Annotating and forecasting a long‑tail distribution

Q: Annotating and forecasting a long‑tail distribution

This question evaluates competence in robust statistical summarization of heavy-tailed, cross-sectional distributions and time-series forecasting that accounts for seasonality and sporadic spikes, emphasizing understanding of percentiles, robustness to outliers, and trend decomposition.

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Question

Scenario

You are analyzing daily share counts across many pages on a social platform. The cross-sectional distribution of per-page shares on any given day is heavy-tailed (most pages get few or zero shares; a small fraction get very large counts). You need to summarize this distribution and forecast future share volumes.

Tasks

For a given day (or a recent rolling window), compute and annotate robust summary statistics across pages: mean, median, P1 (1st percentile), and P99 (99th percentile).
Because the distribution is long-tailed, describe how you would compute and visualize these statistics so they are interpretable (e.g., log scaling, winsorization).
Forecast the trend of share volumes over time, accounting for seasonality and heavy tails. Mention methods such as quantile smoothing and seasonal decomposition.
State any assumptions you make.

Assumptions

You have daily per-page share counts y_{p,t} and a daily total Y_t = ∑ p y {p,t} over a sufficiently long history (e.g., 6–24 months).
The series exhibits weekly seasonality and occasional spikes.
P1 and P99 refer to the 1st and 99th percentiles across pages on a day.

Annotating and forecasting a long‑tail distribution

Scenario

Tasks

Assumptions

Solution

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Annotating and forecasting a long‑tail distribution

Overview

Scenario

Tasks

Assumptions

Solution

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