Abstract
The identification of peaks or maxima in probability densities, by mode testing or bump hunting, has become an important problem in applied fields. For real random variables, this task has been approached in the statistical literature from different perspectives, with the proposal of testing procedures which are based on kernel density estimators or on the quantification of excess mass. However, none of the existing proposals for testing the number of modes provides a satisfactory performance in practice. In this work, a new procedure which combines the previous approaches (smoothing and excess mass) is presented together with a revision on the previous proposals. The new method is compared with the existing ones in an extensive simulation study, showing a superior behaviour, with good calibration and power results. Theoretical justification for its performance is also obtained. A real data example on philatelic data is also included for illustration purposes, revising previous approaches and discussing the results with the new procedure.
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Notes
Note that, although asymptotically the sign of \(\widehat{f}''_{h_{\tiny {\text{ PI }}}}(\widehat{x_i})\) is always correct (under the assumptions of Theorem 1), in the finite-sample case, it may not be negative in the modes or positive in the antimodes. In that case, an abuse of notation will be done, denoting as \(h_{\tiny {\text{ PI }}}\) to the critical or other plug-in bandwidth in order to guarantee that the sign of this second derivative remains correct.
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The authors gratefully acknowledge the support of Projects MTM2016-76969-P (Spanish State Research Agency, AEI) and MTM2013-41383-P (Spanish Ministry of Economy, Industry and Competitiveness), both co-funded by the European Regional Development Fund (ERDF), IAP network from Belgian Science Policy. Work of J. Ameijeiras-Alonso has been supported by the Ph.D. Grant BES-2014-071006 from the Spanish Ministry of Economy, Industry and Competitiveness.
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Ameijeiras-Alonso, J., Crujeiras, R.M. & Rodríguez-Casal, A. Mode testing, critical bandwidth and excess mass. TEST 28, 900–919 (2019). https://doi.org/10.1007/s11749-018-0611-5
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DOI: https://doi.org/10.1007/s11749-018-0611-5