## Doctoral Theses

### Some Contributions to the Design and Analysis of Experiments with Special Reference to Weighing Designs,Partially Balanced Designs and Designs for Two Way Elimination of Heterogensity.

3-28-1967

3-28-1968

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Research and Training School (RTS)

#### Supervisor

Chakrabarti, Mukund Chand (RTS-Kolkata; ISI and Univ of Mumbai)

#### Abstract (Summary of the Work)

Thie dissertation containe the authors vurious contributions to the design and analysia of experimenta. The min topios covered in this theeia are veighing designe, partially balanced designa and designs for tuo-vay elimination of hoterogeneity. Some of the material of the chaptere 2, 3, 4, 5 (marked vith asteriska in the theais) has beon conpiled from the authors publiahed papers. [9],[10].[13],[12],[13].[14].Chapters I and II deal with chemioal balanoe velghing designs. Yates [11] and Hotelling [36] obeerved that the veichta of the objeeta ean be more accurately determined by voighing thon in groupe. I a wodghing operations are made to veighp objecta, the minimun variance that ouch eatimtod vodght may have, will be where o is the variene of each voighing. Sinoe we an interested in the weigh ta of the objecte and not in the estimte of , the niniman number of veighing operations to ved gh n objeata ia n. Lot I= ( y be the vedghing design matrix of order a for vedghing n objecta in a veighinga with a chenieal balanoe having no bias, where z =â€¢1 or -1 ir the j objeat is ineluded in the 1th velghing by being placed 1in the left or right pan and x - o if the jth abjeat is not ineluded in the 1th vedching. It ean be aeon that (x'x)-1 ls the variance oovarianoe mtrix of the estimated velghta. Hotelling has ahoun that the miniman variance for the eatimsted weight is obtained if I ie a mtrix consiating of :1 such that the columne ere or thogonal. Hence, ve oall a vedghing design to be optiaum if x = al 1.o. X is a Hadanard matrix (). It mmy be romrked that a necessary oondition for the existence of Hadamard matrices is nEo mod 4 with the possible exception of a= 2. A eomplote summary of the atatue of the existence of , 1s givon by Bose and Shrikhande [22] and conjectured that for every ardor of no nod 4, existe. In seetion 1.2 of the chapter I, we give a nothod of construetion of H, where a 2 Tag 1 , pk 3 nod 4.In the absense of optiaun volghing deadna, the problom is to find the best veighing design. There are three vell known eriteria to decide thie problem, Thoy are due to (1) Kishen [42] (11) arenfeld [30] and (111) Mood CuJ. Banerjee [7] has dvon an expository artiele reviowing the vork done in veighing dealgns ti11 the yoar 1950, Raghavareo [52] Found beat velghing designs subject to sone restrictiona. No general solution for finding best woighing deuigna when no mod 4 ves obtained in the previous 1itorature in veighing designe.[50] In soctions 1.3, 1.4, 1.5 ve abou that uhen a is odd, P, mtrix (ef. Raghavarao so ar st exdata, is the buet voighing dosign under the three effioieney eriteria. Also, 16 ie ohovn that whenng2 nod 4, Ta (ar, iaghavarao sa 7 u st extata ia the best vedghing design vith Shronfolds definition of efiieieney; and (ef. Ehilieh [29] is it exiata, ia the bout one in Hoods definition of efficioney.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843372

ISILib-C8328