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

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

X

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.

For `PCvM.statistic`

, the value of the statistic. For `Adot`

,
a suitable output to be used in the argument `Adot.vec`

.

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).

Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory, 22, 1030-1051. http://dx.doi.org/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. http://dx.doi.org/10.1080/10618600.2013.812519

# NOT RUN { # 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) # }