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)


Mukherjee, Rahul (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

This thesis deals primarily with the application of the calculus for factorial ar- rangements (Kurkjian and Zelen (1962, 1963)) to various designs. The thesis has been divided into six chapters. We have made extensive use of Kronecker products and various other results from matrix theory. The results in the first two chapters involve the use of projection operators.In the first four chapters, different classes of factorial experiments have been studied by applying the calculus. Chapter 5 deals with another class of designs called repeated measurements designs (RMD's). It has been shown that the calculus for factorial arrangements serves as a powerful tool for studying the optimality properties of such designs under the possible presence of interaction. In Chapter 6, a class of designa very closely related to the RMD's has been investignted.Section 0.2 contains a review of the literature on factorial experiments and RMD39's. While reviewing the literature, we have restricted ourselves to the results pertaining to the topics discussed in this thesis. Section 0.3 presents a detailed chaterwise summary of this thesis. The motivation of the different chapters has been discussed and all main results have been quoted. Where necessary, the relevant definitions have been recalled. For ease of reference, the serial numbers of these theorems and definitions are the same as those in the main chapters.It has been attempted to retain notational uniformity as far as practicable. The relevant notations have been explained in the beginning of each chapter.0.2 A Brief Review of the LiteratureFisher (1935) introduced and popularized factorial designs. Among the early au- thors, Yates (1937) considered both symmetric and asymmetric factorial experiments and Bose and Kishen (1940) and Bose (1947) applied finite geometries to develop a mathematical theory for symmetric prime-powered factorials. Generalizations of the classical method due to Bose (1947) to asymmetrie factorials were considered among others by White and Hultquist (1965), Raktoe (1969, 1970), Worthley and Banerjee (1974) and Sihota and Banerjee (1981). All but the latest of these were reviewed and discussed by Raktoe, Rayner and Chalton (1978). A related development was through the use of the DSIGN algorithm (Patterson (1976), Bailey (1977), Bailey, Gilchrist and Patterson (1977). Recently, Voss (1986) and Voss and Dean (1987) investigated the relationship among these procedures with the objective of integrating them. Nair and Rao (1948) introduced balanced confounded designs for asymmetric factorials. These designs ensure balance with respect to each factorial effect and have orthogonal factorial structure (OFS) in the sense that the best lincar unbiassed estimators (BLUES) of estimable contrasts belonging to different factorial effects are uncorrelated. Further construction procedures for and combinatorial properties of balanced confounded designs were explored among others by Kramer and Bradley (1957), Zelen (1958), Kishen and Srivastava (1959), Das (1960), Paik and Federer (1973), and more recently by Lewis and Tuck (1985), Suen and Chakravarti (1985) and Gupta (1987).Kurkjian and Zelen (1962, 1963) introduced a calculus for factorial arrangements which, as pointed out by Federer (1980), serves as a very powerful analytical tool in the context of factorial designs. The calculus is extremely helpful in deriving characterizationa, in a compact form, for balanos and/or OFS in a general multifuctorontting In turn, these characterizations lead to useful construction procedures retaining high efficiency with respect to the factorial effects of interest.Kurkjian and Zelen (1963) employed the calculus to obtain a sufficient condition for balance together with OFS for factorial experiments in block designs. Zelen and Federer (1964) extended their result to designs for two-way heterogeneity elimination. Kshirsagar (1966) established that the sufficient condition in Kurkjian and Zelen (1963) is also necessary for balance with OFS. The emphasis on balance, however, has a drawback that the resulting designs, although theoretically elegant, may become too large and hence expensive, Because of this reason, since the early seventies, work started on the conditions for OFS alone. John and Smith (1972) and Cotter, John and Smith (1973) obtained a sufficient condition for OFS in terms of a generalized- (g-) inverse of the intrablock matrix. Mukerjee (1979, 1980) gave necessary and sufficient conditions for OFS directly in terms of the intrablock matrix. These conditions, applicable to both block designs and designs for multiway elimination of heterogencity, involve the checking of the commutativity of certain matrices with the intrablock matrix and, therefore, can be easily verified. Chauhan and Dean (1986) extended the results in Mukerjee (1980) to obtain characterizations for partial OFS. Some results from Mukerjee (1979, 1980) have been used in Chapters 1, 3, 4 and 5 of this thesis and there these have been stated as Lemmas 1.2.1, 1.3.4, 4.3.1 and 5.3.1.


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