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E-values for k-Sample Tests with Exponential Families

Author

Listed:
  • Yunda Hao

    (Centrum Wiskunde en Informatica)

  • Peter Grünwald

    (Centrum Wiskunde en Informatica
    Mathematical Institute, Leiden University)

  • Tyron Lardy

    (Centrum Wiskunde en Informatica
    Mathematical Institute, Leiden University)

  • Long Long
  • Reuben Adams

    (University College London)

Abstract

We develop and compare e-variables for testing whether k samples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family. We consider the GRO (growth-rate optimal) e-variables for (1) a ‘small’ null inside the same exponential family, and (2) a ‘large’ nonparametric null, as well as (3) an e-variable arrived at by conditioning on the sum of the sufficient statistics. (2) and (3) are efficiently computable, and extend ideas from Turner et al. (2021) and Wald (1947) respectively from Bernoulli to general exponential families. We provide theoretical and simulation-based comparisons of these e-variables in terms of their logarithmic growth rate, and find that for small effects all four e-variables behave surprisingly similarly; for the Gaussian location and Poisson families, e-variables (1) and (3) coincide; for Bernoulli, (1) and (2) coincide; but in general, whether (2) or (3) grows faster under the alternative is family-dependent. We furthermore discuss algorithms for numerically approximating (1).

Suggested Citation

  • Yunda Hao & Peter Grünwald & Tyron Lardy & Long Long & Reuben Adams, 2024. "E-values for k-Sample Tests with Exponential Families," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(1), pages 596-636, February.
    • Handle: RePEc:spr:sankha:v:86:y:2024:i:1:d:10.1007_s13171-024-00339-9
    DOI: 10.1007/s13171-024-00339-9
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    References listed on IDEAS

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    1. Glenn Shafer, 2021. "Testing by betting: A strategy for statistical and scientific communication," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 184(2), pages 407-431, April.
    2. Alexander Henzi & Johanna F Ziegel, 2022. "Valid sequential inference on probability forecast performance [A comparison of the ECMWF, MSC, and NCEP global ensemble prediction systems]," Biometrika, Biometrika Trust, vol. 109(3), pages 647-663.
    3. Alexander Henzi & Johanna F Ziegel, 2022. "Correction to: ‘Valid sequential inference on probability forecast performance’ [Valid sequential inference on probability forecast performance]," Biometrika, Biometrika Trust, vol. 109(4), pages 1181-1182.
    Full references (including those not matched with items on IDEAS)

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