8 resultados para Non-gaussian Random Functions

em Cochin University of Science


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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.

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The thesis deals with some of the non-linear Gaussian and non-Gaussian time models and mainly concentrated in studying the properties and application of a first order autoregressive process with Cauchy marginal distribution. In this thesis some of the non-linear Gaussian and non-Gaussian time series models and mainly concentrated in studying the properties and application of a order autoregressive process with Cauchy marginal distribution. Time series relating to prices, consumptions, money in circulation, bank deposits and bank clearing, sales and profit in a departmental store, national income and foreign exchange reserves, prices and dividend of shares in a stock exchange etc. are examples of economic and business time series. The thesis discuses the application of a threshold autoregressive(TAR) model, try to fit this model to a time series data. Another important non-linear model is the ARCH model, and the third model is the TARCH model. The main objective here is to identify an appropriate model to a given set of data. The data considered are the daily coconut oil prices for a period of three years. Since it is a price data the consecutive prices may not be independent and hence a time series based model is more appropriate. In this study the properties like ergodicity, mixing property and time reversibility and also various estimation procedures used to estimate the unknown parameters of the process.

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Dental caries persists to be the most predominant oral disease in spite of remarkable progress made during the past half- century to reduce its prevalence. Early diagnosis of carious lesions is an important factor in the prevention and management of dental caries. Conventional procedures for caries detection involve visual-tactile and radiographic examination, which is considered as “gold standard”. These techniques are subjective and are unable to detect the lesions until they are well advanced and involve about one-third of the thickness of enamel. Therefore, all these factors necessitate the need for the development of new techniques for early diagnosis of carious lesions. Researchers have been trying to develop various instruments based on optical spectroscopic techniques for detection of dental caries during the last two decades. These optical spectroscopic techniques facilitate noninvasive and real-time tissue characterization with reduced radiation exposure to patient, thereby improving the management of dental caries. Nonetheless, a costeffective optical system with adequate sensitivity and specificity for clinical use is still not realized and development of such a system is a challenging task.Two key techniques based on the optical properties of dental hard tissues are discussed in this current thesis, namely laser-induced fluorescence (LIF) and diffuse reflectance (DR) spectroscopy for detection of tooth caries and demineralization. The work described in this thesis is mainly of applied nature, focusing on the analysis of data from in vitro tooth samples and extending these results to diagnose dental caries in a clinical environment. The work mainly aims to improve and contribute to the contemporary research on fluorescence and diffuse reflectance for discriminating different stages of carious lesions. Towards this, a portable and compact laser-induced fluorescence and reflectance spectroscopic system (LIFRS) was developed for point monitoring of fluorescence and diffuse reflectance spectra from tooth samples. The LIFRS system uses either a 337 nm nitrogen laser or a 404 nm diode laser for the excitation of tooth autofluorescence and a white light source (tungsten halogen lamp) for measuring diffuse reflectance.Extensive in vitro studies were carried out on extracted tooth samples to test the applicability of LIFRS system for detecting dental caries, before being tested in a clinical environment. Both LIF and DR studies were performed for diagnosis of dental caries, but special emphasis was given for early detection and also to discriminate between different stages of carious lesions. Further the potential of LIFRS system in detecting demineralization and remineralization were also assessed.In the clinical trial on 105 patients, fluorescence reference standard (FRS) criteria was developed based on LIF spectral ratios (F500/F635 and F500/F680) to discriminate different stages of caries and for early detection of dental caries. The FRS ratio scatter plots developed showed better sensitivity and specificity as compared to clinical and radiographic examination, and the results were validated with the blindtests. Moreover, the LIF spectra were analyzed by curve-fitting using Gaussian spectral functions and the derived curve-fitted parameters such as peak position, Gaussian curve area, amplitude and width were found to be useful for distinguishing different stages of caries. In DR studies, a novel method was established based on DR ratios (R500/R700, R600/R700 and R650/R700) to detect dental caries with improved accuracy. Further the diagnostic accuracy of LIFRS system was evaluated in terms of sensitivity, specificity and area under the ROC curve. On the basis of these results, the LIFRS system was found useful as a valuable adjunct to the clinicians for detecting carious lesions.

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We propose a novel, simple, efficient and distribution-free re-sampling technique for developing prediction intervals for returns and volatilities following ARCH/GARCH models. In particular, our key idea is to employ a Box–Jenkins linear representation of an ARCH/GARCH equation and then to adapt a sieve bootstrap procedure to the nonlinear GARCH framework. Our simulation studies indicate that the new re-sampling method provides sharp and well calibrated prediction intervals for both returns and volatilities while reducing computational costs by up to 100 times, compared to other available re-sampling techniques for ARCH/GARCH models. The proposed procedure is illustrated by an application to Yen/U.S. dollar daily exchange rate data.

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The classical methods of analysing time series by Box-Jenkins approach assume that the observed series uctuates around changing levels with constant variance. That is, the time series is assumed to be of homoscedastic nature. However, the nancial time series exhibits the presence of heteroscedasticity in the sense that, it possesses non-constant conditional variance given the past observations. So, the analysis of nancial time series, requires the modelling of such variances, which may depend on some time dependent factors or its own past values. This lead to introduction of several classes of models to study the behaviour of nancial time series. See Taylor (1986), Tsay (2005), Rachev et al. (2007). The class of models, used to describe the evolution of conditional variances is referred to as stochastic volatility modelsThe stochastic models available to analyse the conditional variances, are based on either normal or log-normal distributions. One of the objectives of the present study is to explore the possibility of employing some non-Gaussian distributions to model the volatility sequences and then study the behaviour of the resulting return series. This lead us to work on the related problem of statistical inference, which is the main contribution of the thesis

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This study is about the stability of random sums and extremes.The difficulty in finding exact sampling distributions resulted in considerable problems of computing probabilities concerning the sums that involve a large number of terms.Functions of sample observations that are natural interest other than the sum,are the extremes,that is , the minimum and the maximum of the observations.Extreme value distributions also arise in problems like the study of size effect on material strengths,the reliability of parallel and series systems made up of large number of components,record values and assessing the levels of air pollution.It may be noticed that the theories of sums and extremes are mutually connected.For instance,in the search for asymptotic normality of sums ,it is assumed that at least the variance of the population is finite.In such cases the contributions of the extremes to the sum of independent and identically distributed(i.i.d) r.vs is negligible.

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The present study gave emphasis on characterizing continuous probability distributions and its weighted versions in univariate set up. Therefore a possible work in this direction is to study the properties of weighted distributions for truncated random variables in discrete set up. The problem of extending the measures into higher dimensions as well as its weighted versions is yet to be examined. As the present study focused attention to length-biased models, the problem of studying the properties of weighted models with various other weight functions and their functional relationships is yet to be examined.

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The present work is intended to discuss various properties and reliability aspects of higher order equilibrium distributions in continuous, discrete and multivariate cases, which contribute to the study on equilibrium distributions. At first, we have to study and consolidate the existing literature on equilibrium distributions. For this we need some basic concepts in reliability. These are being discussed in the 2nd chapter, In Chapter 3, some identities connecting the failure rate functions and moments of residual life of the univariate, non-negative continuous equilibrium distributions of higher order and that of the baseline distribution are derived. These identities are then used to characterize the generalized Pareto model, mixture of exponentials and gamma distribution. An approach using the characteristic functions is also discussed with illustrations. Moreover, characterizations of ageing classes using stochastic orders has been discussed. Part of the results of this chapter has been reported in Nair and Preeth (2009). Various properties of equilibrium distributions of non-negative discrete univariate random variables are discussed in Chapter 4. Then some characterizations of the geo- metric, Waring and negative hyper-geometric distributions are presented. Moreover, the ageing properties of the original distribution and nth order equilibrium distribu- tions are compared. Part of the results of this chapter have been reported in Nair, Sankaran and Preeth (2012). Chapter 5 is a continuation of Chapter 4. Here, several conditions, in terms of stochastic orders connecting the baseline and its equilibrium distributions are derived. These conditions can be used to rede_ne certain ageing notions. Then equilibrium distributions of two random variables are compared in terms of various stochastic orders that have implications in reliability applications. In Chapter 6, we make two approaches to de_ne multivariate equilibrium distribu- tions of order n. Then various properties including characterizations of higher order equilibrium distributions are presented. Part of the results of this chapter have been reported in Nair and Preeth (2008). The Thesis is concluded in Chapter 7. A discussion on further studies on equilib- rium distributions is also made in this chapter.