[School of Natural Sciences PhD Scholarships] Advances in Simultaneous Tolerance Intervals and Bands

About the project

About the Project This project will develop a new generation of simultaneous tolerance intervals, bands and regions for reliable prediction, calibration and anomaly detection in complex modern data settings. Classical simultaneous tolerance procedures typically rely on restrictive assumptions such as homoscedasticity, normality and simple independent data structures. These assumptions are often violated in applications involving biomedical monitoring, industrial quality control, environmental surveillance and financial risk assessment, where variability may change with covariates, errors may be heavy-tailed, and repeated or hierarchical measurements are common. The central objective is to construct statistically rigorous and computationally practical simultaneous tolerance procedures that remain valid under heteroscedasticity, non-normality and hierarchical dependence. The project will address three related methodological challenges. First, Wild Bootstrap techniques will be developed for simultaneous tolerance bands and regions, with the aim of achieving robust simultaneous coverage when the error variance is non-constant or the error distribution is non-Gaussian. Second, a conditional generalised pivotal quantity (GPQ) framework will be investigated to construct narrower, subject-specific simultaneous tolerance bands by incorporating historical or repeated measurements. Third, variance-adjusted inverse simultaneous tolerance interval procedures will be developed for anomaly and outlier detection, allowing observations from heterogeneous noise regimes to be assessed fairly rather than against a common fixed threshold. Theoretical properties of the proposed procedures will be established where possible, including simultaneous coverage guarantees, finite-sample behaviour and asymptotic properties. Extensive Monte Carlo simulation studies will evaluate coverage accuracy, efficiency, robustness and computational performance under a wide range of model configurations. The methods will also be demonstrated using real datasets from relevant application areas and compared systematically with existing approaches. The project is expected to produce new methodological results in simultaneous inference, peer-reviewed publications, and practical statistical tools. An open-source R package will be developed to implement the proposed methods, together with an interactive Shiny application to make the methodology accessible to practitioners. Existing links with external organisations, including Novartis and the UK Health Security Agency (UKHSA), will provide opportunities for dissemination, discussion of practical challenges and potential knowledge transfer. The student will receive close supervision and structured training throughout the project. Training will include statistical theory for simultaneous inference and tolerance procedures, bootstrap and resampling methods, generalised pivotal quantities, Monte Carlo simulation, numerical optimisation, reproducible statistical computing in R, package development and scientific visualisation. The student will also be supported in developing academic writing, presentation and research communication skills, and encouraged to present their work at research seminars and international conferences. Regular supervisory meetings will be used to review progress, identify methodological challenges and plan successive stages of the research. This project is expected to start in September 2027. Before you apply: We strongly recommend that you contact the supervisors for this project before you apply. How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Statistics, Mathematics, Data Science or Biostatistics OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Statistics, Mathematics, Data Science or Biostatistics. Experience with statistical programming, particularly in R, would be advantageous, as would familiarity with simulation studies, bootstrap or resampling methods, and computational statistics. Previous research experience in statistical methodology is desirable but not essential. Candidates should have strong analytical and problem-solving skills, an interest in developing new statistical methodology, and the ability to engage with both theoretical and computational aspects of the project. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

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