Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].

Reliability Engineering and System Safety

In this article, we present a concise review of developments on discrete multivariate distributions. We first present some basic definitions and notations. Then, we present several important discrete multivariate distributions and list their significant properties and characteristics.


Keywords:

generating function;
moments;
stirling numbers;
regression;
inflated distributions;
truncated forms;
compound distributions;
multinomial;
negative multinomial;
multivariate poisson;
multivariate hypergeometric;
multivariate Polya–Eggenberger;
multivariate discrete exponential;
multivariate power series;
multivariate hermite;
multivariate occupancy;
multivariate weighted;
dirichlet;
multivariate run-related distributions

Discrete Multivariate Distributions

(2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

Comparison Methods for Stochastic Models and Risks

Survival Analysis: Techniques for Censored and Truncated Data

The Beta Generalized Weibull distribution: Properties and applications

Relibility measures of weighted distribution of alifeistribution have been derived Sufficientconditions on the weight function have been obtained for the weighted distribution of an IFR distribution to be IFR. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context.

Kanchan Jain

Papers

The Beta Generalized Weibull distribution: Properties and applications

Relations for reliability measures of weighted distributions

Some weighted distribution results on univariate and bivariate cases

Bonferroni Curve and the related statistical inference

Preservation of some partial orderings under the formation of coherent systems