On Truncated Versions of Certain Measures of Inequality and Stability

The present study focuses attention on defining certain measures of income inequality for the truncated distributions and characterization of probability distributions using the functional form of these measures, extension of some measures of inequality and stability to higher dimensions, characterization of bivariate models using the above concepts and estimation of some measures of inequality using the Bayesian techniques. The thesis defines certain measures of income inequality for the truncated distributions and studies the effect of truncation upon these measures. An important measure used in Reliability theory, to measure the stability of the component is the residual entropy function. This concept can advantageously used as a measure of inequality of truncated distributions. The geometric mean comes up as handy tool in the measurement of income inequality. The geometric vitality function being the geometric mean of the truncated random variable can be advantageously utilized to measure inequality of the truncated distributions. The study includes problem of estimation of the Lorenz curve, Gini-index and variance of logarithms for the Pareto distribution using Bayesian techniques.

Characterization of Probability Distributions Using the Residual Entropy Function

The present study on the characterization of probability distributions using the residual entropy function. The concept of entropy is extensively used in literature as a quantitative measure of uncertainty associated with a random phenomenon. The commonly used life time models in reliability Theory are exponential distribution, Pareto distribution, Beta distribution, Weibull distribution and gamma distribution. Several characterization theorems are obtained for the above models using reliability concepts such as failure rate, mean residual life function, vitality function, variance residual life function etc. Most of the works on characterization of distributions in the reliability context centers around the failure rate or the residual life function. The important aspect of interest in the study of entropy is that of locating distributions for which the shannon’s entropy is maximum subject to certain restrictions on the underlying random variable. The geometric vitality function and examine its properties. It is established that the geometric vitality function determines the distribution uniquely. The problem of averaging the residual entropy function is examined, and also the truncated form version of entropies of higher order are defined. In this study it is established that the residual entropy function determines the distribution uniquely and that the constancy of the same is characteristics to the geometric distribution

‘Reliability Concepts in Quantile-based Analysis of Lifetime Data

Reliability analysis is a well established branch of statistics that deals with the statistical study of different aspects of lifetimes of a system of components. As we pointed out earlier that major part of the theory and applications in connection with reliability analysis were discussed based on the measures in terms of distribution function. In the beginning chapters of the thesis, we have described some attractive features of quantile functions and the relevance of its use in reliability analysis. Motivated by the works of Parzen (1979), Freimer et al. (1988) and Gilchrist (2000), who indicated the scope of quantile functions in reliability analysis and as a follow up of the systematic study in this connection by Nair and Sankaran (2009), in the present work we tried to extend their ideas to develop necessary theoretical framework for lifetime data analysis. In Chapter 1, we have given the relevance and scope of the study and a brief outline of the work we have carried out. Chapter 2 of this thesis is devoted to the presentation of various concepts and their brief reviews, which were useful for the discussions in the subsequent chapters .In the introduction of Chapter 4, we have pointed out the role of ageing concepts in reliability analysis and in identifying life distributions .In Chapter 6, we have studied the first two L-moments of residual life and their relevance in various applications of reliability analysis. We have shown that the first L-moment of residual function is equivalent to the vitality function, which have been widely discussed in the literature .In Chapter 7, we have defined percentile residual life in reversed time (RPRL) and derived its relationship with reversed hazard rate (RHR). We have discussed the characterization problem of RPRL and demonstrated with an example that the RPRL for given does not determine the distribution uniquely

Social Justice and Weaker Sections: Role of the Judiciary

The Constitution of India. which has been described by an eminent writer as a "Corner stone of a nation". Has bestowed sufficient thought on the underprivileged. A number of provisions incorporated in it for their benefit tell the tale of statesmanship of the framers of the Constitution. for the vitality of a Constitution depends on the extent to which it affords protection to the under—priveleged. One such laudable provision in the Constitution relates to "weaker sections of the people", which has directed the State to promote with special care the educational and economic interests of such people. Besides. the Constitution has laid great stress on social justice. No comprehensive analysis in a single work seems to have been made so far of the connotations of social justice and the scope of the constitutional safeguards provided in favour of the weaker sections of the people. This thesis is the result of an attempt to analyse the connotations of social justice and the scope of the constitutional provisions made for the benefit of the weaker sections and the role played by the judiciary in this field The weaker sections, which are sought to be covered in this work, are "Backward C1asses". socially and educationally Backward Classes", "Scheduled Castes and Scheduled Tribes" and women. The first two categories of weaker sections have not been defined in the Constitution. So, their meaning and the criteria to determine them have to be gathered from the reports submitted by various Backward Class Commissions and judicial decisions rendered in a number of cases. The main thrust in this work is to understand the meaning and contents of social justice, identify the relevant weaker sections and to examine the extent to which the social justice has been rendered to the said weaker sections. The scope of this thesis is confined to the examination of the role of the judiciary in this field. So, the enquiry has been focussed mainly on the decisions of the judiciary bearing on the subject with a view to assessing the role of the judiciary in rendering social justice meaningful to the weaker sections in particular and to the Indian Society in general.

Characterizations of Life Distributions Using Conditional Expectations of Doubly (Interval) Truncated Random Variables

In this article, we study reliability measures such as geometric vitality function and conditional Shannon’s measures of uncertainty proposed by Ebrahimi (1996) and Sankaran and Gupta (1999), respectively, for the doubly (interval) truncated random variables. In survival analysis and reliability engineering, these measures play a significant role in studying the various characteristics of a system/component when it fails between two time points. The interrelationships among these uncertainty measures for various distributions are derived and proved characterization theorems arising out of them