#### Date of Submission

8-19-1987

#### Date of Award

8-19-1988

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Dasgupta, Somesh (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

The works in this dissertation are primarily baset ur. difterent concepts of majorization and the results thereof. True first part of this dissertation is a study on dirfferent concepts of uni variate and multivariate majorization. The latter part of this dissertation includes studies on some problems in Sample Survey, and problems relating to ranking and selection with the use of some results, old and new, in majorization.The concept of univariate majorization has been consi- dered by economists in relation to Lorenz ourve, as well as by mathematicians and statisticians, especially in the field of reliability. It appears that the results relating the different concepts that are available in the literature are not widely known; as a matter of fact, it appear often from some popers ir economics that the re spective authors are not familiar with some of the relevant results published earlier in journals of Iisthematics or statistics. In the first chap ter we ha ve a brief raview of the results in uni variate majorization ant brought out a unified relstionship among different concep ts of majorization a vailable in the literature. The extension of these concep ta to the multivariate case is then studied. Certain concep ts on multivariate majorizatibn have been presented along with some new results. The se results can he released to problem ia economics; wi th that in view some sufficient condltions o concave utility function have been presented.An important tool in the theory of majorization is a theorem due to Hardy, Littlewood and Polya (1929), which says that for P:nxn, yP in majorized by y for all ye R"n if and only if P is a doubly stochestic matrix. But similar results on weak supermajorization was an open question [Marshall and Olkin (1979)). Such a result has been deve loped in the first part of Chap ter 2. In particular, it has been proved that a non-negati ve matrix P:nxn is doubly super- stochastic if and only if yP is weakly supermajorized by y, for all y with all components positive. This result is based on the following fact that a non-negative matrix P:n xn is doubly superstochasticir ond or.ly if it satisfies the following condition.Condition: For 1 s k, k s n, and any kx/ su brnu trix of P, total sum of the entries in B is greater than or equal to (k+A -n).The latter part of Chapter 2 is devoted to some mathematical problÄ™ms in multivariate majorization. For two matrices X : axn and Y:mxn, Marshall and 0lkin (1979) defined X to be majorized by Y, if X - YP, for some doubly stochastic matrix P. Following Marahall and o1xr. (p.433) we define X to be directionally majorized by Y, aX 18 majorized aY for all a E R". They have posed tne open question whether these two types of ma trix ma jorizetiona are equi valent. Here we give some sufficient conditions under which directional majorization implies multivariate msjerizatson. In particular, for m = 2, it has heen proved that if all the column vectors of Y: 2xn are boundary points in the convex hull of the column vectors of Y and this convex hu22 has two dimenasional positive volume, then directional ma corization implies aultivariate majorization.Chapter 3 is devoted to some inequelities relating to random replacement schemes introduced by Karlin (1974). In particular, a conjecture of Karlin (1974) has been studied in this context. The theory of majorization plays a centra;re le in those problems. Neither part of Karlin;conjecture words to be true, as has been observed by different authors Krafft and Schaefer (1984), Schaefer (1987)]. In the first pert we give short and elegant proofs of some of the existing results Krafft and Schaefer (1984)). In the latter, we anely se the problem from a Ä‘ifferent view point and give a large class of Sotur-concave: (convex) functions for which the conjecture holds. Some other related inequalities have also been derived.

#### Control Number

ISILib-TH113

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Bhandari, Subir Kumar Dr., "Some Aspects of Majorization and their Applications in Statistics." (1988). *Doctoral Theses*. 401.

https://digitalcommons.isical.ac.in/doctoral-theses/401

## Comments

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:28843764