Abstract
We consider the estimation of error variance in the analysis of experiments using two level orthogonal arrays. We address the estimator which is the minimum of all the estimators which we obtain by pooling some sums of squares for factorial effects. Under squared error loss, we discuss whether or not this estimator uniformly improves upon the best positive multiple of error sum of squares. We show that when we have two factorial effects, we obtain uniform improvement. However, we show that when we have more than two factorial effects, we cannot necessarily obtain uniform improvement. Further, the above results are applied to the problem of estimating the smallest scale parameter of chi-square distributions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barlow, R.E., Bartholomew, D.J., Bremner, J.M., Brunk, H.D. (1972). R.A. Bradley, D.G. Kendall, J.S. Hunter, G.S. Watson (Eds.), Statistical inference under order restrictions: The theory and application of isotonic regression. New York: Wiley
Brewster J.F., Zidek J.V. (1974). Improving on equivariant estimators. The Annals of Statistics 2:21–38
Brown L. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. The Annals of Mathematical Statistics 39:29–48
Chang Y-T., Shinozaki N. (2002). A comparison of restricted and unrestricted estimators in estimating linear functions of ordered scale parameters of two gamma distributions. Annals of the Institute of Statistical Mathematics 54:848–860
Cohen A., Kemperman J.H.B., Sackrowitz H.B. (2000). Properties of likelihood inference for order restricted models. Journal of Multivariate Analysis 72:50–77
Efron B., Morris C. (1976). Families of minimax estimators of the mean of a multivariate normal distributions. The Annals of Statistics 4:11–21
Gelfand A.E., Dey D.K. (1988). Improved estimation of the disturbance variance in a linear regression model. Journal of Econometrics 39:387–395
George E.I. (1990). Developments in decision-theoretic variance estimation:comment. Statistical Science 5:107–109
Hwang J.T.G., Peddada S.D. (1994). Confidence interval estimation subject to order restrictions. The Annals of Statistics 22:67–93
Iliopoulos G., Kourouklis S. (2000). Interval estimation for the ratio of scale parameters and for ordered scale parameters. Statistics & Decisions 18:169–184
Kaur A., Singh H. (1991). On the estimation of ordered means of two exponential populations. Annals of the Institute of Statistical Mathematics 43:347–356
Kourouklis S. (2001). Estimating the smallest scale parameter: universal domination results. In: Charalambides Ch.A., Koutras M.V., Balakrishnan N. (eds) Probability and Statistical Models with Applications. Chapman & Hall/CRC, London/Boca Raton
Kubokawa T., Morita K., Makita S., Nagakura K. (1993). Estimation of the variance and its applications. Journal of Statistical Planning and Inference 35:319–333
Kushary D., Cohen A. (1989). Estimating ordered location and scale parameters. Statistics & Decisions 7:201–213
Lee C.I.C. (1988). Quadratic loss of order restricted estimators for treatment means with a control. The Annals of Statistics 16:751–758
Lehmann, E.L., Casella, G. (1998). G. Casella, S. Fienberg, I. Olkin (Eds.), Theory of Point Estimation New York: Springer
Maatta J.M., Casella G. (1990). Developments in decision-theoretic variance estimation. Statistical Science 5:90–120
Nagata Y. (1989). Estimation with pooling procedures of error variance in ANOVA for orthogonal arrays. Journal of the Japanese Society for Quality Control 19:12–19 (in Japanese).
Oono, Y. (2005). On the improved estimation of error variance and order restricted normal means and variances, PhD. dissertation, Keio University.
Oono, Y., Shinozaki, N. (2004). Estimation of error variance in the analysis of experiments using two-level orthogonal arrays. Communications in Statistics – Theory and Methods 33:75–98
Oono Y., Shinozaki N. (2006). On a class of improved estimators of variance and estimation under order restriction. Journal of Statistical Planning and Inference 136:2584–2605
Robertson, T., Wright, F.T., Dykstra, R.L. (1988). In: V. Barnett, R.A. Bradley, J.S. Hunter, D.G. Kendall, R.G. Miller, A.F.M. Smith, S.M. Stigler, G.S. Watson, (Eds.), Order restricted statistical inference. New York: Wiley
Shinozaki N. (1995). Some modifications of improved estimators of a normal variance. Annals of the Institute of Statistical Mathematics 47:273–286
Stein C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Annals of the Institute of Statistical Mathematics 16:155–160
Vijayasree G., Singh H. (1993). Mixed estimators of two ordered exponential means. Journal of Statistical Planning and Inference 35:47–53
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Oono, Y., Shinozaki, N. Estimation of error variance in ANOVA model and order restricted scale parameters. AISM 58, 739–756 (2006). https://doi.org/10.1007/s10463-005-0025-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-005-0025-5