Estimate delayed CVR nonparametrically with censored data

Today is 2025-09-01. We need the 14-day conversion rate (CVR14) for impressions served between 2025-08-18 and 2025-09-01, but many conversions occur with unknown delays up to 14 days, so recent impressions are right-censored. You cannot assume any parametric delay distribution.

Tasks:

1. Suggest a nonparametric estimator of CVR14 that learns the time-to-convert survival function from historical cohorts and then applies it to the current cohort, which is only partially observed (for instance, Kaplan–Meier on the conversion delay under right-censoring, followed by inverse-probability weighting to correct the converts observed to date). Provide the formulas for the estimator and state which data each term draws on.

2. Build a 95% confidence interval from Greenwood's formula for the KM variance together with the delta method applied to the transformed CVR, and list the assumptions. Describe how you would widen the interval if you suspect the delays are non-stationary.

3. Give a distribution-free conservative bound on CVR14 that rests on as few assumptions as possible (for example, the DKW inequality applied to the empirical CDF of delays, or Clopper–Pearson on the conversions observed plus a worst-case treatment of impressions that have not yet finished). Explain how to compute it from the raw counts you have today.

4. Outline diagnostics for judging whether the historical delay distribution still applies (for instance, checking covariate shift through PSI or KS tests on the traffic mix, day-of-week patterns, or device breakdowns), and how you would stratify or reweight when shift is detected.

5. If the only data you can observe are aggregated daily counts of impressions and same-day conversions, with nothing at the user level, sketch an identifiable approach and the additional assumptions required to estimate or bound CVR14.

Overview:

This question assesses competence in handling right-censored time-to-event data, nonparametric estimation and inference for delayed conversions, construction of confidence intervals and distribution-free bounds, diagnostic checks for nonstationarity, and reasoning about identifiability under aggregated data.