Theses
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Item Some Contributions to the Asymptotic Theory of Estimation in Non-regular Case(1989-02-28) Samanta, TapasThe introduction of the concept of onymptotie efficiency of statistical eatimators in connection with proposing and developing she method of maximu, 1ikelihood by . Fislior (Fisher (1922, 1929)) 1s really the atar ting point of the asymptotie theory of es timation. -istorieally, however, Laplace (1774) and Gausa (1009) had made tuo irferent studieo oarlier then Fisher bo ti connected d th as ymp to tie theory of estination. Fisher considered only consistent a symptotically rewnal untinotoro and meured the asyaptotie performance of an anti- mator by ite aayep totle variaree. Thus, a coreistent aeynp to tieally nomal eetimoator with let possible aayno to tie voriance wes defined to be an erricient estimator. Fisher also clained to have proved that under certain regulari ty condi tiona the naxinum likelhaod estinatar (MLE) 1s offieient in the above sense. In the thirties and forties several authors (Dugue (1936a, 1936b, 1937), Wlks (1938), Neynan (1949) ond ethere) attompted to a btain a Figorous proof of the erricieney of the MLE and there uas a genet al teller that there exinte an afrieient es timator in the general case unich nay be obtained by the method of maximum likelihood. This belief enieted until 3.. Hodges produced in 1951 the " revolutionary exanples of;auper ofricient osti- ma tore (ane ean se, for example, Ghonh (1905) or Le Can (1953) hera It rirot appenred). Hodgen exanple nhowe that in the unual rogular casee there axist neymptotically nomal estinatere uhone aaymptotic variances are alunye lees than or equal to that of tha MLE and are trietly less than that of the MLE at partieuler values of the para- ter and at these particular valuen the asymp to tie variance may even e made equal to zero. Thus, the MLE is net errieient in tha above sense and indeed, within the elass of all symptotically normal esti- natera re entinator ui th minimal symptotic variance exists.However, the ideas of Fiaher and the existence of " super ofricient " otinators greatly influenced the developement of the theory of efficient entimation and a modern approach to the theory of synap totie efrrietent entimation emerged in the fundamantal paper of Le Can (1983). The theory vos further developed in the worke of Le Can (1960, 1964, 1972), Hajek (1970, 1972), unlfouitz (1965), Millar (1903) and othere. Thie approach which renched nore ar lese itarinal form in the papere of Hajek (1972) and Le Can (1972) consi dere all e tina tore in oteed of rontricting to the class of naynptotically nornal en tinn tora only, tut the ofricieney or the performance of these estinn tors is meanur ed in a slightly different uay. Mmiller (1983) prenenta a very clenr expoeitian of thie theory extending sone of the basie results of Le Can (1072). A lucid aneount of this developrent of kajek - La Can theory of afrieient estination is available in Choah (1905).Item Development of Some Scalable Pattern Recognition Algorithms for Real Life Data Analysis(2017-11-20) Garai, ParthaA huge amount of data is being generated continuously as a result of recent advancement and wide use of high-throughput technologies. With the rapid increase in size of data distributed worldwide, understanding the data has become critical. In this regard, dimensionality reduction and clustering have become the necessary preprocessing steps of multiple research areas and applications. One of the important problems of real life large data sets is uncertainty. Some of the sources of this uncertainty include imprecision in computation and vagueness in class denitions. The uncertainty may also be present in the denition of class membership function. In this background, the thesis addresses the problem of dimensionality reduction and clustering of real life data sets, in the presence of noise and uncertainty. The thesis rst presents the problem of feature selection using both type-1 and interval type-2 fuzzyrough sets, which are eective for dimensionality reduction of real life data sets when uncertainty is present in the data set. The properties of fuzzy-rough sets allow greater exibility in handling noisy and real valued data. While the concept of lower approximation and boundary region of rough sets deals with uncertainty, incompleteness, and vagueness in class denition, the use of either type-1 or interval type-2 fuzzy sets enables ecient handling of overlapping classes in uncertain environment. Moreover, a new concept of \simultaneous attribute selection and feature extraction" is introduced for dimensionality reduction, integrating judiciously the merits of both feature selection and extraction. A scalable rough-fuzzy clustering algorithm is introduced for large real life data sets, where the theory of rough hypercuboid approach, interval type-2 fuzzy sets, and c-means algorithm are integrated judiciously to handle the uncertainty present in a data set. While the concept of rough hypercuboid approach deals with uncertainty, incompleteness, and vagueness in cluster denition, the use of fuzzy membership of interval type-2 fuzzy sets in the boundary region of a cluster enables ecient handling of overlapping partitions in uncertain environment. Finally, the application of both clustering and feature selection algorithms is demonstrated by grouping functionally similar microRNAs from microarray data. The proposed approach can automatically select the optimum set of features while clustering the microRNAs, making the complexity of the algorithm lower.Item A Study of the SHA-2 Cryptographic Hash Family(Indian Statistical Institute, Kolkata, 2009-02-01) Sanadhya Somitra KumarItem Contributions to Analysis of Consumer Expenditure(Indian Statistical Institute, Kolkata, 1964-04) Iyengar, N SreenivasaAbstractItem Some Results on Combinatorial Batch Codes and Permutation Binomials over Finite Fields(Indian Statistical Institute,Kolkata, 2015) Bhattacharya, SrimantaIn this thesis,we study combinatorial batch codes (CBCs) and permutation binomials (PBs) over �nite �elds of even characteristic. Our primary motivation for considering these problems comes from their importance in cryptography. CBCs are replication based variants of batch codes, which were introduced in [IKOS04a] as a tool for reducing the computational overhead of private information retrieval protocols (a cryptographic primitive). On the other hand, permutation polynomials, with favourable cryptographic properties, have applications in symmetric key encryption schemes, especially in block ciphers. Moreover, these two objects are interesting in their own right, and they have connections with other important combinatorial objects. CBCs are much similar to unbalanced expanders, a much studied combinatorial object having numerous applications in theoretical computer science. On the other hand, the speci�c class of PBs that we consider in this work, are intimately related to orthomorphisms. Orthomorphisms are relevant in the construction of mutually orthogonal latin squares, a classical combinatorial objects having applications in design of statistical experiments. These aspects motivate us to explore theoretical properties of CBCs and PBs over �nite �elds. However, these two objects are inherently widely di�erent; CBCs are purely combinatorial objects, and PBs are algebraic entities. So, we explore these two objects independently in two di�erent parts, where our entire focus lies in exploring theoretical aspects of these objects. In Part I, we consider CBCs. There, we provide bounds on the parameters of CBCs and obtain explicit constructions of optimal CBCs. In Part II, we consider PBs over �nite �elds. There, we obtain explicit characterization and enumeration of subclasses of PBs under certain restrictions. Next, we describe these two parts in more detail.Item Orbit Spaces of Unimodular rows over Smooth Real Affine Algebras(Indian Statistical Institute, Kolkata, 2018) Tikader, SoumiItem Infinite Mode Quantum Gaussian States(Indian Statistical Institute, New Delhi, 2018) John, Tiju CherianItem Higher Chow Cycles on the Jacobian of curves(Indian Statistical Institute, Bangalore, 2019-03) Sarkar, SubhamItem A study of operators on the discrete analogue of Hardy spaces on homogeneous trees and on other structures(Indian Statistical Institute, Bangalore, 2019-01) MUTHUKUMAR, PItem On free-type rigid C*-tensor categories and their annular representations(Indian Statistical Institute, Kolkata, 2018-07) MADHAV REDDY, B