Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Mukerjee, Rahul (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

This thesis primarily deals with search designs (see Srivastava (1975)) of various types for general symmetric and asymmetric factorials. The thesis also considers some other types on fractional factorials. We have made use of Kronecker products and various other results from matrix theory.In the first four chapters, different types of search designs have been studied. Chapter 5A considers the construction of orthogonal main-effect plans for general factorial experiments under the presence of a linear time trend. Chapter 5B extends the calculus for factorial arrangements to a fractional factorial set-up and examines critically the proportional frequency plans from the point of view of estimability and efficiency.Section 0.2 contains a review of the literature on search designs and also on trend-free plans and proportional frequency plans. While reviewing the liter- ature, we have restricted ourselves to the results pertaining to the topics discussed in this thesis. Section 0.3A,B present the main notations, plan of the thesis and a detailed chapterwise summary. The motivation for the different chapters has been discussed and all main results have been recalled. For ease of reference, the serial numbers of these theorems and definitions are the same as those in the main chapters.0.2 Review of the LiteratureThrough the contributions of numerous researchers, the area of fractional replication is now so rich that it is difficult to make a comprehensive review of the entire literature in a limited space. We, therefore, begin with a brief overview of the general area of fractional replication and indicate some of the key references including review articles and monographs. This has been done in the next paragraph. The rest of this section presents a detailed review of the literature on the topics pertaining to this thesis.The idea of fractional replication in factorial experiments was introduced and popularized by Finney (1945). Among the early authors, Plackett and Burman (1946), Kempthorne (1947) and Rao (1947) studied the problem in more detail and gave several fractional plans. Rao (1947) developed the theory of fractional repli- cation for symmetric factorials through the use of orthogonal arrays. Box and Hunter (1961a,b) introduced the notion of resolution which was subsequently gener- alized by Webb (1968). The development of orthogonal fractional factorial plans, including studies on their optimality properties (vide Cheng (1980)), attracted the attention of many authors over the last four decades and for a review, together with a comprehensive list of references, we refer to Dey (1985). Also, we refer to Raghavarao (1971), Rao (1973) and Hedayat and Stufken (1988) (and the refer- ences therein) for discussion on orthogonal arrays which play a crucial role in the context of orthogonal fractional factorial plans. Another very important area was opened up by Bose and Srivastava (1964) who considered fractional factorial plans based on balanced arrays - for further results in this line, together with studies on optimality, see Srivastava and Chopra (1971), Shirakura (1976) and Yamamoto, Shirakura and Kuwada (1976), among others. A detailed list of references on fractional factorial plans based on balanced arrays is available in Kuwada and Nishii (1988). As for review articles on fractional replication, reference is made to Addelman (1963, 1972) for informative reviews up to those stages. For more up- to-date excellent reviews, together with detailed lists of references and discussion on open problems, we refer to Srivastava (1978), Raktoe, Hedayat and Federer (1981), Dey (1985) and Srivastava (1987).


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