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## Über dieses Buch

​The book dwells mainly on the optimality aspects of mixture designs. As mixture models are a special case of regression models, a general discussion on regression designs has been presented, which includes topics like continuous designs, de la Garza phenomenon, Loewner order domination, Equivalence theorems for different optimality criteria and standard optimality results for single variable polynomial regression and multivariate linear and quadratic regression models. This is followed by a review of the available literature on estimation of parameters in mixture models. Based on recent research findings, the volume also introduces optimal mixture designs for estimation of optimum mixing proportions in different mixture models, which include Scheffé’s quadratic model, Darroch-Waller model, log- contrast model, mixture-amount models, random coefficient models and multi-response model. Robust mixture designs and mixture designs in blocks have been also reviewed. Moreover, some applications of mixture designs in areas like agriculture, pharmaceutics and food and beverages have been presented. Familiarity with the basic concepts of design and analysis of experiments, along with the concept of optimality criteria are desirable prerequisites for a clear understanding of the book. It is likely to be helpful to both theoreticians and practitioners working in the area of mixture experiments.

## Inhaltsverzeichnis

### Chapter 1. Mixture Models and Mixture Designs: Scope of the Monograph

Abstract
We introduce standard mixture models and standard mixture designs as are well known in the literature (vide Cornell2002). Some of the less known models are also introduced briefly. Next we explain the frameworks of exact and approximate [or, continuous] mixture designs. We mention about known applications of mixture experiments in agriculture, food processing, and pharmaceutical studies. We also provide a chapter-wise brief summary of the contents covered in the monograph.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 2. Optimal Regression Designs

Abstract
In this chapter, we review the theory of optimum regression designs. Concept of continuous design and different optimality criteria are introduced. The role of de la Garza phenomenon and Loewner order domination are discussed. Equivalence theorems for different optimality criteria, which play an important role in checking the optimality of a given otherwise prospective design, are presented. These results are repeatedly used in later chapters in the search for optimal mixture designs. We present standard optimality results for single variable polynomial regression model and multivariate linear and quadratic regression model . Kronecker product representation of the model(s) and related optimality results are also discussed.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 3. Parameter Estimation in Linear and Quadratic Mixture Models

Abstract
In this chapter, we present standard mixture models and standard mixture designs as generally applied to such models. These mixture models and mixture designs will occur throughout the monograph. Several generalizations of standard mixture designs are also discussed here. Estimability issues involving the model parameters are addressed at length. In the process, information matrices are worked out and their roles are emphasized. The concept of Loewner domination is also brought in.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 4. Optimal Mixture Designs for Estimation of Natural Parameters in Scheffé’s Models

Abstract
In this chapter, we review the optimality results for the estimation of parameters and subset of parameters of the Scheffé’s mixture models while the factor space is the entire simplex, i.e., while we are in the framework of an unconstrained factor space . Though most of the results are related to quadratic model, optimality results for linear- and higher-degree polynomials are also discussed. Kiefer’s equivalence theorem plays an important role in characterizing optimum designs. It has been observed that the support points of an optimum design under different optimality criteria belong to the subclass of union of barycenters.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 5. Optimal Mixture Designs for Estimation of Natural Parameters in Scheffé’s Model Under Constrained Factor Space

Abstract
Most of the studies on mixture experiments assume that the experimental region is the whole simplex. However, experimentation at the vertices of the simplex is generally not meaningful. One may, therefore, restrict the experiment to a subset of the simplex. In this chapter, an ellipsoid within the simplex is used as the experimental region, and Kiefer optimal designs are determined for both linear and quadratic models due to Scheffé.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 6. Optimal Mixture Designs for Estimation of Natural Parameters in Other Mixture Models

Abstract
In this chapter, we focus on finding optimum mixture designs for the estimation of natural parameters of models other than that of Scheffé viz., Becker’s models, Darroch–Waller [D–W] model and Log-contrast model. It is also equally fascinating to note that so much has been done in these other mixture models as well. We mainly review the results that are already available and some new findings are presented.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 7. Optimal Designs for Estimation of Optimum Mixture in Scheffé’s Quadratic Model

Abstract
This chapter examines the optimum designs for estimating the optimum mixing proportions in Scheffé’s quadratic mixture model with respect to the A-optimality criterion. By optimum mixing proportion, we refer to the one that maximizes the mean response. Since the dispersion matrix of the estimate depends on the unknown model parameters, a pseudo-Bayesian approach is used in defining the optimality criterion. The optimum designs under this criterion have been obtained for two- and three-component mixtures. Further, using Kiefer’s equivalence theorem, it has been shown that under invariant assumption on prior moments, the optimum design for a $$q$$-component mixture is a $$(q, 2)$$ simplex lattice design for $$q = 3, 4.$$
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 8. More on Estimation of Optimum Mixture in Scheffé’s Quadratic Model

Abstract
Chapter 7 discusses the optimum designs for estimating the optimum mixture in Scheffé’s quadratic mixture model, using the trace optimality criterion. In this chapter, we address the problem of finding optimum mixture designs under deficiency and minimax criteria. In most cases, Kiefer’s equivalence theorem plays a key role in identifying the designs.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 9. Optimal Designs for Estimation of Optimum Mixture in Scheffé’s Quadratic Model Under Constrained Factor Space

Abstract
While in Chap. 7, we have discussed determination of optimum designs for the estimation of optimum mixture when the mixing proportions vary in the whole simplex, in the present chapter we address the problem when (i) one of the proportions is bounded above, (ii) there is a cost constraint. Here, again, the trace criterion is used to find the optimum design.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 10. Optimal Designs for Estimation of Optimum Mixture Under Darroch–Waller and Log-Contrast Models

Abstract
This chapter addresses the problem of finding optimum designs for the estimation of optimum mixture combination when the mean response is defined by (i) the additive quadratic mixture model due to Darroch and Waller (1985) and (ii) the quadratic log-contrast model due to Aichison and Bacon-Shoane (1984). Both the models have some advantage over Scheffé quadratic mixture model, in specific situations.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 11. Applications of Mixture Experiments

Abstract
The purpose of this chapter is to describe some application areas of mixture experiments. We present some studies taken up in the context of agricultural/horticultural/pharmaceutical experiments in the form of mixture designs in order to extract meaningful information for specific items of enquiry.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Chapter 12. Miscellaneous Topics: Robust Mixtures, Random Regression Coefficients, Multi-response Experiments, Mixture–Amount Models, Blocking in Mixture Designs

Abstract
In this chapter, we dwell on some mixture design settings and present the underlying optimal designs. The purpose is to acquaint the readers with a variety of interesting and nonstandard areas of mixture designs. The chapter is divided into two parts. In Part A, we cover robust mixture designs and optimality in Scheffé and D–W models with random regression coefficients. In Part B, we discuss mixture–amount model due to Pal and Mandal (Comm Statist Theo Meth 41:665–673, 2012a), multi-response mixture models and mixture designs in blocks. We present the results already available and also some recent findings.
B. K. Sinha, N. K. Mandal, Manisha Pal, P. Das

### Backmatter

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