arXiv 15 Aug 2017 · Mathematics — Statistics Theory · publishedJournal of Econometrics (2018) · 76 citations (OpenAlex)
arXiv:1708.04658 · PDF · DOI · OpenAlex · Extracted main text
When comparing two distributions, it is often helpful to learn at which quantiles or values there is a statistically significant difference. This provides more information than the binary "reject" or "do not reject" decision of a global goodness-of-fit test. Framing our question as multiple testing across the continuum of quantiles $\tau\in(0,1)$ or values $r\in\mathbb{R}$, we show that the Kolmogorov--Smirnov test (interpreted as a multiple testing procedure) achieves strong control of the familywise error rate. However, its well-known flaw of low sensitivity in the tails remains. We provide an alternative method that retains such strong control of familywise error rate while also having even sensitivity, i.e., equal pointwise type I error rates at each of $n\to\infty$ order statistics across the distribution. Our one-sample method computes instantly, using our new formula that also instantly computes goodness-of-fit $p$-values and uniform confidence bands. To improve power, we also propose stepdown and pre-test procedures that maintain control of the asymptotic familywise error rate. One-sample and two-sample cases are considered, as well as extensions to regression discontinuity designs and conditional distributions. Simulations, empirical examples, and code are provided.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Buja, A., Rolke, W (2006) Calibration for simultaneity: (re)sampling methods for simultaneous inference with applications to function estimation and funct… | 1.000 | 15 | 5 | 100% |
| 2 | Lehmann, E. L., Romano, J. P (2005) b | 0.950 | 7 | 5 | 86% |
| 3 | Aldor-Noiman, S., Brown, L. D., Buja, A., Rolke, W., Stine, R. A (2013) The power to see: A new graphical test of normality | 0.941 | 6 | 4 | 83% |
| 4 | Jaeschke, D (1979) The asymptotic distribution of the supremum of the standardized empirical distribution function on subintervals | 0.928 | 4 | 3 | 100% |
| 5 | Cattaneo, M. D., Frandsen, B. R., Titiunik, R (2015) Randomization inference in the regression discontinuity design: An application to party advantages in the U.S | 0.874 | 11 | 2 | 100% |
| 6 | Banks, D. L (1988) Histospline smoothing the Bayesian bootstrap | 0.843 | 3 | 3 | 100% |
| 7 | Eicker, F (1979) The asymptotic distribution of the suprema of the standardized empirical processes | 0.811 | 4 | 2 | 100% |
| 8 | Goldman, M., Kaplan, D. M (2017) b self | 0.737 | 5 | 3 | 40% |
| 9 | Moscovich, A., Nadler, B., Spiegelman, C (2016) On the exact Berk–Jones statistics and their $p$-value calculation | 0.737 | 3 | 2 | 100% |
| 10 | Canay, I. A., Kamat, V (2017) Approximate permutation tests and induced order statistics in the regression discontinuity design | 0.725 | 7 | 2 | 57% |
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