Statistics and Probability Letters 82 (2012) 1088–1094 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Pseudo-Bayesian A-optimal designs for estimating the point of maximum in component-amount Darroch–Waller mixture model N.K. Mandal a,1 , Manisha Pal a,∗ , M.L. Aggarwal b a University of Calcutta, India b University of Memphis, USA article info Article history: Received 14 May 2011 Received in revised form 11 February 2012 Accepted 12 February 2012 Available online 17 February 2012 MSC: 62K99 62J05 Keywords: Mixture experiment Darroch–Waller model Non-linear parametric function Asymptotic efficiency A-optimal design abstract In the analysis of experiments with mixture, quadratic models have been widely used. Several authors considered finding optimum designs for the estimation of the parameters of the model. The optimum designs for the estimation of optimum mixing proportions in Scheffé’s quadratic mixture model has been studied by Pal and Mandal (2006) and Mandal et al. (2008a,b) using a pseudo-Bayesian approach. In this paper, we consider an additive quadratic mixture model, proposed by Darroch and Waller (1985), when the amount of mixture is taken into account, and obtain the A-optimal designs for the estimation of optimum proportions, adopting the approach of Pal and Mandal (2006). We show that, besides other support points, the origin and the vertices of the simplex are necessarily the support points of the optimum design. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Design of experiments with mixtures aims at finding the composition of products so as to maximize their properties. There are many products formed by mixing different components, whose quality depends on the mixing proportions, like food, polymers, paint, concrete, glass, etc. The mixture design approach is used in such industries for obtaining the optimum formulation of the products. The response in mixture experiments primarily depends on the mixing proportions. Scheffé (1958, 1963) was the first to introduce canonical models of different degrees for representing the response function in terms of only the relative proportions of the components in the mixture. He proposed designs suitable for estimation of the regression coefficients in such models. He also introduced the Simplex Lattice Designs and Simplex Centroid Designs in such situations. Several authors, like Kiefer (1961), Farrel et al. (1967), Atwood (1969), Galil and Kiefer (1977) and Liu and Neudecker (1997), to name a few, studied the optimality of mixture designs for the estimation of the parameters of the response function. Draper and Pukelsheim (1999) established the optimality of Weighted Centroid Designs with respect to Partial Loewner Ordering (PLO) in two and three component mixtures, for first and second degree models. Optimum designs for the estimation of some specific non-linear functions of the parameters have also been studied (see, Pal and Mandal, 2006, 2007, 2008, 2009; Mandal and Pal, 2008; Mandal et al., 2008a,b). ∗ Corresponding author. E-mail address: manishapal2@gmail.com (M. Pal). 1 The work was carried out when the author was visiting the University of Memphis, USA. 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.02.011