This paper presents the Bayes Fisher information measures, defined by the expected Fisher information under a distribution for the parameter, for the arithmetic, geometric, and generalized mixtures of two probability density functions. The Fisher information of the arithmetic mixture about the mixing parameter is related to chi-square divergence, Shannon entropy, and the Jensen-Shannon divergence. The Bayes Fisher measures of the three mixture models are related to the Kullback-Leibler, Jeffreys, Jensen-Shannon, Rényi, and Tsallis divergences. These measures indicate that the farther away are the components from each other, the more informative are data about the mixing parameter. We also unify three different relative entropy derivations of the geometric mixture scattered in statistics and physics literatures. Extensions of two of the formulations to the minimization of Tsallis divergence give the generalized mixture as the solution.

The objective of this paper is to provide a hybrid algorithm for non-negative matrix factorization based on a symmetric version of Kullback-Leibler divergence, known as intrinsic information. The convergence of the proposed algorithm is shown for several members of the exponential family such as the Gaussian, Poisson, gamma and inverse Gaussian models. The speed of this algorithm is examined and its usefulness is illustrated through some applied problems.

Recent advances in nanotechnology have provided the opportunity to significantly enhance the performance of hydrogen gas nanosensors. Our research focuses on the reliability of one particular nanosensor, a network of ultra small palladium nanowires, which detects hydrogen gas through a change in resistivity. The discrete random variable, representing the lifetime of the nanosensor, is defined as the number of exposures to, or cycles of, hydrogen gas that the nanosensor can withstand before it no longer functions. The nanosensor is modeled, and the reliability is analyzed under the assumption that the nanosensor is performing in an environment where the probability of a nanowire breaking changes after each cycle of hydrogen gas. Nanoscale components present unique difficulties when evaluating the reliability of any device. We attempt to resolve some of these issues by creating a flexible model that allows for the unknown characteristics of the nanosensor to be accounted for. Although this work is motivated by one particular nanosensor, our results can also be applied to assess the reliability of any nanodevice where our proposed model is a reasonable choice.

Rapid development in genomics in recent years has allowed the simultaneous measurement of the expression levels of thousands of genes using DNA microarrays. This has offered tremendous potential for growth in our understanding of the pathophysiology of many diseases. When microarray studies also contain information about an outcome variable such as time to an event or death, one of the goals of an investigator is to understand how the expression levels of genes (covariates) relate to the time-to-event (referred to as survival time) in the course of a disease.In this article, we consider the case where the number of covariates, p, exceeds the number of observations, N, a setting typical of microarray gene expression data. For a given vector of responses representing survival times of N subjects and the corresponding p × N gene expression matrix, we examine the problem of predicting the survival probability when N ≪ p. This is an ill-conditioned problem further compounded by the presence of possibly censored survival times. We propose a model that combines the partial least squares approach for dimensionality reduction with the accelerated failure time model, a widely used log-linear model for linking censored survival time to covariates. We develop parametric methods to account for censoring as well as for predicting patient survival probabilities. We illustrate the applicability of our methods using cancer microarray data and explore the biological relevance of our results using pathway analysis. Finally, we evaluate the performance of our methods using extensive simulation studies.

Nanocomponents are devices that are in the size range of billionth of a meter (1×10-9) , and therefore are built necessarily from individual atoms. A nanocomponent fails if the component stops performing its intended function. If one dimension of a component is reduced to the nanorange (below 100 nanometers) while the other two dimensions remains large, then we obtain a nanocomponent with a structure known as a quantum well. If two dimensions of a component are reduced to nanorange and one remains large, then we obtain a nanocomponent with a structure known as a quantum wire. If all three dimensions are reduced to nanorange, then we obtain a nanocomponent with a structure known as a quantum dot. The family of two dimensional nanocomponents (2-D nanocomponents) encompasses all the nanocomponents where their individual atoms can move in two directions. Usually, a nanocomponent with the quantum well structure is a 2-D nanocomponent. However, there are nanocomponents with a quantum wire or a quantum dot structure that can be 2-D nanocomponents. In this article, we assess the 2-D nanocomponent's reliability, which is the probability of the nanocomponent not failing. We consider the 2-D nanocomponent at a fixed moment of time, say the present moment. We assume that the present state of the 2-D nanocomponent depends only on the present states of its atoms, and focus on two types of 2-D nanocomponents. Both cases describe the relationship between failure of a 2-D nanocomponent and its atoms. Our results indicate that there are situations where both types of nanocomponents have reliability virtually zero, i.e., they are highly unreliable. Also, we provide situations where both types have reliability virtually unity, i.e. they are highly reliable.

We explore properties of the entropy, Kullback-Leibler information, and mutual information for order statistics. The probability integral transformation plays a pivotal role in developing our results. We provide bounds for the entropy of order statistics and some results that relate entropy ordering of order statistics to other well-known orderings of random variables. We show that the discrimination information between order statistics and data distribution, the discrimination information among the order statistics, and the mutual information between order statistics are all distribution free and are computable using the distributions of the order statistics of the samples from the uniform distribution. We also discuss information properties of spacings for uniform and exponential samples and provide a large sample distribution-free result on the entropy of spacings. The results show interesting symmetries of information orderings among order statistics.

Assessing system-reliability has been a major goal of many people in many walks of life, including those that develop and deploy systems. For that purpose many methods and techniques have been published. However, most of these techniques and methods rely heavily on failure data. Because the design of high-reliability systems generally requires that the individual system components have extremely high reliability, even after long periods of time, and because short product-development times and tightened-budgets imply that reliability tests must be conducted with severe time-constraints, frequently no failures occur during such tests. Thus, there is a growing need to assess reliabilities for complex systems for which it is difficult to collect failure data. This paper focuses on a method that expresses the failure time and consequently the reliability of a system as a function of several explanatory variables (referred to as covariates). Then, the system reliability is assessed indirectly by using information on these covariates. From a practical viewpoint, this method might be more complicated and difficult to apply. The author still recommends this methodology because there are many situations where it is difficult to have data on failure times of expensive and complex systems, and this method can handle such situations.

In this correspondence, we examine the relation between two random variables with common entropy function. The entropy has been used to measure dispersion, uncertainty, risk, and volatility. There is no general relationship between two random variables having equal entropies. We present results that identify certain conditions based on the entropy under which two random variables are stochastically equal. We also obtain similar results for the relative entropy.

One of software engineering's long-standing problems is to estimate the cost of a software project. Using the volume or size of a program to estimate the cost is a common practice in many software development organizations. However, in many situations one is unable to observe the value of this variable at the beginning of the project; one has to estimate it. In this paper we introduce a fairly general model which permits a surrogate or a proxy variable to be observed instead of the actual size. Under this model we obtain estimation of software cost at the given value of a surrogate variable. A confidence interval is also provided under this model.

Sometimes complex software systems fail because of faults introduced in the requirements and design stages of the development process. To improve the reliability of a software, several reviewers inspect documents related to requirements and design. Some faults are detected in this process but often a few remain undetected until the software is developed. Ebrahimi (1997) developed frequentist methods to estimate the number of faults which are nor discovered. Later, Basu and Ebrahimi and Basu developed different Bayesian models. These different Bayesian models yield different estimates of the number of undetected errors. It was further found that changes in the prior parameters often result in measurable changes in the estimates of undetected errors. The authors address the issue of model selection among these different models. They advocate two model selection methods. The first method uses marginal likelihood and Bayes factor whereas the second method is based on cross-validated likelihood. These methods are illustrated in the software review data of AT&T 5 ESS switches.

Sometimes, complex software systems fail because of faults introduced in the requirements and design stages of the development process. Reviewing documents related to requirements and design by several reviewers can remove some of these faults, but often a few remain undetected until the software is developed. In this paper, we propose a procedure leading to the estimation of the number of faults which are not discovered. The main advantage of our procedure is that we do not need the standard assumption of independence among reviewers.

Engineers often make changes during system development in order to correct design weaknesses. If done well this results in reliability growth as development continues. Since no reliability-growth model is adequate for all time (any growth model is valid only for some finite amount of time) it is prudent to use a different reliability model between each two design modifications. This paper assumes that the reliability growth model between each two design modifications is either a Poisson process or power-law process. Maximum likelihood estimates of the parameters are obtained for two cases: (1) no constraints on parameters; and (2) some constraints on parameters.

This article overviews some foundation issues in reliability and argues that generally it does not make sense to use classical notions of the hazard rate (HR) and mean residual life (MRL) to describe the HR and MRL of a device (the HR and failure rate are interchangeable terms). It provides a new methodology, using engineering notions of HR and MRL, to describe the HR and MRL of a device. and discusses the relevance of these methods in comparison to existing methods. In practice, the engineering notions of HR and MRL can be more complicated and more difficult to compute than the classical notions of these functions. Nevertheless, one should adhere to the procedures in this paper insofar as feasible because, as the examples indicate, the engineering notions of these functions dramatically differ from the classical notions. The finite populations have specified joint distributions.