Kernel-based Estimation of Ageing Intensity Function: Properties and Applications

The progressively first failure censored (PFFC) data have become very popular in the past decade due to its usefulness in life testing experiments and in reliability theory. The PFFC data record the number of failures and increase the efficiency of the estimators. The inverse Pareto distribution (IPD) is useful when empirical data suggest a decreasing or upside-down bathtub-shaped failure rate functions. In this article, we consider the classical and Bayesian estimation of the model parameter and the reliability characteristics of the IPD using the PFFC data. The maximum likelihood estimators, asymptotic confidence, and bootstrap confidence intervals are considered in the classical estimation. Under the Bayesian paradigm, Bayes estimators based on non-informative and the gamma informative priors under the squared error loss function using Tierney-Kadane approximation, importance sampling, and Metropolis-Hasting (M-H) algorithm are assessed. In addition, the highest probability density (HPD) intervals based on the M-H algorithm are also constructed. To investigate the efficacy of each of the estimation procedures, numerical computations are performed based on a simulation study. Finally, a real data set is re-analyzed to show the applicability of the IPD model under a censoring scheme.

Recently, Bar-Lev, Bshouty and Van der Duyn Schouten [Math. Methods Stat. 25 (2016) 79–980] developed a systematic method, called operator-based intensity function, for constructing huge classes of nonmonotonic intensity functions (convex or concave) for the nonhomogeneous Poisson process, all of which are suitable for modeling bathtub data. Each class is parametrized by several parameters (as scale and shape parameters) in addition to the operator index parameter [Formula: see text]. For the sake of demonstration only, we focus in this paper on a special subclass called the exponential power law process (EXPLP[Formula: see text]) whose base function is the intensity function of the power-law process. We describe various properties of such a subclass and use one of its special case, namely EXPLP[Formula: see text] intensity function, to analyze failure data which lack monotonicity. Maximum likelihood estimation of the parameters involved and relevant functions thereof is discussed with respect various aspects as existence, uniqueness, asymptotic behavior and statistical inference facets. Using two real datasets from the literature we provide evidence that the EXPLP[Formula: see text] intensity function is well suited to analyze data which exhibit a bathtub behavior.

Network topology can be used to simplify the complexity of the datasets. We are exploring its function in performing survival analysis to identify the most important factors that contributed to the survival time from diagnosis to death. This technique has the potential to illustrate easily some types of complex interactions in dataset. Then, based on those interactions, the most important factors in survival analysis are identified. However, network topology based on classical estimator is extremely sensitive to outlying observations, i.e., conclusions drawn from contaminated network topology could be misleading. Hence, in this paper, the classical estimator, i.e., classical correlation matrix of network topology is substituted with robust estimator, i.e., robust correlation matrix which is developed based on Index Set Equality (ISE). Then, the interpretation of that robust network topology is delivered by using centrality measure, i.e., degree centrality. A case study of the survival time for cervical cancer patients is presented and discussed. Robust network topology revealed that the most important factors that influence the survival of cervical cancer patients is stage at diagnosis (STG). The higher stage of cervical cancer led to shorter survival time of cervical cancer patients. Consequently, early diagnosis is very important. Early diagnosis of cancer allows early intervention to try to slow or prevent cancer development and lethality, hence, the survival improves.

SummaryThe standard maximum likelihood estimator of censored regression models is constructed under the assumption of the normal distribution and it yields inefficient results in case of non‐normal asymmetric data or/and presence of outlier(s). In the analysis of censored real‐life data encountered in many disciplines such as finance, medical sciences, and engineering, there is often a violation of the assumption of normality. In this paper, we propose a generalization of the censored normal regression based on the extended normal distribution (EGT). The generalized model, with two additional shape parameters as a and b, includes classical Tobit model for a = 0 and b = 0 and Alpha‐Power Tobit model for a = 1. It provides very flexible estimation in cases symmetric‐asymmetric data and especially in case non‐normal asymmetric data or/and presence of outlier(s). In addition, Lehmann type II‐G Tobit model (LTII‐GT) is also proposed for b = 1 as a special case of generalization. The performance of proposed two new generalizations of Tobit, namely EGT and LTII‐GT has been compared with Tobit, Alpha‐Power Tobit, and classical estimators by means of a comprehensive Monte Carlo simulation study. The performance of proposed models' ML estimators is illustrated on a real data set. The superiority of the proposed estimators is shown.

Reliability analysis is the key to evaluate software’s quality. Since the early 1970s, the Power Law Process, among others, has been used to assess the rate of change of software reliability as time-varying function by using its intensity function. The Bayesian analysis applicability to the Power Law Process is justified using real software failure times. The choice of a loss function is an important entity of the Bayesian settings. The analytical estimate of likelihood-based Bayesian reliability estimates of the Power Law Process under the squared error and Higgins-Tsokos loss functions were obtained for different prior knowledge of its key parameter. As a result of a simulation analysis and using real data, the Bayesian reliability estimate under the Higgins-Tsokos loss function not only is robust as the Bayesian reliability estimate under the squared error loss function but also performed better, where both are superior to the maximum likelihood reliability estimate. A sensitivity analysis resulted in the Bayesian estimate of the reliability function being sensitive to the prior, whether parametric or non-parametric, and to the loss function. An interactive user interface application was additionally developed using Wolfram language to compute and visualize the Bayesian and maximum likelihood estimates of the intensity and reliability functions of the Power Law Process for a given data.

Nonparametric estimation of averages and errors for small data-sets in isotope geoscience: a proposal

There are many measures for qualitative and quantitative analysis of ageing phenomena of a system with a well defined statistical distribution, namely, hazard rate, ageing intensity (AI) function etc. The extensive usage of Weibull distributions in various fields has prompted us to explore the ageing property of these models. The present work derives the AI function of recent continuous Weibull distributions. In the process, we obtain generalizations of a few recent Weibull models. As noted, the characteristics of a distribution described by hazard rate and ageing intensity function differ altogether in many cases. Some of the results presented in this work have been analyzed through a simulation study using R software.

The mean residual lifetime (MRL) of a unit is its expected additional lifetime provided that it has survived until time t. The MRL estimation problem has been frequently addressed in the literature since it has wide applications in statistics, reliability and survival analysis. In this paper, we consider the problem of estimating the MRL in ranked set sampling when actual quantifications of a concomitant variable are available. To exploit the additional information of the concomitant variable, we introduce several MRL estimators based on some regression techniques. We then compare them with the standard MRL estimator in simple random sampling using Monte Carlo simulation and a real dataset from the Surveillance, Epidemiology, and End Results Program. Our results indicate the superiority of the procedures that we have developed when the quality of ranking is fairly good.

Seed germination experiments are conducted in a wide variety of biological disciplines. Numerous methods of analysing the resulting data have been proposed, most of which fall into three classes: intuition-based germination indexes, classical non-linear regression analysis and time-to-event analysis (also known as survival analysis, failure-time analysis and reliability analysis). This paper briefly reviews all three of these classes, and argues that time-to-event analysis has important advantages over the other methods but has been underutilized to date. It also reviews in detail the types of time-to-event analysis that are most useful in analysing seed germination data with standard statistical software. These include non-parametric methods (life-table and Kaplan–Meier estimators, and various methods for comparing two or more groups of seeds) and semi-parametric methods (Cox proportional hazards model, which permits inclusion of categorical and quantitative covariates, and fixed and random effects). Each method is illustrated by applying it to a set of real germination data. Sample code for conducting these analyses with two standard statistical programs is also provided in the supplementary material available online (athttp://journals.cambridge.org/). The methods of time-to-event analysis reviewed here can be applied to many other types of biological data, such as seedling emergence times, flowering times, development times for eggs or embryos, and organism lifetimes.

This paper presents an effort to investigate the estimations of the Weibull distribution using an improved adaptive Type-II progressive censoring scheme. This scheme effectively guarantees that the experimental time will not exceed a pre-fixed time. The point and interval estimations using two classical estimation methods, namely maximum likelihood and maximum product of spacing, are considered to estimate the unknown parameters as well as the reliability and hazard rate functions. The approximate confidence intervals of these quantities are obtained based on the asymptotic normality of the maximum likelihood and maximum product of spacing methods. The Bayesian estimations are also considered using MCMC techniques based on the two classical approaches. An extensive simulation study is implemented to compare the performance of the different methods. Further, we propose the use of various optimality criteria to find the optimal sampling scheme. Finally, one real data set is applied to show how the proposed estimators and the optimality criteria work in real-life scenarios. The numerical outcomes demonstrated that the Bayesian estimates using the likelihood and product of spacing functions performed better than the classical estimates.

The Burr-XII distribution has been widely applied in engineering, reliability, and survival analysis. Due to its importance, in this article, statistical inferences for Burr-XII distribution under a joint type-II censoring scheme is discussed. The classical likelihood estimation of unknown model parameters is studied via different calculating approaches, such as the expectation-maximization (EM) algorithm and approximate confidence intervals (ACIs) using the observed Fisher information matrix are obtained. The asymptotic bootstrap confidence intervals are also computed. In the sequel, Bayesian estimations of unknown parameters with a gamma prior distribution are considered under squared error, linear-exponential, and generalized entropy loss functions. Subsequently, we calculate the Bayesian credible interval using the importance sample. The performance of the developed methods is investigated through a Monte Carlo simulation study and two real-life examples. The results showed that the proposed estimation strategies have satisfactory results. However, Bayesian approaches were preferable to EM in terms of lower mean square error and higher coverage probability.

Preface. 1 Introduction. 1.1 The Importance of Risk and Uncertainty Assessments. 1.2 The Need to Develop a Proper Risk Analysis Framework. Bibliographic Notes. 2 Common Thinking about Risk and Risk Analysis. 2.1 Accident Risk. 2.1.1 Accident Statistics. 2.1.2 Risk Analysis. 2.1.3 Reliability Analysis. 2.2 Economic Risk. 2.2.1 General Definitions of Economic Risk in Business and Project Management. 2.2.2 A Cost Risk Analysis. 2.2.3 Finance and Portfolio Theory. 2.2.4 Treatment of Risk in Project Discounted Cash Flow Analysis. 2.3 Discussion and Conclusions. 2.3.1 The Classical Approach. 2.3.2 The Bayesian Paradigm. 2.3.3 Economic Risk and Rational Decision-Making. 2.3.4 Other Perspectives and Applications. 2.3.5 Conclusions. Bibliographic Notes. 3 How to Think about Risk and Risk Analysis. 3.1 Basic Ideas and Principles. 3.1.1 Background Information. 3.1.2 Models and Simplifications in Probability Considerations. 3.1.3 Observable Quantities. 3.2 Economic Risk. 3.2.1 A Simple Cost Risk Example. 3.2.2 Production Risk. 3.2.3 Business and Project Management. 3.2.4 Investing Money in a Stock Market. 3.2.5 Discounted Cash Flow Analysis. 3.3 Accident Risk. Bibliographic Notes. 4 How to Assess Uncertainties and Specify Probabilities. 4.1 What Is a Good Probability Assignment? 4.1.1 Criteria for Evaluating Probabilities. 4.1.2 Heuristics and Biases. 4.1.3 Evaluation of the Assessors. 4.1.4 Standardization and Consensus. 4.2 Modelling. 4.2.1 Examples of Models. 4.2.2 Discussion. 4.3 Assessing Uncertainty of Y. 4.3.1 Assignments Based on Classical Statistical Methods. 4.3.2 Analyst Judgements Using All Sources of Information. 4.3.3 Formal Expert Elicitation. 4.3.4 Bayesian Analysis. 4.4 Uncertainty Assessments of a Vector X. 4.4.1 Cost Risk. 4.4.2 Production Risk. 4.4.3 Reliability Analysis. 4.5 Discussion and Conclusions. Bibliographic Notes. 5 How to Use Risk Analysis to Support Decision-Making. 5.1 What Is a Good Decision? 5.1.1 Features of a Decision-Making Model. 5.1.2 Decision-Support Tools. 5.1.3 Discussion. 5.2 Some Examples. 5.2.1 Accident Risk. 5.2.2 Scrap in Place or Complete Removal of Plant. 5.2.3 Production System. 5.2.4 Reliability Target. 5.2.5 Health Risk. 5.2.6 Warranties. 5.2.7 Offshore Development Project. 5.2.8 Risk Assessment: National Sector. 5.2.9 Multi-Attribute Utility Example. 5.3 Risk Problem Classification Schemes. 5.3.1 A Scheme Based on Potential Consequences and Uncertainties. 5.3.2 A Scheme Based on Closeness to Hazard and Level of Authority. Bibliographic Notes. 6 Summary and Conclusions. Appendix A: Basic Theory of Probability and Statistics. A.1 Probability Theory. A.1.1 Types of Probabilities. A.1.2 Probability Rules. A.1.3 Random Quantities (Random Variables). A.1.4 Some Common Discrete Probability Distributions (Models). A.1.5 Some Common Continuous Distributions (Models). A.1.6 Some Remarks on Probability Models and Their Parameters. A.1.7 Random Processes. A.2 Classical Statistical Inference. A.2.1 Non-Parametric Estimation. A.2.2 Estimation of Distribution Parameters. A.2.3 Testing Hypotheses. A.2.4 Regression. A.3 Bayesian Inference. A.3.1 Statistical (Bayesian) Decision Analysis. Bibliographic Notes. Appendix B: Terminology. Bibliography. Index.

A point process is a collection of positional variables in a given space, while a marked point process (MPP) is basically a point process that measures some additional feature or value at each point. Existing applications such as financial market models and hospitalization models have demonstrated that MPPS are suitable for modeling random events with meaningful values. A big challenge is to study the relationship between different types of events, or how different types of markers affect each other. While Hawkes procedures are an effective and efficient way to model point processes with internal influences, inferential methods of intensity functions can be expensive, especially for multi-dimensional processes. Graph convolutional Network (GCN) is a powerful neural network designed to directly process graphs. A GCN-based model is presented to study potential patterns in the marking process and to predict the location of future events without estimating the Hawkes intensity function. A real data set of water pipe leakage and rupture records over the last 50 years was used in the experiment, modeled as a marker point process, where leakage and rupture are two marker forms. Experimental results on synthetic data sets and real water pipe data sets show that the proposed model is superior to recent state-of-the-art methods.

Biological networks provide additional information for the analysis of human diseases, beyond the traditional analysis that focuses on single variables. Gaussian graphical model (GGM), a probability model that characterizes the conditional dependence structure of a set of random variables by a graph, has wide applications in the analysis of biological networks, such as inferring interaction or comparing differential networks. However, existing approaches are either not statistically rigorous or are inefficient for high-dimensional data that include tens of thousands of variables for making inference. In this study, we propose an efficient algorithm to implement the estimation of GGM and obtain p-value and confidence interval for each edge in the graph, based on a recent proposal by Ren et al., 2015. Through simulation studies, we demonstrate that the algorithm is faster by several orders of magnitude than the current implemented algorithm for Ren et al. without losing any accuracy. Then, we apply our algorithm to two real data sets: transcriptomic data from a study of childhood asthma and proteomic data from a study of Alzheimer’s disease. We estimate the global gene or protein interaction networks for the disease and healthy samples. The resulting networks reveal interesting interactions and the differential networks between cases and controls show functional relevance to the diseases. In conclusion, we provide a computationally fast algorithm to implement a statistically sound procedure for constructing Gaussian graphical model and making inference with high-dimensional biological data. The algorithm has been implemented in an R package named “FastGGM”.

The study of the First Passage Time (FPT) problem (also known as first passage problem, FPP) started more than a century ago, but its diverse applications in science and engineering mostly emerged in the last two to three decades. Assuming that X(t) is a one-dimensional stochastic process, the First Passage Time is defined as the time (T) when X(t) first crosses a threshold. Engineering reliability is obviously a suitable application domain, and indeed applications such as optimal dam design in hydrology and analysis of structural failure in civil and mechanical engineering are typical examples. Although we envision that the FPT problem has great potential in network and software reliability, it should be more useful for network security and survivability because the approaches developed for the FPT problem are mostly analytical. The assumption for this inference is that in reliability analysis, experimental or historical data are often more readily available, which makes statistical approaches such as survival analysis more convenient and likely more realistic. In contrast, data is generally more difficult to obtain in security and survivability analyses, and analytical approaches can be leveraged to play more important roles. Furthermore, security and survivability often have to deal with malicious actions that may be driven by sophisticated cognition and behavioral processes, which are highly variable over time and very difficult to detect with short term data. If the behavior of an intruder can be characterized with some stochastic process such as Brownian motion, then the FPT approach may be applied to find the closed-form solution of the probability density function (PDF) of the first passage time, which can be the time when the system breaks down or when the hacker is successful in compromising a network. In addition, the solutions to FPT depend on boundary and initial conditions of the corresponding partial differential equations, and they also describe the evolution of PDF over time. This may suggest that it is possible to model the behavior changes of an intruder over time and circumstances. Another advantage of FPT analysis is that it may help solve some non-Markov stochastic process problems in reliability analysis and survival analysis. In this article, we first briefly introduce the FPT problem with Brownian motion as an example, and then suggest its potential applications in software reliability and network security.