Summary
A test due to B.W. Silverman for modality of a probability density is based on counting modes of a kernel density estimator, and the idea of critical smoothing. An asymptotic formula is given for the expected number of modes. This, together with other methods, establishes the rate of convergence of the critically smoothed bandwidth. These ideas are extended to provide insight concerning the behaviour of the test based on bootstrap critical values.
Article PDF
Similar content being viewed by others
References
Cramér, H., Leadbetter, M.R.: Stationary and related processes. New York: Wiley 1967
Cuevas, A., Gonzalez Manteiga, W.: Data—driven smoothing based on convexity properties. In: Nonparametric functional estimation and related topics Roussas, G. (ed.) pp. 225–240. Dordrecht: Kluwer 1991
Fisher, N.I., Mammen, E., Marron, J.S.: Testing for multimodality. Technical Report 1990
Good, I.J., Gaskins, R.A.: Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data. J. Am. Stat. Assoc.75, 42–73 (1980)
Hall, P.: Edgeworth expansions for nonparametric density estimators, with applications. Technical Report 1989
Hartigan, J.A., Hartigan, P.M.: The DIP test of unimodality. Ann. Stat.13, 70–84 (1985)
Mammen, E.: Estimating a smooth monotone regression function. Ann. Stat.19, 724–740 (1991a)
Mammen, E.: Nonparametric regression under qualitative smoothness assumptions. Ann. Stat.19, 741–759 (1991b)
Mammen, E.: On qualitative smoothness of kernel density estimates. Technical Report 1991c
Müller, D.W., Sawitzki, G.: Excess mass estimates and tests for multimodality. J. Am. Stat. Assoc.86, 738–746 (1991)
Silverman, B.W.: Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Stat.6, 177–184 (1978)
Silverman, B.W.: Using kernel estimates to investigate multimodality. J. R. Stat. Soc., Ser. B43, 97–99 (1981)
Silverman, B.W.: Some properties of a test for mulitmodality based on kernel density estimates. In: Kingman, J.F.C., Reuter, G.E.H. (eds.) Probability, statistics and analysis, pp. 248–259 Cambridge: Cambridge University Press 1983
Silverman, B.W.: Density estimation for statistics and data analysis. London: Chapman and Hall 1986
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mammen, E., Marron, J.S. & Fisher, N.I. Some asymptotics for multimodality tests based on kernel density estimates. Probab. Th. Rel. Fields 91, 115–132 (1992). https://doi.org/10.1007/BF01194493
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01194493