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Abstract: | 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 |
URI: | http://dyuthi.cusat.ac.in/purl/1002 |
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Dyuthi-T0052.pdf | (1.421Mb) |
Abstract: | 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. |
URI: | http://dyuthi.cusat.ac.in/purl/37 |
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Dyuthi-T0212.pdf | (1.641Mb) |
URI: | http://dyuthi.cusat.ac.in/purl/1679 |
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Dyuthi-T0153.pdf | (4.461Mb) |
Abstract: | It is highly desirable that any multivariate distribution possessescharacteristic properties that are generalisation in some sense of the corresponding results in the univariate case. Therefore it is of interest to examine whether a multivariate distribution can admit such characterizations. In the exponential context, the question to be answered is, in what meaning— ful way can one extend the unique properties in the univariate case in a bivariate set up? Since the lack of memory property is the best studied and most useful property of the exponential law, our first endeavour in the present thesis, is to suitably extend this property and its equivalent forms so as to characterize the Gumbel's bivariate exponential distribution. Though there are many forms of bivariate exponential distributions, a matching interest has not been shown in developing corresponding discrete versions in the form of bivariate geometric distributions. Accordingly, attempt is also made to introduce the geometric version of the Gumbel distribution and examine several of its characteristic properties. A major area where exponential models are successfully applied being reliability theory, we also look into the role of these bivariate laws in that context. The present thesis is organised into five Chapters |
Description: | Department of Mathematics and Statistics, Cochin University of Science and Technology |
URI: | http://dyuthi.cusat.ac.in/purl/3657 |
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Dyuthi-T1577.pdf | (3.283Mb) |
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