{
  "abstract": "Expected Shortfall (ES) measures the average size of tail losses. Its sequential backtests typically also require a Value-at-Risk (VaR) forecast. We develop two e-backtesting methods using only observed losses and ES forecasts: one for iid losses only and the other for time series under an AR-GARCH conditional VaR structure. Under the stated assumptions, both control false rejections at any stopping time. For the time-series method, validity requires a fixed parameter--quantile pair representing the conditional VaR path and a positive report margin. The time-series statistic takes an infimum over possible parameter and VaR values. We choose the betting process $(λ_t)$ by maximizing an empirical growth rate averaged across candidate parameters. We compute the statistic by continuous VaR profiling on a dense parameter grid. For grid approximations, we prove convergence to the continuous infimum, together with convergence of the probability of false rejection, as the grid is refined over a fixed monitoring period. Lower and upper bounds for the VaR infimum on the parameter grid control the accuracy of the numerical profiling. Simulations, including ES reports that change from over-reporting to under-reporting, and an application to daily NASDAQ losses provide numerical comparisons with VaR--ES e-processes and fixed-sample ES backtests, assessing detection frequencies and the timing of rejections.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17681",
  "author_total": 2,
  "authors": [
    {
      "name": "Ruodu Wang"
    },
    {
      "name": "Jingcheng Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a8w5q3rqsbjx3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17681",
  "fulltext_url": null,
  "id": "a8w5q3rqsbjx3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17681",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "q-fin.RM"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "openalex",
      "2003|Financial Risk and Volatility Modeling"
    ]
  ],
  "openalex": "W7221771297",
  "pdf_url": "https://arxiv.org/pdf/2610.17681",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a8w5q3rqsbjx3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:26Z",
      "native_id": "2610.17681",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17681"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "E-Backtesting Expected Shortfall without Value-at-Risk Forecasts under Time-series Models",
  "title_text": "E-Backtesting Expected Shortfall without Value-at-Risk Forecasts under Time-series Models",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/a8w5q3rqsbjx3",
  "venue": "arXiv",
  "versions": []
}