Delsol, Ferraty and Vieu test for no functional-scalar interaction
The function dfv.test tests the null hypothesis of no interaction between a functional covariate and a scalar response in a general framework. The null hypothesis is
H_0: m(X)=0,
where m(.) denotes the regression function of the functional variate X over the centred scalar response Y (E[Y]=0). The null hypothesis is tested by the smoothed integrated square error of the response (see Details).
dfv.statistic( X.fdata, Y, h = quantile(x = metric.lp(X.fdata), probs = c(0.05, 0.1, 0.15, 0.25, 0.5)), K = function(x) 2 * dnorm(abs(x)), weights = rep(1, dim(X.fdata$data)[1]), d = metric.lp, dist = NULL ) dfv.test( X.fdata, Y, B = 5000, h = quantile(x = metric.lp(X.fdata), probs = c(0.05, 0.1, 0.15, 0.25, 0.5)), K = function(x) 2 * dnorm(abs(x)), weights = rep(1, dim(X.fdata$data)[1]), d = metric.lp, verbose = TRUE )
X.fdata |
Functional covariate. The object must be in the class |
Y |
Scalar response. Must be a vector with the same number of elements as functions are in |
h |
Bandwidth parameter for the kernel smoothing. This is a crucial parameter that affects the power performance of the test. One possibility to choose it is considering the Cross-validatory bandwidth of the nonparametric functional regression, given by the function |
K |
Kernel function. If no specified it is taken to be the rescaled right part of the normal density. |
weights |
A vector of weights for the sample data. The default is the uniform weights |
d |
Semimetric to use in the kernel smoothers. By default is the L^2 distance given by |
dist |
Matrix of distances of the functional data, used to save time in the bootstrap calibration. If not given, the matrix is automatically computed using the semimetric |
B |
Number of bootstrap replicates to calibrate the distribution of the test statistic. |
verbose |
Either to show or not information about computing progress. |
The Delsol, Ferraty and Vieu statistic is defined as
T_n=\int(∑_{i=1}^n(Y_i-m(X_i))K(\frac{d(X,X_i)}{h}))^2ω(X)dP_X(X)
and in the case of no interaction with centred scalar response (when H_0: m(X)=0 holds), its sample version is computed from
T_n=\frac{1}{n}∑_{j=1}^n(∑_{i=1}^n Y_iK(\frac{d(X_j,X_i)}{h}))^2ω(X_j).
The sample version implemented here does not consider a splitting of the sample, as the authors
comment in their paper. The statistic is computed by the function dfv.statistic and, before
applying the test, the response Y is centred. The distribution of the test statistic
is approximated by a wild bootstrap on the residuals, using the golden section bootstrap.
Please note that if a grid of bandwidths is passed, a harmless warning message will prompt at the
end of the test (it comes from returning several p-values in the htest class).
The value of dfv.statistic is a vector of length length(h) with the values
of the statistic for each bandwidth. The value of dfv.test is an object with class
"htest" whose underlying structure is a list containing the following components:
statistic The value of the Delsol, Ferraty and Vieu test statistic.
boot.statistics A vector of length B with the values of the bootstrap test statistics.
p.value The p-value of the test.
method The character string "Delsol, Ferraty and Vieu test for no functional-scalar interaction".
BThe number of bootstrap replicates used.
hBandwidth parameters for the test.
KKernel function used.
weightsThe weights considered.
dMatrix of distances of the functional data.
data.nameThe character string "Y=0+e"
No NA's are allowed neither in the functional covariate nor in the scalar response.
Eduardo Garcia-Portugues. Please, report bugs and suggestions to egarcia@math.ku.dk
Delsol, L., Ferraty, F. and Vieu, P. (2011). Structural test in regression on functional variables. Journal of Multivariate Analysis, 102, 422-447. http://dx.doi.org/10.1016/j.jmva.2010.10.003
Delsol, L. (2013). No effect tests in regression on functional variable and some applications to spectrometric studies. Computational Statistics, 28(4), 1775-1811. http://dx.doi.org/10.1007/s00180-012-0378-1
## Not run: ## Simulated example ## X=rproc2fdata(n=50,t=seq(0,1,l=101),sigma="OU") beta0=fdata(mdata=rep(0,length=101)+rnorm(101,sd=0.05), argvals=seq(0,1,l=101),rangeval=c(0,1)) beta1=fdata(mdata=cos(2*pi*seq(0,1,l=101))-(seq(0,1,l=101)-0.5)^2+ rnorm(101,sd=0.05),argvals=seq(0,1,l=101),rangeval=c(0,1)) # Null hypothesis holds Y0=drop(inprod.fdata(X,beta0)+rnorm(50,sd=0.1)) # Null hypothesis does not hold Y1=drop(inprod.fdata(X,beta1)+rnorm(50,sd=0.1)) # We use the CV bandwidth given by fregre.np # Do not reject H0 dfv.test(X,Y0,h=fregre.np(X,Y0)$h.opt,B=100) # dfv.test(X,Y0,B=5000) # Reject H0 dfv.test(X,Y1,B=100) # dfv.test(X,Y1,B=5000) ## End(Not run)
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