Distributional Fitting and Tail Analysis of Lead-Time Compositions: Nights vs. Revenue on Airbnb

TL;DR

We address how lead-time distributions differ between Nights Booked (volume) and Gross Booking Value (revenue) for Airbnb by treating daily allocations as compositional vectors on the 0–365 horizon. The analysis combines Wasserstein-distance comparisons, Bai–Perron structural breaks, interval-censored cross-entropy parametric fitting (Gamma, Weibull, Lognormal), tail analysis with tail-mass ratios and generalized Pareto models, and nonparametric GAM benchmarks, all while rigorously handling truncation at 365 days. The key findings are that GBV concentrates more in mid-range horizons than Nights, Gamma and Weibull provide comparable two-parameter fits with Gamma slightly favored, GAM yields superior CRPS in-sample at the cost of interpretability, and tail inference is reliable only below about 150 days due to truncation. Practically, ignoring the volume–revenue divergence biases revenue timing forecasts, and tail analyses require stability diagnostics when data are truncated; methodologically, the study demonstrates robust, truncation-aware approaches for compositional lead-time data in tourism analytics.

Abstract

We analyze daily lead-time distributions for two Airbnb demand metrics, Nights Booked (volume) and Gross Booking Value (revenue), treating each day's allocation across 0-365 days as a compositional vector. The data span 2,557 days from January 2019 through December 2025 in a large North American region. Three findings emerge. First, GBV concentrates more heavily in mid-range horizons: beyond 90 days, GBV tail mass typically exceeds Nights by 20-50%, with ratios reaching 75% at the 180-day threshold during peak seasons. Second, Gamma and Weibull distributions fit comparably well under interval-censored cross-entropy. Gamma wins on 61% of days for Nights and 52% for GBV, with Weibull close behind at 38% and 45%. Lognormal rarely wins (<3%). Nonparametric GAMs achieve 18-80x lower CRPS but sacrifice interpretability. Third, generalized Pareto fits suggest bounded tails for both metrics at thresholds below 150 days, though this may partly reflect right-truncation at 365 days; above 150 days, estimates destabilize. Bai-Perron tests with HAC standard errors identify five structural breaks in the Wasserstein distance series, with early breaks coinciding with COVID-19 disruptions. The results show that volume and revenue lead-time shapes diverge systematically, that simple two-parameter distributions capture daily pmfs adequately, and that tail inference requires care near truncation boundaries.

Figures (8)

• Figure 1: Aggregated lead-time distributions for Nights (blue) and GBV (red), 2019--2025. Distributions are day-weighted averages of daily pmfs: $\bar{p}(\ell) = D^{-1} \sum_d x_{d,\ell}$. Both peak near $\ell = 0$ and decline rapidly. The curves cross around $\ell = 30$ days: below, Nights slightly exceeds GBV; above, GBV exceeds Nights. The gap persists to about 200 days.
• Figure 2: Daily Wasserstein-1 distance between Nights and GBV, with structural breakpoints (dashed vertical lines) identified via Bai--Perron with HAC standard errors. The series averages 8.67 (95% CI: 8.42--8.92). Seasonality peaks in summer. Early breaks (2020--2021) align with COVID disruptions; later breaks may reflect post-pandemic normalization or other market shifts.
• Figure 3: Tail-mass ratios (GBV/Nights) at thresholds 7, 30, 60, 90, and 180 days. At 7 days, the ratio hovers near 1.0. At longer horizons, GBV increasingly dominates: ratios reach 1.2--1.4 at 90 days and 1.5+ at 180 days during summer peaks. COVID (early 2020) briefly inverts the pattern.
• Figure 4: Proportion of daily bookings beyond 30, 90, and 180 days for Nights (blue) and GBV (red). GBV exceeds Nights at all thresholds. COVID shows a collapse in tail mass for both metrics as booking windows shortened dramatically.
• Figure 5: GPD shape parameter $\xi$ for Nights at thresholds 60--270 days. Counts shown are synthetic exceedances (from 1,000 draws/day pooled across all days), not raw booking counts. Negative values suggest bounded tails within the observed window. Estimates are stable for $u \leq 150$ days; above, truncation effects dominate.