PCvM statistic for the Functional Linear Model with scalar response
Projected Cramer-von Mises statistic (PCvM) for the Functional Linear Model with scalar response (FLM): Y=<X,β>+ε.
Adot(X, inpr) PCvM.statistic(X, residuals, p, Adot.vec)
X |
Functional covariate for the FLM.
The object must be either in the class |
inpr |
Matrix of inner products of |
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 |
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 x 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((n^3-n^2)/2). The memory requirement can also be reduced
to O((n^2-n+2)/2). The value of Adot is a vector of
length (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).
For PCvM.statistic, the value of the statistic. For Adot,
a suitable output to be used in the argument Adot.vec.
No NA's are allowed in the functional covariate.
Eduardo Garcia-Portugues. Please, report bugs and suggestions to egarcia@math.ku.dk
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
# 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)
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