Study-Period Estimation Under Imperfect Infection Testing

Definitions and mathematical statement

Let $\mathcal O$ denote the available observations: known admission and discharge data together with imperfect observations of colonization status. Let $\mathcal S$ denote the testing scheme, on which the colonization observations depend. These symbols describe the supplied information without imposing a probability model, a sampling design, or an error mechanism.

Write $T_{\mathrm{inf}}$ and $T_{\mathrm{non}}$ provisionally for the quantities called the study periods for infectious and non-infectious diseases, respectively. If these quantities are defined and $T_{\mathrm{non}}\ne 0$, their formal ratio is \[ \theta=\frac{T_{\mathrm{inf}}}{T_{\mathrm{non}}}. \] The supplied record does not define these periods mathematically. In particular, it does not specify whether they represent study durations, accumulated observation times, or another quantity; nor does it identify colonization status with infectiousness.

Requested methodological construction. Construct a simple method, expressible schematically as an estimator \[ \widehat\theta=g(\mathcal O,\mathcal S), \] for the ratio of study periods for infectious and non-infectious diseases when admission and discharge data are known but colonization status is observed imperfectly and depends on the testing scheme.

Unresolved qualifications. This is a methodological question rather than a fully specified statistical proposition. The estimand, observation model, admissible testing schemes, treatment of observation boundaries, meaning of simplicity, and required estimation guarantees are unspecified. Consequently, the supplied evidence does not determine a unique formal estimation problem or justify an assertion of identifiability, unbiasedness, or consistency.

Short English statement

The question asks for a simple estimator of a ratio of study periods using known admission and discharge data and imperfect, testing-dependent colonization observations. The periods, testing-error model, relationship between colonization and infectiousness, and required accuracy are not defined, so a precise statistical target remains unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001365$, source problem code catalogue entry, entitled Study-Period Estimation Under Imperfect Infection Testing; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2009.
• Supplied source citation: Design and Analysis of Infectious Disease Studies (2010), Oberwolfach Reports, pp. 2673--2698, DOI 10.4171/owr/2009/48. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2009/48.
• The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The registry labels the statement exact and records a prior check against the official report, with notation, context, and question form normalized without changing the source-level claim. These are supplied provenance assertions, not independent verification in this draft.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, states that the report explicitly identifies the requested estimator as an open methodological problem, and reports that no general solution matching the setup was found. The only listed source checked is the original OWR report linked above. This is supplied dated curation, not an independently verified literature assessment; the background explicitly calls for independent verification of time-sensitive status and cited claims.