Kybernetika 54 no. 4, 798-814, 2018

Estimation and bimodality testing in the cusp model

Jan VoříšekDOI: 10.14736/kyb-2018-4-0798


The probability density function of the stochastic cusp model belongs to the class of generalized exponential distributions. It accommodates variable skewness, kurtosis, and bimodality. A statistical test for bimodality of the stochastic cusp model using the maximum likelihood estimation and delta method for Cardan's discriminant is introduced in this paper, as is a necessary condition for bimodality, which can be used for simplified testing to reject bimodality. Numerical maximum likelihood estimation of the cusp model is simplified by analytical reduction of the parameter space dimension, and connection to the method of moment estimates is shown. A simulation study is used to determine the size and power of the proposed tests and to compare pertinence among different tests for various parameter settings.


multimodal distributions, cusp model, bimodality test, reduced maximum likelihood estimation




  1. V. I. Arnold: Catastrophe Theory. Springer-Verlag, Berlin 1992.   DOI:10.1007/978-3-642-58124-3
  2. J. Barunik and J. Kukačka: Realizing stock market crashes: stochastic cusp catastrophe model of returns under time-varying volatility. Quantitative Finance 15 (2015), 959-973.   DOI:10.1080/14697688.2014.950319
  3. J. Barunik and M. Vošvrda: Can a stochastic cusp catastrophe model explain stock market crashes? J. Economic Dynamics Control 33 (2009), 1824-1836.   DOI:10.1016/j.jedc.2009.04.004
  4. L. Cobb: Stochastic catastrophe models and multimodal distributions. Behavioral Sci. 23 (1978), 360-374.   DOI:10.1002/bs.3830230407
  5. L. Cobb and B. Watson: Statistical catastrophe theory: An overview. Math. Modell. 1 (1980), 311-317.   DOI:10.1016/0270-0255(80)90041-x
  6. L. Cobb: Parameter estimation for the cusp catastrophe model. Behavioral Sci. 26 (1981), 75-78.   DOI:10.1002/bs.3830260107
  7. L. Cobb, P. Koppstein and N. H. Chen: Estimation and moment recursion relations for multimodal distributions of the exponential family. J. Amer. Statist. Assoc. 78 (1983), 124-130.   DOI:10.2307/2287118
  8. J. Creedy, J. Lye and V. Martin: A non-linear model of the real US/UK exchange rate. Econom. Modell. 11 (1996), 669-686.   DOI:10.1002/(sici)1099-1255(199611)11:6<669::aid-jae415>;2-5
  9. C. Diks and J. Wang: Can a stochastic cusp catastrophe model explain housing market crashes? J. Econom. Dynamics Control 69 (2016), 68-88.   DOI:10.1016/j.jedc.2016.05.008
  10. M. Fernandes: Financial crashes as endogenous jumps: estimation, testing and forecasting. J. Econom. Dynamics Control 30 (2006), 111-141.   DOI:10.1016/j.jedc.2004.11.005
  11. R. P. P. P. Grasman, H. L. J. van der Maas and E. J. Wagenmakers: Fitting the cusp catastrophe in R: A cusp package primer. J. Statist. Software 32 (2009), 1-28.   DOI:10.18637/jss.v032.i08
  12. J. A. Hartigan and P. M. Hartigan: The dip test of unimodality. Ann. Statist. 13 (1985), 70-84.   DOI:10.1214/aos/1176346577
  13. D. A. Kodde and F. C. Palm: Wald criteria for jointly testing equality and inequality restrictions. Econometrica 54 (1986), 1243-1248.   DOI:10.2307/1912331
  14. S. K. Koh, W. M. Fong and F. Chan: A Cardans discriminant approach to predicting currency crashes. J. Int. Money Finance 26 (2007), 131-148.   DOI:10.1016/j.jimonfin.2006.08.001
  15. E. L. Lehman and J. P. Romano: Testing Statistical Hypotheses. Third edition. Springer-Verlag, New York 2005.   DOI:10.1007/0-387-27605-x
  16. A. W. Matz: Maximum likelihood parameter estimation for the quartic exponential distribution. Technometrics 20 (1978), 475-484.   DOI:10.1080/00401706.1978.10489702
  17. R. Thom: Structural Stability and Morpohogenesis. W. A. Benjamin, New York 1975.   CrossRef