Poisson 2021
Lead Research Organisation:
Middlesex University
Department Name: Faculty of Science & Technology
Abstract
Probability Theory is a branch of science dealing with random events. It provides mathematical tools for evaluating chances, forecasting possible outcomes, and suggesting optimal choices in the presence of uncertainty.
The theory of sums of random variables lies at the heart of research in Probability Theory. It is truly fundamental when dealing with the aggregate effect of random events, e.g., when observing the combined effect of a large number of small or rare random contributions.
The work on the theory of sums of random variables started in the 17th century, and continued ever since. Among famous contributors are Bernoulli, de Moivre, Laplace, and other well-known mathematicians. Their results have shaped modern Probability Theory.
The actual distribution of a sum of random variables is typically complex, and one would prefer using a simpler and more tractable approximate distribution (e.g., normal, Poisson or compound Poisson). However, one can only substitute a complex actual distribution by an approximate one if there is a sharp estimate of the accuracy of approximation indicating the error would be "small".
As a result, a lot of work in Probability Theory has been devoted to evaluating the accuracy of approximation. In particular, work on the accuracy of normal approximation to the distributions of sums of random variables started in the late 19th century, and still continues.
However, in situations where one deals with rare events a natural approximating distribution is Poisson (or, more generally, compound Poisson).
The class of compound Poisson laws is so general that the class of all possible limit laws to the distributions of sums of asymptotically "small" random variables (the class of so-called infinitely divisible distributions) coincides with the class of weak limits of compound Poisson distributions.
Poisson and compound Poisson approximations naturally arise when one deals with the number of long head runs is discrete random sequences, the number of long match patterns in DNA sequences, aggregate claims to a (re)insurance company, the number of exceedances of high thresholds in extreme value theory, etc..
The approximation of a complex actual distribution by a Poisson and compound Poisson one is only justified if there is a sharp estimate of the accuracy of approximation indicating the error is "small". Hence the need of sharp estimates of the accuracy of approximation.
The proposed research will concentrate on establishing sharp estimates of the accuracy of Poisson and compound Poisson approximation to the distribution of a sum of random variables. In particular, we aim to address a long-standing open question concerning establishing an estimate of the accuracy of (compound) Poisson approximation with a correct (the best possible) constant at the leading term.
The theory of sums of random variables lies at the heart of research in Probability Theory. It is truly fundamental when dealing with the aggregate effect of random events, e.g., when observing the combined effect of a large number of small or rare random contributions.
The work on the theory of sums of random variables started in the 17th century, and continued ever since. Among famous contributors are Bernoulli, de Moivre, Laplace, and other well-known mathematicians. Their results have shaped modern Probability Theory.
The actual distribution of a sum of random variables is typically complex, and one would prefer using a simpler and more tractable approximate distribution (e.g., normal, Poisson or compound Poisson). However, one can only substitute a complex actual distribution by an approximate one if there is a sharp estimate of the accuracy of approximation indicating the error would be "small".
As a result, a lot of work in Probability Theory has been devoted to evaluating the accuracy of approximation. In particular, work on the accuracy of normal approximation to the distributions of sums of random variables started in the late 19th century, and still continues.
However, in situations where one deals with rare events a natural approximating distribution is Poisson (or, more generally, compound Poisson).
The class of compound Poisson laws is so general that the class of all possible limit laws to the distributions of sums of asymptotically "small" random variables (the class of so-called infinitely divisible distributions) coincides with the class of weak limits of compound Poisson distributions.
Poisson and compound Poisson approximations naturally arise when one deals with the number of long head runs is discrete random sequences, the number of long match patterns in DNA sequences, aggregate claims to a (re)insurance company, the number of exceedances of high thresholds in extreme value theory, etc..
The approximation of a complex actual distribution by a Poisson and compound Poisson one is only justified if there is a sharp estimate of the accuracy of approximation indicating the error is "small". Hence the need of sharp estimates of the accuracy of approximation.
The proposed research will concentrate on establishing sharp estimates of the accuracy of Poisson and compound Poisson approximation to the distribution of a sum of random variables. In particular, we aim to address a long-standing open question concerning establishing an estimate of the accuracy of (compound) Poisson approximation with a correct (the best possible) constant at the leading term.
Publications
Cekanavicius V
(2022)
Compound Poisson approximation
in Probability Surveys
Novak S
(2022)
On the T-test
in Statistics & Probability Letters
| Description | This is an ongoing project; however, some of the award objectives have been already met, a number of results have been obtained, two articles have been published thus making an academic impact. For sums of independent integer-valued random variables an estimate of the accuracy of Poisson approximation in terms of the point metric has been established. The estimate has a correct (the best possible) constant at the leading term. This result provides a solution to a long-standing problem concerning the accuracy of Poisson approximation. A lower bound to the accuracy of estimation in the T-test and an estimate of the accuracy of non-normal approximation in the T-test have been established. These results reveal problems with the traditional T-test, and open a path to improvements in the T-test. The survey on compound Poisson approximation for sums of random variables has consolidated numerous results on the topic scattered in many separate publications, provided a systematic overview of results and methods, and formulated a number of open problems. |
| Exploitation Route | A similar follow-up grant would greatly help to finish the research. |
| Sectors | Digital/Communication/Information Technologies (including Software) Financial Services and Management Consultancy |
| Description | An invitation from an influential journal to write an article on the topic. |
| Geographic Reach | Multiple continents/international |
| Policy Influence Type | Contribution to new or improved professional practice |
| Impact | My presentations attaracted a considerable audience, and participants were telling me that those presentations changed their views on the topic. |
| Title | Research helped polishing existing tools and developing further tools. |
| Description | During the project and after it, during collaboration with professor Chekanavichius, I have polished existing tools of poisson and compound poisson approximations, and worked on developing further tools. |
| Type Of Material | Improvements to research infrastructure |
| Year Produced | 2024 |
| Provided To Others? | Yes |
| Impact | We are awaiting to see the impact of the recent publications. |
| Title | Novel data analysis technique |
| Description | The grant helped developing a new data analysis technique that led to advanced results in consequent publications. |
| Type Of Material | Data analysis technique |
| Year Produced | 2024 |
| Provided To Others? | Yes |
| Impact | These are recent publications, I am awaiting to see their impact. |
| Description | Work led to new collaboration |
| Organisation | Vilnius University |
| Country | Lithuania |
| Sector | Academic/University |
| PI Contribution | Professor V. Chekanavichius. Work led to two new publications written jointly. |
| Collaborator Contribution | Collaboration led to two further publications. |
| Impact | Collaboration started after the grant ended but was influenced by it. |
| Start Year | 2022 |
| Description | Further dissemination of the results |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Presentations at a number of international research seminars. The presentations were video recorded and are now publicly available. |
| Year(s) Of Engagement Activity | 2024 |