Analyze regression to mean in heavy-tailed shares
Overview
This question evaluates statistical reasoning about heavy-tailed metrics and cohort dynamics, covering concepts like regression to the mean, variance differences in extreme quantile cohorts, time-series seasonality, and temporal stability of quantiles.
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Oct 13, 2025, 9:49 PM
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Cohort Dynamics of a Right-Skewed Daily Shares Metric
Context
- You have a right-skewed metric: daily shares per user, with a long tail.
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On Day 1, you form two cohorts based on each user's Day-1 value:
- p50 cohort: users around the 50th percentile of the Day-1 distribution.
- p95 cohort: users around the 95th percentile of the Day-1 distribution.
- You then track each cohort’s mean daily shares for the next 14 days (e.g., Days 2–15), counting zeros for users who do not share.
- Assume cohorts are formed using narrow bands around the quantiles to ensure sufficient sample size (e.g., ±1 percentile point).
Questions
(a) Predict how each cohort’s mean daily shares will evolve over the next 14 days and explain the roles of regression to the mean vs genuine behavioral differences.
(b) Which cohort should exhibit larger day-to-day variance and why?
(c) Describe how weekly seasonality (e.g., weekends) will appear in the cohort time series and how you would adjust for it.
(d) Outline an analysis to test whether quantiles (e.g., p50, p95) are temporally stable for this metric.
Solution
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