Understanding the shape of the mixture failure rate (with engineering and demographic applications)

Mixtures of distributions are usually effectively used for modeling heterogeneity. It is well known that mixtures of DFR distributions are always DFR. On the other hand, mixtures of IFR distributions can decrease, at least in some intervals of time. As IFR distributions often model lifetimes governed by ageing processes, the operation of mixing can dramatically change the pattern of ageing. Therefore, the study of the shape of the observed (mixture) failure rate in a heterogeneous setting is important in many applications. We study discrete and continuous mixtures, obtain conditions for the mixture failure rate to tend to the failure rate of the strongest populations and describe asymptotic behavior as t tends to infty. Some demographic and engineering examples are considered. The corresponding inverse problem is discussed.

A New Class of Generalized Modified Weibull Distribution with Applications

A new five parameter gamma-generalized modified Weibull (GGMW) distribution which includes exponential, Rayleigh, modified Weibull, Weibull, gamma-modified Weibull, gamma-modified Rayleigh, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh, and gamma-exponential distributions as special cases is proposed and studied. Some mathematical properties of the new class of distributions including moments, distribution of the order statistics, and Renyi entropy are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to a real datasets to illustrates the usefulness of the proposed class of models are presented

Interrelations among child mortality, breastfeeding, and fertility in Egypt, 1975-80

This report examines the relationship between reproductive behavior and child survivalin Egypt. The relationship is of fundamental importance to an understanding of demographic dynamics and for the formulation of population policies. Using Egyptian data from 1975-80, it was found that weaning children in infancy increases ths risk of death for children under five. Early weaning is responsible for up to 29 percent of Egyptian children's deaths. Children whose mothers become pregnant again are more likely to die if the pregnancy begins while the child is still an infant. Ending breastfeeding is responsible for up to 41 percent of pregnancies - 52 percent among women who do not use contraceptives. Breastfeeding lasts an average of 17 to 18 months in Egypt, so policy probably should not encourage all women to breastfeed longer, but women who breastfeed for only short periods should probably be encouraged to breastfeed longer. Parents should be encouraged to be more careful about childcare and children's diet and hygiene after weaning. An important feature of this analysis is that fertility was examined simultaneously with child survival and breastfeeding, as three components of a system. The analysis involved regression models for the hazard, or risk, of three events occurring after a live birth: another pregnancy, weaning, or the death of the child.Health Monitoring&Evaluation,Early Child and Children's Health,Adolescent Health,Gender and Health,Reproductive Health

Bayes approach to explore the mixture failure rate model

This thesis has two folds: Firstly, designing mixture failure rate functions by combing few other existing failure rate functions to obtain desirable mixture failure rate functions. The first proposed mixture failure rate is the non-linear failure rate. This failure rate is a mixture of the exponential and Weibull failure rate functions. It was designed for modeling data sets in which failures result from both random shock and wear out or modeling a series system with two components, where one component follows an exponential distribution and the other follows a Weibull distribution. The second proposed mixture failure rate is the additive Chen-Weibull failure rate. This failure rate is considered a mixture of the Chen and Weibull failure rates. It is decided for modeling lifetime data with flexible failure rate including bathtub-shaped failure rate. The final proposed mixture failure rate is the improvement of new modified Weibull failure rate. This failure rate is a mixture of the Weibull and modified Weibull failure rates. It is also decided for modeling lifetime data with flexible failure rate including bathtub-shaped failure rate. The superiority of the proposed models have been demonstrated by fitting to many well-known lifetime data sets. And secondly, applying modern statistical methods and techniques, such as the maximum likelihood estimation, Bayesian inference, cross-entropy method, adaptive Markov chain Monte Carlo, Hamiltonian Monte Carlo and bootstrapping, for analyzing failure time distributions which result from those mixture failure rate functions.Tato disertační práce byla vyvíjena dvěma směry: Za prvé, návrh směsových funkcí intenzity poruch, vycházející z kombinování několika existujících intenzit poruch s cílem získat požadovanou směsovou funkci intenzity poruch. První navržená směsová intenzita poruch je nelineární intenzita poruch. Tato funkce intenzity poruch je směsí exponenciální a Weibullovy funkce intenzity poruch. Byla navržena pro účely modelování datových souborů, ve kterých poruchy jsou výsledkem jak náhodných šoků, tak i procesu opotřebení, neboli modelování poruch lze popsat jako sériový systém se dvěma komponentami, kde jedna komponenta se řídí exponenciálním rozdělením a druhá Weibullovým. Druhá navrhovaná směsová funkce intenzity poruch je aditivní Chen-Weibullova intenzita poruch. Tato intenzita poruch je uvažována jako směs Chenovy a Weibullovy intenzity poruch. Je navržena pro účely modelování dat, popisujících životnost nějakých objektů, kdy intenzita poruch vykazuje velmi flexibilní chování, včetně průběhu ve tvaru vanové křivky. Poslední navržená směsová intenzita poruch představuje inovaci nově modifikované Weibullovy intenzity poruch, což je směs Weibullovy a nově modifikované Weibullovy intenzity poruch. Je to další alternativa pro modelování dat popisujících životnost, kdy intenzita poruch vykazuje velmi flexibilní chování, včetně průběhu ve tvaru vanové křivky. Vysoká kvalita navržených modelů byla demonstrována na mnoha známých datových souborech, vybraných ze světové literatury. Za druhé, byly aplikovány a programově implementovány moderní metody a techniky teorie odhadu, vycházející z Bayesova přístupu, používané pro analýzu takových pravděpodobnostních rozdělení doby do poruchy, která jsou založena na směsové funkci intenzity poruch.470 - Katedra aplikované matematikyvyhově

"Nonlinear" covariance matrix and portfolio theory for non-Gaussian multivariate distributions

This paper offers a precise analytical characterization of the distribution of returns for a portfolio constituted of assets whose returns are described by an arbitrary joint multivariate distribution. In this goal, we introduce a non-linear transformation that maps the returns onto gaussian variables whose covariance matrix provides a new measure of dependence between the non-normal returns, generalizing the covariance matrix into a non-linear fractional covariance matrix. This nonlinear covariance matrix is chiseled to the specific fat tail structure of the underlying marginal distributions, thus ensuring stability and good-conditionning. The portfolio distribution is obtained as the solution of a mapping to a so-called phi-q field theory in particle physics, of which we offer an extensive treatment using Feynman diagrammatic techniques and large deviation theory, that we illustrate in details for multivariate Weibull distributions. The main result of our theory is that minimizing the portfolio variance (i.e. the relatively ``small'' risks) may often increase the large risks, as measured by higher normalized cumulants. Extensive empirical tests are presented on the foreign exchange market that validate satisfactorily the theory. For ``fat tail'' distributions, we show that an adequete prediction of the risks of a portfolio relies much more on the correct description of the tail structure rather than on their correlations.Comment: Latex, 76 page

Extreme value distributions and Renormalization Group

In the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent, identically distributed random variables are studied. So far, only affine rescalings have been considered. We show, however, that more general rescalings are natural and lead to new limit distributions, apart from the Gumbel, Weibull, and Fr\'echet families. The problem is approached using the language of Renormalization Group transformations in the space of probability densities. The limit distributions are fixed points of the transformation and the study of the differential around them allows a local analysis of the domains of attraction and the computation of finite-size corrections.Comment: 16 pages, 5 figures. Final versio

New survival distributions that quantify the gain from eliminating flawed components

A general method for deriving new survival distributions from old is presented. This yields a class of useful mixture distributions. Fitting such distributions to failure-time data allows estimation of the improvement in reliability that could be gained from eliminating ‘frail’ components. One model parameter is the proportional increase of expected survival time that could be achieved. Some 2 and 3 parameter distributions in this class are described, which are extensions of the Weibull, exponential, gamma and lognormal distributions. The methodology is illustrated by fitting some well travelled datasets. Keywords: Weibull distribution, gamma distribution, mixture distribution, hazard function, partial integration, frailty mode