Quantile based entropy function

Quantile functions are efficient and equivalent alternatives to distribution functions in modeling and analysis of statistical data (see Gilchrist, 2000; Nair and Sankaran, 2009). Motivated by this, in the present paper, we introduce a quantile based Shannon entropy function. We also introduce residual entropy function in the quantile setup and study its properties. Unlike the residual entropy function due to Ebrahimi (1996), the residual quantile entropy function determines the quantile density function uniquely through a simple relationship. The measure is used to define two nonparametric classes of distributions

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.

Service quality and reliability assessment of mobile communication infrastructure

The service quality of any sector has two major aspects namely technical and functional. Technical quality can be attained by maintaining technical specification as decided by the organization. Functional quality refers to the manner which service is delivered to customer which can be assessed by the customer feed backs. A field survey was conducted based on the management tool SERVQUAL, by designing 28 constructs under 7 dimensions of service quality. Stratified sampling techniques were used to get 336 valid responses and the gap scores of expectations and perceptions are analyzed using statistical techniques to identify the weakest dimension. To assess the technical aspects of availability six months live outage data of base transceiver were collected. The statistical and exploratory techniques were used to model the network performance. The failure patterns have been modeled in competing risk models and probability distribution of service outage and restorations were parameterized. Since the availability of network is a function of the reliability and maintainability of the network elements, any service provider who wishes to keep up their service level agreements on availability should be aware of the variability of these elements and its effects on interactions. The availability variations were studied by designing a discrete time event simulation model with probabilistic input parameters. The probabilistic distribution parameters arrived from live data analysis was used to design experiments to define the availability domain of the network under consideration. The availability domain can be used as a reference for planning and implementing maintenance activities. A new metric is proposed which incorporates a consistency index along with key service parameters that can be used to compare the performance of different service providers. The developed tool can be used for reliability analysis of mobile communication systems and assumes greater significance in the wake of mobile portability facility. It is also possible to have a relative measure of the effectiveness of different service providers.

Studies on optical attenuation in sea water using dye laser and laser propagation in a turbulent medium

The central theme of the work presented in this thesis is a careful investigation of the factors influencing the attenuation of laser beam through sea water. The thesis presents a detailed report of the work done by the author on the attenuation studies in sea water and on laser propagation through a turbulent medium. The thesis contains six chapters which are more or less self-contained with separate abstracts and references. The first chapter is divided into two parts. The first part introduces the subject of laser propagation through sea water. It includes a brief description of optical properties of sea water followed by a review of the earlier works on attenuation studies in water. The second part gives the theoretical background of the problem of laser propagation through a turbulent medium.

A study on fuzzy semi inner product spaces

Mathematical models are often used to describe physical realities. However, the physical realities are imprecise while the mathematical concepts are required to be precise and perfect. Even mathematicians like H. Poincare worried about this. He observed that mathematical models are over idealizations, for instance, he said that only in Mathematics, equality is a transitive relation. A first attempt to save this situation was perhaps given by K. Menger in 1951 by introducing the concept of statistical metric space in which the distance between points is a probability distribution on the set of nonnegative real numbers rather than a mere nonnegative real number. Other attempts were made by M.J. Frank, U. Hbhle, B. Schweizer, A. Sklar and others. An aspect in common to all these approaches is that they model impreciseness in a probabilistic manner. They are not able to deal with situations in which impreciseness is not apparently of a probabilistic nature. This thesis is confined to introducing and developing a theory of fuzzy semi inner product spaces.

Quantile based entropy function in past lifetime

Di Crescenzo and Longobardi (2002) introduced a measure of uncertainty in past lifetime distributions and studied its relationship with residual entropy function. In the present paper, we introduce a quantile version of the entropy function in past lifetime and study its properties. Unlike the measure of uncertainty given in Di Crescenzo and Longobardi (2002) the proposed measure uniquely determines the underlying probability distribution. The measure is used to study two nonparametric classes of distributions. We prove characterizations theorems for some well known quantile lifetime distributions

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

An Extended Pearson System Useful In Reliability Analysis

In the present environment, industry should provide the products of high quality. Quality of products is judged by the period of time they can successfully perform their intended functions without failure. The cause of the failures can be ascertained through life testing experiments and the times to failure due to different cause are likely to follow different distributions. Knowledge of this distribution is essential to eliminate causes of failures and thereby to improve the quality and the reliability of products. The main accomplishment expected to the study is to develop statistical tools that could facilitate solution to lifetime data arising in such and similar contexts

Some bivariate life time models in discrete time"

The term reliability of an equipment or device is often meant to indicate the probability that it carries out the functions expected of it adequately or without failure and within specified performance limits at a given age for a desired mission time when put to use under the designated application and operating environmental stress. A broad classification of the approaches employed in relation to reliability studies can be made as probabilistic and deterministic, where the main interest in the former is to device tools and methods to identify the random mechanism governing the failure process through a proper statistical frame work, while the latter addresses the question of finding the causes of failure and steps to reduce individual failures thereby enhancing reliability. In the probabilistic attitude to which the present study subscribes to, the concept of life distribution, a mathematical idealisation that describes the failure times, is fundamental and a basic question a reliability analyst has to settle is the form of the life distribution. It is for no other reason that a major share of the literature on the mathematical theory of reliability is focussed on methods of arriving at reasonable models of failure times and in showing the failure patterns that induce such models. The application of the methodology of life time distributions is not confined to the assesment of endurance of equipments and systems only, but ranges over a wide variety of scientific investigations where the word life time may not refer to the length of life in the literal sense, but can be concieved in its most general form as a non-negative random variable. Thus the tools developed in connection with modelling life time data have found applications in other areas of research such as actuarial science, engineering, biomedical sciences, economics, extreme value theory etc.