Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics

Let A be p×p positive definite matrix. A p-vector x such that Ax=λx is called an eigenvector with the associated with eigenvalue λ. Equivalent characterizations are: (i) cosθ=1, where θ is the angle between x and Ax. (ii) (x'Ax)-1=xA-1x. (iii) cosΦ=1, where φ is the angle between A½x and A-½x. We ask the question what is x such that cosθ as defined in (i) is a minimum or the angle of separation between x and Ax is a maximum. Such a vector is called an anti-eigenvector and cosθ an anti-eigenvalue of A. This is the basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao (1997), Numerical Range: The Field of Values of Linear Operators and Matrices, Springer. We may define a measure of departure from condition (ii) as min[(x'Ax)(x'A-1x)]-1 which gives the same anti-eigenvalue. The same result holds if the maximum of the angle Φ between A½x and A-½x as in condition (iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider optimization problems associated with measures of separation between an r(<p) dimensional subspace S and its transform AS. Similar problems are considered for a general matrix A and its singular values leading to anti-singular values. Other possible definitions of anti-eigen and anti-singular values, and applications to problems in statistics will be presented

Graphical representation of blood group data of human populations

A general survey of various measures of diversity within and distance between populations in gene frequencies of blood group systems is given. A method of ordering populations by an overall measure of diversity within and of clustering populations in terms of differences in pattern of diversity in different blood group systems is developed. Principal coordinate analysis and multidimensional scaling are used to represent populations with given distances between them graphically in an appropriate dimensional Euclidean space. Such graphical representations together with dendrograms are of great help in studying interrelationships between populations. The methods are illustrated using blood group data on some human populations

Conditions for optimality and validity of simple least squares theory

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Estimation and Determinants of Chronic Poverty in India : An Alternative Approach

The paper conceptualizes chronic poverty by using the spaces of income and nutrition and estimates its incidence among states and social groups. It also aims to improve our understanding of the determinant of chronic poverty by considering economic, demographic and social factors. It attempts to answer the following questions : How important a determinant of chronic poverty is household income? What factors inhibit escape from chronic poverty? How different are the other poor from chronic poor? The analysis uses the unit level NSS and NFHS data.Chrinic Poverty, India

Generalized inverse of a matrix and its applications

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Estimation and determinants of chronic poverty in India: An Alternative approach

The paper conceptualizes chronic poverty by using the spaces of income and nutrition and estimates its incidence among states and social groups. It also aims to improve our understanding of the determinant of chronic poverty by considering economic, demographic and social factors. It attempts to answer the following questions: How important a determinant of chronic poverty is household income? What factors inhibit escape from chronic poverty? How different are the other poor from chronic poor? The analysis uses the unit level NSS and NFHS data.

On linear estimation and testing of hypothesis

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Familial correlations or the multivariate generalisations of the intraclass correlation

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