China
Africa
Explore chapters and articles related to this topic
Natural Disasters and Structural Survivability
Published in Vladimir Raizer, Isaac Elishakoff, Philosophies of Structural Safety and Reliability, 2022
Vladimir Raizer, Isaac Elishakoff
This law determines the number of occurrences of rare events, while the theory of extreme values (Gumbel, 1967) considers their values. For random events where extreme values play major role, the asymptotic theory of extreme order statistics provides in some cases relatively accurate but mostly approximate probabilistic models. Therefore, if the basic assumptions of the model have a similarity to the main conditions of a real situation of a catastrophic action, the complex real conditions can be simulated by a considerably simple asymptotic model. Areas, where dangerous phenomena can occur at intensity levels exceeding those on the record (earthquake exceeding the design level, etc.), can be determined and assessed by the test’s observations of similar but less intensive occurrences.
Bayesian Quickest Change Detection in a Single Population
Published in Alexander G. Tartakovsky, Sequential Change Detection and Hypothesis Testing, 2019
In this chapter, we provide the asymptotic Bayesian theory of change detection for the composite post-change hypothesis where the post-change parameter is unknown. We assume that the observations can have a very general structure, i.e., can be dependent and non-identically distributed. The key assumption in the general asymptotic theory is a stability property of the log-likelihood ratio process between the “change” and “no-change” hypotheses, which can be formulated in terms of a Law of Large Numbers and rates of convergence, e.g., as the r-complete convergence of the properly normalized log-likelihood ratio and its adaptive version in the vicinity of the true parameter value.
Nonlinear Time Series Analysis
Published in Tucker S. McElroy, Dimitris N. Politis, Time Series, 2019
Tucker S. McElroy, Dimitris N. Politis
Exercise 11.32. Testing Entropy of Residuals [♠] Write code for the kurtosis estimator, using the asymptotic theory of Exercise 11.31. Apply this test statistic, along with the test of total variation (Remark 10.8.9), to determine whether a given Exponential NoVaS transformation is successful at yielding a residual of higher entropy (Exercise 11.30). Apply to a simulation from the ARCH(1) process of Example 11.12, finding choices of p and r such that maximum entropy residuals are obtained.
Degradation in Common Dynamic Environments
Published in Technometrics, 2018
A notable feature of the variance-gamma and the NIG models is that each subject possesses an independent realization of the subordinated stochastic process. This enables the two models to explain large variations in the observed data. Our study also uses the IG process as the subordinator, whereas the purpose is different from the two models in that: (a) Instead of independent subordinators among subjects, we allow a number of subjects to share the same subordinator, that is, a common time scale; and (b) our main purpose is to capture the temporal correlation of subjects in the same group, though explanation of large variations is a by-product of our formulation. Therefore, our model arises more from consideration of the underlying degradation physics than mathematical tractability. The existing NIG model (e.g., Barndorff-Nielsen 1997; Wang 2009) is called the iid NIG process in our study, and ours is called the group NIG process in contrast. The common subordinator introduces correlation among subjects in the same group, thereby making statistical inference for the model challenging. As the subordinator is nondecreasing, it involves constrained optimization problems in the nonparametric estimation of the mean subordinator. Particularly, the estimation would be further complicated if the observation times for different groups are nonidentical. To tackle these problems, an EM algorithm is developed for the maximum likelihood (ML) estimation. The large sample properties of the estimator are established using asymptotic theory while the finite-sample properties are justified by simulations. In addition, parametric models with specified forms for the mean degradation are also studied.