fda.usc (version 1.2.3)

flm.Ftest: F-test for the Functional Linear Model with scalar response

Description

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): $Y=+\epsilon$. 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).

Usage

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

Arguments

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.

Value

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=+e"

Details

The Functional Linear Model with scalar response (FLM), is defined as $Y=+\epsilon$, 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 $||.||$ 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.

References

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

See Also

rwild, flm.test, dfv.test

Examples

## 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))

## Not run: 
# # 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)
# ## End(Not run)