Open research study · Scientific basis

Literature & how it applies

The method is grounded in peer-reviewed research on tail distributions, probabilistic forecast evaluation, and out-of-sample validation. Each reference below is mapped to the specific part of this study it supports. Published results from other studies are not evidence of this model's performance — only the live ledger is.

Core: implemented as described
Related: related work; this study differs in scope
Background: context that motivates a design choice

Core distribution

Chen, Q. & Gerlach, R. (2013). The two-sided Weibull distribution and forecasting financial tail risk. International Journal of Forecasting, 29(4), 527–540.

Introduces the two-sided Weibull distribution for financial returns, with separate shape and scale parameters for the upper and lower tails. The motivation is empirical: returns are not symmetric, and a single distribution cannot capture crash risk and rally risk with equal fidelity. The authors apply the model to Value-at-Risk and Expected Shortfall forecasting and report competitive tail-risk forecasting performance.

In this study: This study adopts the defining idea directly. Upside and downside returns are fitted separately, each with its own shape (k) and scale (λ), and implied probabilities are computed from the resulting two-sided distribution. The corridor construction and the C+cap rule both operate on this two-sided representation, not on a symmetric approximation.

Core

Wang, C., Chen, Q. & Gerlach, R. (2019). Bayesian realized-GARCH models for financial tail risk forecasting incorporating the two-sided Weibull distribution. Quantitative Finance, 19(6), 1017–1042.

Embeds the two-sided Weibull in a time-varying volatility framework (realized-GARCH), so that the return distribution evolves with the volatility state and adapts more quickly to changing conditions than a static fit.

In this study: This study uses static rolling-window fits: Weibull parameters are estimated on a 365-day window and held fixed for each issue day. Regime effects are monitored diagnostically — the Calibration page splits results by 30-day realized volatility — rather than modelled within the distribution. A volatility layer is a separate research question; the comparison is a difference in scope, not a claim of equivalence.

Laherrère, J. & Sornette, D. (1998). Stretched exponential distributions in nature and economy: "fat tails" with characteristic scales. European Physical Journal B, 2(4), 525–539.

Argues that stretched-exponential (Weibull-type) tails often describe heavy-tailed data, including financial returns, as well as or better than pure power laws, while retaining a characteristic scale.

In this study: Supports the choice of a Weibull tail with an explicit scale (λ) and shape (k) rather than a Gaussian or a pure power law.

Background

Crypto evidence

Silahli, B., Dingec, K. D., Cifter, A. & Aydin, N. (2021). Portfolio value-at-risk with two-sided Weibull distribution: Evidence from cryptocurrency markets. Finance Research Letters, 38, 101425.

Applies the two-sided Weibull to a portfolio of cryptocurrencies and reports that it captures the stylized facts of crypto returns — extreme volatility, heavy tails and asymmetry — and delivers competitive VaR accuracy. It is the most direct empirical support for using a two-sided Weibull on digital assets.

In this study: The setting differs: the paper estimates portfolio VaR, while this study emits single-asset corridors and event probabilities. What transfers is the finding that crypto returns are well described by separate upper and lower Weibull tails; what does not transfer is any specific accuracy figure.

Kończal, J. & Wronka, M. (2024). Tail and memory behaviour of the cryptocurrency prices and stock market indices. Mathematica Applicanda, 52(1–2).

Reports a good fit of double-Weibull-type distributions to BTC and ETH returns and examines the memory properties of their tails.

In this study: Independent support that the distribution family used here is appropriate for the two largest crypto assets.

Background

Calibration theory

Brier, G. W. (1950). Verification of forecasts expressed in terms of probability. Monthly Weather Review, 78(1), 1–3.

Defines the Brier score: the mean squared error of probability forecasts. It remains a standard scoring rule because it is strictly proper — a forecaster cannot improve the expected score by misreporting their true belief.

In this study: The Calibration page reports the Brier score overall and on the informative 10–90% subset. The overall score includes near-certain predictions, which can dominate the average; the informative subset isolates predictions where the model makes a discriminating claim.

Core

Gneiting, T., Balabdaoui, F. & Raftery, A. E. (2007). Probabilistic forecasts, calibration and sharpness. JRSS Series B, 69(2), 243–268.

Formalises the principle that probabilistic forecasts should maximise sharpness subject to calibration. A forecast can be calibrated yet uninformative: an always-99% forecaster is calibrated if the event happens 99% of the time, but it carries no information.

In this study: Basis for the reliability chart and for separating trivial near-certain calls from the informative subset. Both the overall and the informative-only Brier score are shown because a single number can hide the difference between a sharp model and a trivial one.

Core

Validation

Tashman, L. J. (2000). Out-of-sample tests of forecasting accuracy: an analysis and review. International Journal of Forecasting, 16(4), 437–450.

Reviews rolling-origin evaluation: at each forecast origin the model is re-fitted using only data available at that time and evaluated on the subsequent window — the standard way to avoid look-ahead bias in time-series forecasting.

In this study: Corridor widths are calibrated walk-forward: parameters for each issue day use only prior data, and reported accuracy comes only from windows the calibration did not see. The live append-only ledger is the stricter test — every estimate is locked before its outcome is known, and no record can be edited or deleted afterwards.

Core

Dawid, A. P. (1984). Statistical theory: The prequential approach. Journal of the Royal Statistical Society Series A, 147(2), 278–292.

Proposes prequential (predictive sequential) evaluation: a forecasting system is judged by the sequence of forecasts it actually issued, each made before the outcome, compared with what then happened.

In this study: The ledger is a prequential record. Each day's probabilities are issued and stored before the outcome, and evaluation uses only those issued forecasts — never re-fitted or retrospectively generated ones.

Core

Evaluation philosophy

Bailey, D. H., Borwein, J. M., López de Prado, M. & Zhu, Q. J. (2014). Pseudo-mathematics and financial charlatanism: The effects of backtest overfitting on out-of-sample performance. Notices of the American Mathematical Society, 61(5), 458–471.

Shows that when many strategy configurations are tried on the same history, an impressive backtest is easy to obtain by chance and is likely to underperform out of sample.

In this study: Motivates why backtest figures are not presented as evidence of performance here. Model parameters are fixed in advance, every candidate is recorded, and only the forward ledger counts.

Background

Harvey, C. R., Liu, Y. & Zhu, H. (2016). … and the cross-section of expected returns. Review of Financial Studies, 29(1), 5–68.

Argues that with extensive data mining across many tested factors, conventional significance thresholds are too lenient and many published findings are likely false discoveries.

In this study: Motivates the non-selective ledger: selection affects only what the overview shows, never what is recorded, so results cannot be improved by keeping only the hits.

Background

Regime diagnostics

Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2), 223–236.

Documents the stylized facts of asset returns, including volatility clustering, heavy tails and the absence of linear autocorrelation in returns.

In this study: Because volatility clusters, calibration measured over a single calm period can look better or worse than long-run behaviour. Calibration is therefore split by 30-day realized volatility (High/Low regime). The split is diagnostic: it shows where the model is well calibrated without modifying the model.

Background

Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2), 357–384.

Formalises regime-switching behaviour in time series: the parameters governing a process can change discretely between states, so a model that assumes a single state can be misleading across state changes.

In this study: This study does not fit a switching model; the regime split is a diagnostic, not a generative component. The reference establishes the conceptual basis for evaluating calibration conditionally on regime.

Background