In this article, we study the applicability of Benford’s law and Zipf’s law to national COVID-19 ... more In this article, we study the applicability of Benford’s law and Zipf’s law to national COVID-19 case figures with the aim of establishing guidelines upon which methods of fraud detection in epidemiology, based on formal statistical analysis, can be developed. Moreover, these approaches may also be used in evaluating the performance of public health surveillance systems. We provide theoretical arguments for why the empirical laws should hold in the early stages of an epidemic, along with preliminary empirical evidence in support of these claims. Based on data published by the World Health Organization and various national governments, we find empirical evidence that suggests that both Benford’s law and Zipf’s law largely hold across countries, and deviations can be readily explained. To the best of our knowledge, this paper is among the first to present a practical application of Zipf’s law to fraud detection.
ESAIM: Control, Optimisation and Calculus of Variations
In our present article, we follow our way of developing mean field type control theory in our ear... more In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussan et al., Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gau...
ESAIM: Control, Optimisation and Calculus of Variations
In this article, we study a control problem in an appropriate space of random variables; in fact,... more In this article, we study a control problem in an appropriate space of random variables; in fact, in our set up, we can consider an arbitrary Hilbert space, yet we specialize only to a Hilbert space of square-integrable random variables. We see that the control problem can then be related to a mean field type control problem. We explore here a suggestion of Lions in (Lectures at College de France, urihttp://www.college-de-france.frhttp://www.college-de-france.fr) and (Seminar at College de France). Mean field type control problems are control problems in which functionals depend on probability measures of the underlying controlled process. Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643] considered this type of problem in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient; their work provides a completely rigorous treatment, but it is quite intricate, because metric spaces are not vector spaces. The approach sug...
In this article, we study the applicability of Benford’s law and Zipf’s law to national COVID-19 ... more In this article, we study the applicability of Benford’s law and Zipf’s law to national COVID-19 case figures with the aim of establishing guidelines upon which methods of fraud detection in epidemiology, based on formal statistical analysis, can be developed. Moreover, these approaches may also be used in evaluating the performance of public health surveillance systems. We provide theoretical arguments for why the empirical laws should hold in the early stages of an epidemic, along with preliminary empirical evidence in support of these claims. Based on data published by the World Health Organization and various national governments, we find empirical evidence that suggests that both Benford’s law and Zipf’s law largely hold across countries, and deviations can be readily explained. To the best of our knowledge, this paper is among the first to present a practical application of Zipf’s law to fraud detection.
ESAIM: Control, Optimisation and Calculus of Variations
In our present article, we follow our way of developing mean field type control theory in our ear... more In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussan et al., Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gau...
ESAIM: Control, Optimisation and Calculus of Variations
In this article, we study a control problem in an appropriate space of random variables; in fact,... more In this article, we study a control problem in an appropriate space of random variables; in fact, in our set up, we can consider an arbitrary Hilbert space, yet we specialize only to a Hilbert space of square-integrable random variables. We see that the control problem can then be related to a mean field type control problem. We explore here a suggestion of Lions in (Lectures at College de France, urihttp://www.college-de-france.frhttp://www.college-de-france.fr) and (Seminar at College de France). Mean field type control problems are control problems in which functionals depend on probability measures of the underlying controlled process. Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643] considered this type of problem in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient; their work provides a completely rigorous treatment, but it is quite intricate, because metric spaces are not vector spaces. The approach sug...
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