00121-5
  2. ALDRIN, M. (1997). 'Length modified ridge regression', Computational Statistics and Data Analysis, 25, 377-398 https://doi.org/10.1016/S0167-9473(97)00015-7
  3. ANDERSON, T. W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd ed., John Wiley & Sons, New York
  4. BANCROFT, T. A. (1944). 'On biases in estimation due to the use of preliminary tests of significance', The Annals of Mathematical Statistics, 15, 190-204 https://doi.org/10.1214/aoms/1177731284
  5. BANCROFT, T. A. (1964). 'Analysis and inference for incompletely specified models involving the use of preliminary teste(s) of significance', Biometrics, 20, 427-442 https://doi.org/10.2307/2528486
  6. BERNDT, E. R. AND SAVIN, N. E. (1977). 'Conflict among criteria for testing hypotheses in the multivariate linear regression model', Econometrica, 45, 1263-1277 https://doi.org/10.2307/1914072
  7. BILLAH, B. AND SALEH, A. K. MD. E. (2000). 'Performance of the PTE based on the conflicting W, LR and LM tests in regression model', In Advances on Methodological and Applied Aspects of Probability and Statistics (N. Balakrishnan, ed.), Gordon and Breach Science Publishers
  8. BLATTBERG, R. C. AND GONEDES, N. J. (1974). 'A comparison of the stable and Student distributions as statistical models for stock prices', Journal of Business, 47, 244-280 https://doi.org/10.1086/295634
  9. EVANS, G. B. A. AND SAVIN, N. E. (1982). 'Conflict among the criteria revisited; The W, LR and LM tests', Econometrica, 50, 737-748 https://doi.org/10.2307/1912611
  10. FOUCART, T. (1999). 'Stability of the inverse correlation matrix. Partial ridge regression', Journal of Statistical Planning and Inference, 77, 141-154 https://doi.org/10.1016/S0378-3758(98)00195-5
  11. GIBBONS, D. G. (1981). 'A simulation study of some ridge estimators', Journal of the American Statistical Association, 76, 131-139 https://doi.org/10.2307/2287058
  12. GILES, J. A. (1991). 'Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances', Journal of Econometrics, 50, 377-398 https://doi.org/10.1016/0304-4076(91)90026-A
  13. GNANADESIKAN, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations, John Wiley & Sons, New York
  14. GUNST, R. (2000). 'Classical studies that revolutionized the practice of regression analysis', Technometrics, 42, 62-64 https://doi.org/10.2307/1271433
  15. HAN, C-P. AND BANCROFT, T. A. (1968). 'On pooling means when variance is unknown', Journal of the American Statistical Association, 63, 1333-1342 https://doi.org/10.2307/2285888
  16. HAQ, M. S. AND KIBRIA, B. M. G. (1996). 'A shrinkage estimator for the restricted linear regression model: Ridge regression approach', Journal of Applied Statistical Science, 3, 301-316
  17. HOERL, A. E. AND KENNARD, R. W. (1970). 'Ridge regression : Biased estimation for nonorthogonal problems', Technometrics, 12, 55-67 https://doi.org/10.2307/1267351
  18. JUDGE, G. G. AND BOCK, M. E. (1978). The Statistical Implications of Pre-test and Stein-rule Estimators in Econometrics, North-Holland Publishing Company, Amsterdam
  19. KELKER, D. (1970). 'Distribution theory of spherical distributions and a location-scale parameter generalization', Sankhya, A32, 419-438
  20. KENNEDY, P. (1998). A Guide to Econometrics, 4th ed., The MIT Press, Cambridge, Massachusetts
  21. KIBRIA, B. M. G. (2003). 'Performance of some new ridge regression estimators', Communications in Statistics-Simulation and Computation, 32, 419-435 https://doi.org/10.1081/SAC-120017499
  22. KIBRIA, B. M. G. AND AHMED, S. E. (1997). 'Shrinkage estimation for the multicollinear observations in a regression model with multivariate t disturbances', Journal of Statistical Research, 31, 83-102
  23. KIBRIA, B. M. G. AND SALEH, A. K. MD. E. (1993). 'Performance of shrinkage preliminary test estimator in regression analysis', Jahangirnagar Review, A17, 133-148
  24. KIBRIA, B. M. G. AND SALEH, A. K. MD. E. (2003a). 'Estimation of the mean vector of a multivariate normal distribution under various test statistics', Journal of Probability and Statistical Science, 1, 141-155
  25. KIBRIA, B. M. G. AND SALEH, A. K. MD. E. (2003b). 'Effect of W, LR, and LM tests on the performance of preliminary test ridge regression estimators', Journal of the Japan Statistical Society, 33, 119-136 https://doi.org/10.14490/jjss.33.119
  26. KING, M. L. (1980). 'Robust tests for spherical symmetry and their application to least squares regression', The Annals of Statistics, 8, 1265-1271 https://doi.org/10.1214/aos/1176345199
  27. SALEH, A. K. MD. E. AND HAN, C-P. (1990). 'Shrinkage estimation in regression analysis', Estadistica, 42, 40-63
  28. SALEH, A. K. MD. E. AND KIBRIA, B. M. G. (1993). 'Performances of some new preliminary test ridge regression estimators and their properties', Communications in StatisticsTheory and Methods, 22, 2747-2764 https://doi.org/10.1080/03610929308831183
  29. SARKAR, N. (1992). 'A new estimator combining the ridge regression and the restricted least squares methods of estimation', Communications in Statistics-Theory and Methods, 21, 1987-2000 https://doi.org/10.1080/03610929208830893
  30. SAVIN, N. E. (1976). 'Conflict among testing procedures in a linear regression model with autoregressive disturbances', Econometrica, 44, 1303-1315 https://doi.org/10.2307/1914262
  31. SEN, P. K. AND SALEH, A. K. MD. E. (1987). 'On preliminary test and shrinkage Mestimation in linear models', The Annals of Statistics, 15, 1580-1592 https://doi.org/10.1214/aos/1176350611
  32. ULLAH, A. AND ZINDE-WALSH, V. (1984). 'On the robustness of LM, LR and W tests in regression models', Econometrica, 52, 1055-1066 https://doi.org/10.2307/1911199
  33. ZELLNER, A. (1976). 'Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms', Journal of the American Statistical Association, 71, 400-405 https://doi.org/10.2307/2285322