fda.usc (version 2.1.0)

PCvM.statistic: PCvM statistic for the Functional Linear Model with scalar response

Description

Projected Cramer-von Mises statistic (PCvM) for the Functional Linear Model with scalar response (FLM): \(Y=\big<X,\beta\big>+\varepsilon\).

Usage

Adot(X, inpr)
PCvM.statistic(X, residuals, p, Adot.vec)

Value

For PCvM.statistic, the value of the statistic. For Adot, a suitable output to be used in the argument Adot.vec.

Arguments

X

Functional covariate for the FLM. The object must be either in the class fdata or in the class fd. It is used to compute the matrix of inner products.

inpr

Matrix of inner products of X. Computed if not given.

residuals

Residuals of the estimated FLM.

p

Number of elements of the functional basis where the functional covariate is represented.

Adot.vec

Output from the Adot function (see Details). Computed if not given.

Author

Eduardo Garcia-Portugues. Please, report bugs and suggestions to

Details

In order to optimize the computation of the statistic, the critical parts of these two functions are coded in FORTRAN. The hardest part corresponds to the function Adot, which involves the computation of a symmetric matrix of dimension \(n\times n\) where each entry is a sum of \(n\) elements. As this matrix is symmetric, the order of the method can be reduced from \(O(n^3)\) to \(O\big(\frac{n^3-n^2}{2}\big)\). The memory requirement can also be reduced to \(O\big(\frac{n^2-n+2}{2}\big)\). The value of Adot is a vector of length \(\frac{n^2-n+2}{2}\) where the first element is the common diagonal element and the rest are the lower triangle entries of the matrix, sorted by rows (see Examples).

References

Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory, 22, 1030-1051. tools:::Rd_expr_doi("10.1017/S0266466606060506")

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. tools:::Rd_expr_doi("10.1080/10618600.2013.812519")

See Also

flm.test

Examples

# Functional process
X=rproc2fdata(n=10,t=seq(0,1,l=101))
# Adot
Adot.vec=Adot(X)

# Obtain the entire matrix Adot
Ad=diag(rep(Adot.vec[1],dim(X$data)[1]))
Ad[upper.tri(Ad,diag=FALSE)]=Adot.vec[-1]
Ad=t(Ad)
Ad=Ad+t(Ad)-diag(diag(Ad))
Ad
# Statistic
PCvM.statistic(X,residuals=rnorm(10),p=5)