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Mode testing, critical bandwidth and excess mass

Jose Ameijeiras-Alonso, Rosa M. Crujeiras, Alberto Rodríguez-Casal

arXiv:1609.05188v3stat.ME

TL;DR

Identifying density modes matters in applied fields, yet existing tests based on critical bandwidth or excess mass are reported to perform unsatisfactorily. The paper combines these approaches in a new testing procedure, which it reports outperforms existing procedures; its scope for general mode counts is limited to testing exactly k modes against more than k.

  • Problem

    Applied analyses need to identify density modes, but existing critical-bandwidth and excess-mass tests are reported to lack satisfactory performance in practice.

    Source: §1 · pp. 2–3

  • Method

    The paper proposes a testing procedure combining critical bandwidth and excess mass ideas, with bootstrap calibration.

    Source: §1 · p. 3

  • Results

    The new procedure is reported to outperform existing procedures for testing unimodality and more general hypotheses.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The totally nonparametric procedure has reasonable behavior for general mode-count testing and can extend to contexts with a natural nonparametric estimator under the null.

    Source: §5 · p. 31

    For general k, the proposal tests H0: j = k against j > k, not the composite null H0: j ≤ k.

    Source: SM3 New proposal when the support is… · p. 41

Abstract

from arXiv · show

The identification of peaks or maxima in probability densities, by mode testing or bump hunting, has become an important problem in applied fields. 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 provides a satisfactory performance in practice. In this work, a new procedure which combines the previous approaches (smoothing and excess mass) is presented and compared with the existing methods, showing a superior behaviour. A real data example on philatelic data is also included for illustration purposes.

1 Introduction

Identifying density modes matters across applied sciences, where unimodal models may miss features of complex measurements. The paper combines critical bandwidth and excess mass ideas in a new test, compares it with existing procedures, and revisits stamp data.

1 Introduction

  • Identifying the number and location of density modes is important in applied fields where unimodal models may not capture complex measurements.Examples include geology, neurology, economics, ecology, and astronomy.Source: §1 · p. 2
  • The new procedure combines critical bandwidth and excess mass ideas and is reported to outperform existing procedures for testing unimodality and more general hypotheses.The paper reviews existing tests and compares procedures in simulations and a stamp-data analysis.Source: §1 · pp. 3–4
  • Existing mode tests use critical bandwidth or excess mass, while kernel density estimation offers a complementary, nonparametric view of density structure.Kernel estimation requires a bandwidth choice that controls smoothing.Source: §1 · p. 3
  • Different histogram bin widths and kernel bandwidths can lead to different conclusions about the number of modes in the stamp data.The 1872 Hidalgo issue motivates an objective criterion because catalogs do not distinguish groups of stamps printed on mixed paper types.Source: §1 · pp. 2–3
  • The paper reviews mode tests, compares their empirical size and power in simulations, and revisits the philatelic stamp dataset.Further technical and simulation details are provided in supplementary material.Source: §1 · pp. 3–4

2 A review on multimodality tests

This section reviews critical-bandwidth and excess-mass approaches to multimodality testing and introduces a data-driven method combining an excess-mass statistic with critical-bandwidth-based bootstrap calibration. It describes the method’s use of local link functions to preserve a hypothesized k-mode structure, along with density normalization and conditions for inference and asymptotic calibration.

2.1 Tests based on the critical bandwidth

  • Silverman (1981) used the Gaussian-kernel critical bandwidth to test H0: j ≤ k against j > k, rejecting for large h_k and calibrating its distribution by bootstrap; Hall and York (2001) found this bootstrap inconsistent under the null.They suggested accurate calibration for k = 1 on a closed interval I where f has a single mode. For k modes, calibration depends on 2k−2 unknown quantities, complicating computation.Source: §2.1 · pp. 6–8

2.2 Tests based on excess mass

  • Müller and Sawitzki (1991) test for k modes using excess mass: large differences between the k+1- and k-cluster statistics argue against the null, and the statistic needs only the sample.For unimodality, their statistic coincides up to a factor with Hartigan and Hartigan’s (1985) dip test; Cheng and Hall (1998) developed a different calibration because earlier proposals were extremely conservative.Source: §2.2 · pp. 9–10

2.3 A new proposal

  • The proposed method uses a modified critical-bandwidth kernel estimate as a bootstrap calibration function for an excess-mass statistic, addressing earlier tests’ practical drawbacks with a data-driven procedure.Its construction is designed to preserve regularity conditions needed for correct calibration; the method also extends to testing at most k modes within a known interval.Source: §2.3 · p. 11
  • The new method’s calibration function modifies the critical-bandwidth estimate near estimated modes and antimodes, preserving their locations and density values while removing saddle points.It is constructed to satisfy regularity conditions and make estimates of |g′′(x̂_i)|/g^3(x̂_i) converge to the corresponding density quantities.Source: §2.3 · pp. 12–13
  • Under the stated regularity conditions, the new method’s bootstrap excess-mass statistic has the same asymptotic distribution as the original statistic when the critical bandwidth satisfies the CBC.The paper presents the method as addressing the poor level accuracy of Fisher and Marron (2001) and the generally conservative behavior of Silverman (1981), while allowing tests for general k.Source: §2.3 · pp. 11–14
  • The J function modifies the estimate around each mode or antimode using a selected height, nearby level-crossing points, and a neighborhood constrained to lie within those crossings.The height ranges from near the estimated mode height to the midpoint with the nearest neighboring extremum, depending on ς_i.Source: §2.3 · p. 16
  • The L function removes saddle points outside the J-function intervals by applying a link function over neighborhoods whose widths are determined by nearby saddle points and J-interval boundaries.The neighborhood endpoints are ζ_p ± ϖξ, with ϖ between 0 and 1/4.Source: §2.3 · pp. 17–18
  • The calibration function g satisfies the specified regularity conditions, has a curvature ratio converging in probability to d_i at each mode, and preserves the data structure under the hypothesis of k modes.It is constructed using the J function at modes and antimodes and the L function to remove saddle points; g may not initially integrate to one.Source: §2.3 · pp. 16, 18
  • A search for ς values satisfying q(ς) = 1 can ensure that g is a density, and convergence is guaranteed when the J-function neighborhoods are small enough.In the simulation study, ς_i values close to zero are used to limit the impact on the integral.Source: §2.3 · p. 18
  • Under regularity conditions, Hall and York’s critical bandwidth can yield better asymptotic results; if modes and antimodes lie in a known closed interval, it can be employed, and its density estimate can locate them.The proposal also reports satisfactory results for unbounded support.Source: §2.3 · p. 19
  • Unlike the Silverman (1981) and Fisher and Marron (2001) proposals, this method tests H0: j = k rather than H0: j ≤ k.Source: §2.3 · p. 19

3 Simulation study

Across the simulation scenarios, the new proposal (NP) generally had satisfactory calibration, while competing methods showed conservative, erratic, or model-dependent behavior. Power increased with sample size, but no correctly calibrated method was clearly more powerful than the others.

Simulation design

  • The simulations used 25 distributions and sample sizes of 50, 200, and 1000; power studies used n = 100 instead of 1000.For each distribution and sample size, 500 samples were generated, with 500 resamples per sample; the study considered significance levels α = 0.01, 0.05, and 0.10.Source: §3 · pp. 19–20

Testing unimodality vs. multimodality and bimodality

  • SI was quite conservative, while FM had no systematic calibration pattern, with rejection rates above or below the significance level depending on the model.For bimodality, FM was below α in M17–M19 in specified sample sizes and above α for M11, M12, M15, M16, or M20 in specified cases.Source: §3 · pp. 20, 22
  • HY calibrated well at n = 1000 when the support interval was known, but rejection rates were usually below the significance level for smaller samples.The support of the density must be known for this approach; its bimodality results were also good except for slight deviations in specified models and sample sizes.Source: §3 · pp. 20, 22

Testing unimodality vs. multimodality

  • CH calibration appeared correct in simple models but could deviate in more complex or asymmetric models because its parametric calibration distributions do not capture features such as skewness.In the reported scenarios, rejection rates were below α for M3, M7, M9, and M10 in specified cases, and considerably above it for M6.Source: §3 · p. 21

Testing unimodality vs. multimodality and bimodality

  • NP calibration was generally satisfactory, including in complicated models, though some cases remained conservative or difficult to calibrate even at n = 1000.Using known support improved results in difficult cases; for bimodality, results at n = 1000 were good except for M16, with the calibration issue corrected using I = [0, 1].Source: §3 · pp. 21–22

Power analysis

  • Power increased with sample size for NP, detecting all alternatives except M21; among correctly calibrated methods, none was clearly more powerful overall.HY detected the small second mode in M13, while CH had the highest empirical power for M11, M12, M14, and M15 at n = 50.Source: §3 · pp. 21–22

4 Real data analysis

Applying the new procedure to the 1872 Hidalgo stamp issue yields four groups, while earlier exploratory estimates and tests gave less consistent group counts. The analysis relates the inferred groups to paper types and notes that thickness classifications were not standardized.

4 Real data analysis

  • Paper thickness mattered to stamp value because scarcity determined price, and paper manufacture before the 1940s involved substantial thickness fluctuations.Mixtures of paper types further complicate thickness-based grouping.Source: §4 · pp. 22, 28
  • Histogram and kernel-density conclusions about the number of modes vary with bin width and bandwidth, limiting their use as a formal group-count criterion.The issue lacked a standard thickness classification in catalogues.Source: §4 · p. 28
  • At α = 0.05, the new procedure rejects through k = 4 but not for larger k, leading to a four-group conclusion for the 1872 Hidalgo Issue.The authors report that SI and FM results were not accurate, while NP had good calibration even at small sample sizes.Source: §4 · p. 29
  • The two final modes under h4, near 0.10 and 0.11 mm, are interpreted as medium- and thick-paper groups in the 1872 stamps.The two minor modes near 0.12 and 0.13 mm under h7 lack an explanation and may be estimation artefacts.Source: §4 · p. 30

5 Discussion

The paper concludes that its proposal performs well for testing a general number of modes and is the only alternative with reasonable behavior in that setting. It is computationally feasible with FDR correction, and an R package was developed for large-scale practical use.

5 Discussion

  • For a general number of modes k, the authors describe their proposal as the only alternative with reasonable behavior and say it can extend to settings with a natural nonparametric estimator under the null.Periodic data are given as an example of a possible extension.Source: §5 · p. 31
  • The procedure combined with FDR correction is computationally feasible for many tests, and the authors developed an R package to support its use in large problems.Source: §5 · p. 31

SM1 Models for simulation study

The simulation supplement specifies model densities using weighted mixtures and organizes the illustrated models by modality. The displayed sets include unimodal, bimodal, and trimodal densities.

SM1 Models for simulation study

  • The simulation models are specified as weighted mixtures of component densities, with component weights and distributions defining each model.The listed component families include Gaussian, Beta, Gamma, and Weibull distributions.Source: SM1 Models for simulation study · p. 32

SM2 Technical proofs

The technical proofs establish asymptotic behavior under regularity conditions, including convergence of estimated local density-curvature quantities. They also identify a more demanding condition when a pilot bandwidth is replaced by the critical bandwidth.

SM2 Technical proofs

  • Under regularity conditions RC1–RC4, the test statistic’s distribution is independent of the underlying density except through the values d_i at its modes and antimodes.Here d_i = |f′′(x_i)|/f^3(x_i).Source: SM2 Technical proofs · p. 34
  • The calibration function is constructed to preserve RC1–RC4 and to make the estimated b_d_i converge in probability to d_i at its modes and antimodes.The calibration distribution must satisfy RC1–RC4 and the estimated values must converge to the corresponding d_i.Source: SM2 Technical proofs · p. 34
  • Replacing h_PI with the critical bandwidth h_k calls for the stronger condition n a_n^5/log n → ∞ for the stated convergence, which the authors say the critical bandwidth may not satisfy.Source: SM2 Technical proofs · p. 37

SM3 New proposal when the support is known

For known support, the proposal modifies its calibration function using Hall and York’s critical bandwidth and adjusts its tails when modes lie outside the specified interval. The section also reports generally satisfactory calibration in simulations while cautioning that testing exactly k modes is not generally a test of at most k modes.

SM3 New proposal when the support is known

  • For known support, the proposal replaces Silverman’s critical bandwidth with Hall and York’s in the calibration function and modifies its tails when modes fall outside the interval.The modification redefines boundary points and selects a and b so the calibration function integrates to one, with normalization as an alternative when needed.Source: SM3 New proposal when the support is… · pp. 39–40
  • The proposal tests H0: j = k against j > k, not the broader H0: j ≤ k; when the true mode count is below k, bootstrap calibration may be unreliable.For practical use, the paper describes a stepwise procedure that starts at one mode and increases k after rejection, and reports generally good calibration even when j < k.Source: SM3 New proposal when the support is… · p. 41
  • NP’s bimodality test had generally satisfactory calibration across the reported unimodal models, though some cases were slightly conservative and correct calibration was not guaranteed.For M26 at n = 1000, testing j = 2 produced rejection rates below α even with the known-support correction.Source: SM3 New proposal when the support is… · p. 42, Table 8 · p. 43

SM5 Numerical approximations

The numerical procedures use binary search for critical bandwidths and an iterative excess-mass calculation for general k. Because the exact excess-mass algorithm is computationally costly, the simulation study uses a grid-based approximation.

SM5 Numerical approximations

  • Critical bandwidths are computed by binary search between bandwidths yielding at most k modes and more than k modes, stopping when their gap is below the specified tolerance.The tolerance is ha0/2^10, where ha0 is the initial lower bandwidth.Source: SM5 Numerical approximations · p. 44
  • For unimodality, the excess-mass statistic is computed as twice Hartigan and Hartigan’s dip statistic.The dip value was obtained using the diptest package.Source: SM5 Numerical approximations · p. 45
  • For k > 1, the excess-mass calculation uses minimum distances of unions of k intervals and iteratively constructs candidate λ values before evaluating the statistic.The algorithm obtains λ candidates for k and k + 1, then uses their minimizing difference to form the test statistic.Source: SM5 Numerical approximations · p. 45
  • The simulation approximation searches a grid of candidate λ values to minimize the excess-mass difference, balancing accuracy against computation time as sample size grows.The grid size decreases from 100 at n = 50 to 5 at n = 1000; the exact algorithm is computationally costly for extensive studies.Source: SM5 Numerical approximations · p. 46

SM6 Further details on real data analysis

Stamp thickness is difficult to classify objectively, and exploratory mode counts vary with histogram bin width and kernel bandwidth. Applying the new procedure to the 1872 Hidalgo Issue yields four groups, rather than the seven previously reported using Silverman’s test.

SM6 Further details on real data analysis

  • Stamp thickness categories are issue-specific and subjective, while catalogues do not distinguish groups, making mixed-paper issues a case for an objective criterion.The issue includes paper types with possible thickness differences.Source: SM6 Further details on real data… · p. 46
  • Exploratory mode counts depend strongly on histogram bin width and kernel bandwidth; an automatic plug-in bandwidth produced nine modes in the stamp data.A histogram with a smaller bin width also showed apparently more modes.Source: SM6 Further details on real data… · p. 48
  • Silverman’s test gave inconsistent conclusions across bootstrap settings, while the paper reports poor calibration for Silverman’s and Fisher–Marron’s procedures in simulations.Prior results included seven groups, whereas other reported analyses did not reject at most two modes but rejected at most six.Source: SM6 Further details on real data… · pp. 48–50
  • The new procedure identifies four groups in the 1872 Hidalgo Issue, rejecting H0: j = 2 but finding no evidence to reject for larger k.The conclusion is similar when the known-support interval [0.04, 0.15] is used.Source: Table 10 · p. 49, SM6 Further details on real data… · p. 50
  • Repeated thickness values were jittered with a small uniform perturbation because the excess-mass statistic requires a non-discrete sample.The artificial sample adds noise from U(−5·10^-4, 5·10^-4).Source: SM6 Further details on real data… · p. 49
  • The authors associate the two modes near 0.10 and 0.11 mm with two paper types used in 1872, while Gaussian mixtures have difficulty capturing the asymmetric first density region.The earlier seven-mode interpretation included two minor modes near 0.12 and 0.13 mm that lacked an explanation.Source: SM6 Further details on real data… · pp. 50–51
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