The function `flm.Ftest`

tests the null hypothesis of no interaction between a functional covariate and a scalar response inside the Functional Linear Model (FLM): \(Y=\big<X,\beta\big>+\epsilon\). The null hypothesis is \(H_0:\,\beta=0\) and the alternative is \(H_1:\,\beta\neq 0\).
The null hypothesis is tested by a functional extension of the classical F-test (see Details).

`Ftest.statistic(X.fdata, Y)`flm.Ftest(X.fdata, Y, B = 5000, verbose = TRUE)

X.fdata

Functional covariate for the FLM. The object must be in the class `fdata`

.

Y

Scalar response for the FLM. Must be a vector with the same number of elements as functions are in `X.fdata`

.

B

Number of bootstrap replicates to calibrate the distribution of the test statistic. `B=5000`

replicates are the recommended for carry out the test, although for exploratory analysis (**not inferential**), an acceptable less time-consuming option is `B=500`

.

verbose

Either to show or not information about computing progress.

The value for `Ftest.statistic`

is simply the F-test statistic. The value for `flm.Ftest`

is an object with class `"htest"`

whose underlying structure is a list containing the following components:

statistic The value of the F-test statistic.

boot.statistics A vector of length

`B`

with the values of the bootstrap F-test statistics.p.value The p-value of the test.

method The character string "Functional Linear Model F-test".

B The number of bootstrap replicates used.

data.name The character string "Y=<X,0>+e"

The Functional Linear Model with scalar response (FLM), is defined as
\(Y=\big<X,\beta\big>+\epsilon\), for a functional process \(X\)
such that \(E[X(t)]=0\), \(E[X(t)\epsilon]=0\) for all \(t\)
and for a scalar variable \(Y\) such that \(E[Y]=0\).
The *functional F-test* is defined as
$$T_n=\bigg\|\frac{1}{n}\sum_{i=1}^n (X_i-\bar X)(Y_i-\bar Y)\bigg\|,$$ where \(\bar X\) is the functional mean of \(X\), \(\bar Y\) is the ordinary mean of \(Y\) and \(\|\cdot\|\) is the \(L^2\) functional norm.
The statistic is computed with the function `Ftest.statistic`

. The distribution of the
test statistic is approximated by a wild bootstrap resampling on the residuals, using the
*golden section bootstrap*.

Garcia-Portugues, E., Gonzalez-Manteiga, W. and Febrero-Bande, M. (2014). A goodness--of--fit test for the functional linear model with scalar response. Journal of Computational and Graphical Statistics, 23(3), 761-778. http://dx.doi.org/10.1080/10618600.2013.812519

Gonzalez-Manteiga, W., Gonzalez-Rodriguez, G., Martinez-Calvo, A. and Garcia-Portugues, E. Bootstrap independence test for functional linear models. arXiv:1210.1072. http://arxiv.org/abs/1210.1072

# 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)) # Do not reject H0 flm.Ftest(X,Y0,B=100) flm.Ftest(X,Y0,B=5000) # Reject H0 flm.Ftest(X,Y1,B=100) flm.Ftest(X,Y1,B=5000) # }