This function tests the number of modes.
modetest(data,mod0=1,method="NP",B=500,full.result=FALSE,lowsup=-Inf,uppsup=Inf,
n=NULL,tol=NULL,tol2=NULL,methodhy=NULL,alpha=NULL,nMC=NULL,BMC=NULL)Sample to be tested.
The maximum number of modes in the null hypothesis. Default mod0=1 (unimodality vs. multimodality test).
The method employed for testing the number of modes. Available methods are: SI (Silverman, 1981), HY (Hall and York, 2001), FM (Fisher and Marron, 2001), HH (Hartigan and Hartigan, 1985), CH (Cheng and Hall,1998), NP (Ameijeiras--Alonso et al., 2016). Default method="NP".
Number of replicates used in the test. Default B=500.
Logical. If TRUE, returns both the test statistic and the p-value. If FALSE, just the p-value is returned. Default is FALSE.
Lower limit for the random variable support in the computation of the critical bandwidth. Default is -Inf.
Upper limit for the random variable support in the computation of the critical bandwidth. Default is Inf.
The number of equally spaced points at which the density is to be estimated. When n > 512, it is rounded up to a power of 2 as for density function. Default is n=2^10 when method is SI, HY or FM and n=2^15 when the method is CH or NP. Argument not used for other methods.
Accuracy for computing the critical bandwidth. Default tol=10^(-5).
Accuracy for integration of the calibration function in the method NP when the support is known. Default tol2=10^(-5).
Different approaches when using Hall and York (2001) method. Default methodhy=1.
Significance level employed for testing unimodality when method 1 of Hall and York (2001) is used. Default alpha=0.05.
Number of Monte Carlo replicates used to approximate the p-value in the method 2 of Hall and York (2001). Default nMC=100.
Number of bootstrap replicas used for computing the p--value in each Monte Carlo replicate of the Hall and York (2001) method 2. Default BMC=100.
P-value obtained after applying the test.
Value of the test statistic. Critical bandwidth if the method is SI or HY; Cramer-von Mises statistic if the method is FM; the dip statistic if the method is HH, and the excess of mass when the method is CH or NP.
The number of modes for the underlying density of a sample given by data can be tested with modetest. The null hypothesis states that the sample has mod0 modes, and the alternative hypothesis is if it has more modes. The test used for calculating the p-value is specified in method. All the available proposals require bootstrap or Monte Carlo resamples (number specified in B).
Except when the support is employed, the typical usages are
modetest(data,mod0=1,...) modetest(data,mod0=1,method="NP",B=500,full.result=FALSE,n=NULL,tol=NULL)
Since a dichotomy algorithm is employed for computing the critical bandwidth (methods SI, HY, FM, NP), the parameter tol is used to determine a stopping time in such a way that the error committed in the computation of the critical bandwidth is less than tol.
The sample data can be perturbed in the methods using the excess of mass or the dip statistic (HH, CH and NP) in order to avoid important effects on the computation of the test statistic. See excessmass.
If a compact support containing a mode is known, it can be used to compute the Hall and York (2001) critical bandwidth. Note that in the case of the test proposed by Hall and York (2001), this support must be known. For their proposal, two methods are implemented. Method 1 is an asymptotic correction of Silverman (1981) test based on the limiting distribution of the test statistic. When methodhy=1, the significance level must be previously determined with alpha. Method 2 is based on Monte Carlo techniques. For this reason, when methodhy=2, the number of replicates nMC and the number of bootstrap replicas used for computing the p--value in each Monte Carlo replicate BMC are needed.
Typical usages are
modetest(data,method="HY",lowsup=-1.5,uppsup=1.5,methodhy=1,alpha=NULL,...) modetest(data,method="HY",B=500,full.result=FALSE,lowsup=-1.5, uppsup=1.5,n=NULL,tol=NULL,methodhy=1,alpha=NULL)modetest(data,method="HY",lowsup=-1.5,uppsup=1.5,methodhy=2,nMC=NULL,BMC=NULL,...) modetest(data,method="HY",B=500,full.result=FALSE,lowsup=-1.5, uppsup=1.5,n=NULL,tol=NULL,methodhy=2,nMC=NULL,BMC=NULL)
A modification of the proposal of Ameijeiras--Alonso et al. (2016) can be also applied, by setting method=NP and including a known compact support for detecting the modes. The parameter tol2 is the accuracy required in the integration of the calibration function. For more information, see Ameijeiras--Alonso et al. (2016), the default approach, when the support is unknown, is given in Section 2.3 and, when it is provided, the approach shown in the Appendix B is employed.
Typical usages are
modetest(data,mod0=1,lowsup=-1.5,uppsup=1.5,...) modetest(data,mod0=1,B=500,full.result=FALSE,lowsup=-1.5,uppsup=1.5, n=NULL,tol=NULL,tol2=NULL)
The NAs will be automatically removed.
Jose Ameijeiras--Alonso, Rosa M. Crujeiras, Alberto Rodr<U+00ED>guez--Casal (2016). Mode testing, critical bandwidth and excess mass, arXiv preprint: 1609.05188.
Cheng, M. Y. and Hall, P. (1998). Calibrating the excess mass and dip tests of modality, Journal of the Royal Statistical Society. Series B, 60, 579--589.
Fisher, N.I. and Marron, J. S. (2001). Mode testing via the excess mass estimate, Biometrika, 88, 419--517.
Hall, P. and York, M. (2001). On the calibration of Silverman's test for multimodality, Statistica Sinica, 11, 515--536.
Hartigan, J. A. and Hartigan, P. M. (1985). The Dip Test of Unimodality, Journal of the American Statistical Association, 86, 738--746.
Silverman, B. W. (1981). Using kernel density estimates to investigate multimodality, Journal of the Royal Statistical Society. Series B, 43, 97--99.
# NOT RUN {
# Testing for unimodality
data(geyser)
data=geyser
modetest(data)
# }
# NOT RUN {
# Testing bimodality using B=100 bootstrap replicas
modetest(data,mod0=2,B=100)
#There is no evidence to reject the null hypothesis of bimodality
locmodes(data,mod0=2,display=TRUE)
# }
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