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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2511.03951 |
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| _version_ | 1866915942668173312 |
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| author | G, Nagananda K Kim, Jong Sung |
| author_facet | G, Nagananda K Kim, Jong Sung |
| contents | A unified framework is presented to study the two-sample Behrens--Fisher problem -- testing equality of means when two normal populations have unequal, unknown variances -- and a compact expression is derived for the null distribution of the classical test statistic. Our new approach involves a Mellin--Barnes factorization that decouples the square root of a weighted sum of independent chi-square variates, thereby collapsing a challenging two-dimensional integral to a tractable single-contour integral. Closing the contour yields a residue series that terminates whenever either sample's degrees of freedom is odd. A complementary Euler--Beta reduction identifies the density as a Gauss hypergeometric function with explicit parameters, yielding a numerically stable form that recovers Student's $t$ under equal variances. Ramanujan's master theorem supplies exact inverse-power tail coefficients, which bound Lugannani--Rice saddle-point approximation errors and support reliable tail analyses. The proposed framework reveals why hypergeometric structure appears, why certain finite-sum cases arise, and how one can pass from the bulk of the distribution to its tails without altering the analytic framework. Finally, it lets us tabulate exact two-sided critical values over a broad grid of sample sizes and variance ratios that reveal the parameter surface on which the well-known Welch's approximation switches from conservative to liberal, quantifying its maximum size distortion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03951 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A unified approach to the Behrens-Fisher problem G, Nagananda K Kim, Jong Sung Statistics Theory A unified framework is presented to study the two-sample Behrens--Fisher problem -- testing equality of means when two normal populations have unequal, unknown variances -- and a compact expression is derived for the null distribution of the classical test statistic. Our new approach involves a Mellin--Barnes factorization that decouples the square root of a weighted sum of independent chi-square variates, thereby collapsing a challenging two-dimensional integral to a tractable single-contour integral. Closing the contour yields a residue series that terminates whenever either sample's degrees of freedom is odd. A complementary Euler--Beta reduction identifies the density as a Gauss hypergeometric function with explicit parameters, yielding a numerically stable form that recovers Student's $t$ under equal variances. Ramanujan's master theorem supplies exact inverse-power tail coefficients, which bound Lugannani--Rice saddle-point approximation errors and support reliable tail analyses. The proposed framework reveals why hypergeometric structure appears, why certain finite-sum cases arise, and how one can pass from the bulk of the distribution to its tails without altering the analytic framework. Finally, it lets us tabulate exact two-sided critical values over a broad grid of sample sizes and variance ratios that reveal the parameter surface on which the well-known Welch's approximation switches from conservative to liberal, quantifying its maximum size distortion. |
| title | A unified approach to the Behrens-Fisher problem |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2511.03951 |