Metrika (2002) 56: 143–161 > Springer-Verlag 2002 Smooth estimators for estimating order restricted scale parameters of two gamma distributions Neeraj Misra1, P. K. Choudhary2, I. D. Dhariyal1, D. Kundu1 1 Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur 208016, India 2 Department of Statistics, Ohio State University, 1958 Neil Avenue, Columbus OH 43210-1247, USA Abstract. We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivar- iant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Keywords: best scale equivariant estimator, mixed estimators, non-informative prior, restricted maximum likelihood estimator, scale equivariant squared error loss function, smooth estimators, unrestricted maximum likelihood estimator 1 Introduction The problem of estimating ordered parameters, when it is known apriori that they are subject to certain order restrictions, is of considerable interest. Sup- pose it is desired to estimate the average yields, say, y 1 and y 2 , under treat- ments t 1 and t 2 respectively, where the treatment t 1 is using certain fertilizer for the crop, while treatment t 2 is not using any fertilizer. In this situation, it is reasonable to assume that y 1 b y 2 . Similarly, in estimating average incomes, say y i , i ¼ 1; ... ; k, of k classes of employees in an establishment, it is quite natural to assume an ordering among the y i s, according to the grade of em- ployees. In the development of a system, engineering changes are made in stages to correct the design deficiencies and thereby increasing reliability. Thus, if we have k stages in which changes are made, then at each stage we expect the reliability to increase. If y i is a measure of the reliability at the i-th stage, then we may assume that y 1 a  a y k .