Abstract
This paper compares the Stein and the usual estimators of the error variance under the Pitman nearness (PN) criterion in a regression model which is mis-specified due to missing relevant explanatory variables. The exact expression of the PN-probability is derived and numerically evaluated. Contrary to the well-known result under mean squared errors (MSE), with the PN criterion the Stein variance estimator is uniformly dominated by the usual estimator when no relevant variables are excluded from the model. With an increased degree of model mis-specification, neither estimator strictly dominates the other.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chaturvedi A, Bhatti MS (1998) On the double k-class estimators in linear regression. J Quant Econ 14:53–58
Gelfand AE, Dey DK (1988) Improved estimation of the disturbance variance in a linear regression model. J Econom 39:387–395
Keating JP, Czitrom V (1989) A comparison of the James–Stein regression with least squares in the Pitman nearness sense. J Stat Comput Simul 34:1–9
Keating JP, Mason RL (1988) James–Stein estimator from an alternative perspective. Am Stat 42:160–164
Keating JP, Mason RL (2005) Pitman nearness comparison of the traditional estimator of the coefficient of determination and its adjusted version in linear regression models. Commun Stat-Theor Meth 34:367–374
Keating JP, Mason RL, Sen PK (1993) Pitman’s measure of closeness: a comparison of statistical estimators. Society for Industrial and Applied Mathematics, Philadelphia
Khattree R (1992) Comparing estimators for population variance using the Pitman nearness. Am Stat 46:214–217
Kumar M, Srivastava VK (2004) Pitman nearness and concentration probability comparisons of the sample coefficient of determination and its adjusted version in linear regression models. Commun Stat-Theor Meth 33:1629–1641
Namba A, Ohtani K (2007) Risk comparison of the Stein-rule estimator in a linear regression model with omitted relevant regressors and multivariate t errors under the Pitman nearness criterion. Stat Pap 48:151–162
Ohtani K (1988) Optimal levels of significance of a pre-test in estimating the disturbance variance after the pre-test for a linear hypothesis on coefficients in a linear regression. Econ Lett 28:151–156
Ohtani K (2002) Exact distribution of a pre-test estimator for regression error variance when there are omitted variables. Stat Probab Lett 60:129–140
Pitman EJG (1937) The closest estimates of statistical parameters. In: Proceedings of the Cambridge philosophical society. 33:212–222
Rao CR (1981) Some comments on the minimum mean square error as a criterion of estimation. In: Csorge M, Dawson DA, Rao JNK, Saleh AKMdE (eds) Statistics and related topics, North Holland, Amsterdam, pp 123–143
Rao CR, Srivastava VK, Toutenburg H (1998) Pitman nearness comparisons of Stein-type estimators for regression coefficients in replicated experiments. Stat Pap 39:61–74
Sen PK, Kubokawa T, Saleh AKME (1989) The Stein paradox in the Pitman closeness. Ann Stat 17:1375–1386
Stein C (1964) Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Ann Instit Stat Math 16:155–160
Sugiura N (1993) Pitman closeness of the equivariant shrinkage estimators of the normal variance. In: Matsusita K, Puru ML, Hayakawa T (eds) Statistical science and data analysis, Proceedings of the third pacific area statistical conference, Interscience, Utrecht
Tran Van Hoa, Chaturvedi A (1997) Performance of the 2SHI estimator under the generalized Pitman nearness criterion. Comm Stat-Theor Meth 26:1227–1238
Tripathi TP, Khattree R (1992) Estimation of a parameter using Pitman nearness criterion. J Stat Plann Infer 32:281–289
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are grateful to two anonymous referees for their valuable comments. Also, the first author is grateful to the Japan Society for the Promotion of Science for partial financial support.
Rights and permissions
About this article
Cite this article
Ohtani, K., Wan, A.T.K. Comparison of the Stein and the usual estimators for the regression error variance under the Pitman nearness criterion when variables are omitted. Stat Papers 50, 151–160 (2009). https://doi.org/10.1007/s00362-007-0047-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-007-0047-6