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): \(Y=\big<X,\beta\big>+\varepsilon\).

Usage

Adot(X, inpr)

PCvM.statistic(X, residuals, p, 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.

Value

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

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).

Note

No NA's are allowed in the functional covariate.

References

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

Author

Eduardo Garcia-Portugues. Please, report bugs and suggestions to eduardo.garcia.portugues@uc3m.es

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
#>           [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]     [,8]
#>  [1,] 34.55752 22.56919 23.42445 22.48372 21.52212 22.14313 22.24761 23.16975
#>  [2,] 22.56919 34.55752 23.11677 23.22390 22.88832 23.43477 23.72314 22.61706
#>  [3,] 23.42445 23.11677 34.55752 22.27545 23.64458 23.07480 23.11521 22.74769
#>  [4,] 22.48372 23.22390 22.27545 34.55752 22.87413 23.14084 22.75209 23.15580
#>  [5,] 21.52212 22.88832 23.64458 22.87413 34.55752 23.21242 24.01934 22.70983
#>  [6,] 22.14313 23.43477 23.07480 23.14084 23.21242 34.55752 23.14678 22.84749
#>  [7,] 22.24761 23.72314 23.11521 22.75209 24.01934 23.14678 34.55752 22.38071
#>  [8,] 23.16975 22.61706 22.74769 23.15580 22.70983 22.84749 22.38071 34.55752
#>  [9,] 23.19679 22.08812 23.39765 22.85707 23.26055 22.92071 22.78013 22.67189
#> [10,] 23.56713 24.23112 23.34407 23.32780 23.06404 23.78524 24.01663 23.20940
#>           [,9]    [,10]
#>  [1,] 23.19679 23.56713
#>  [2,] 22.08812 24.23112
#>  [3,] 23.39765 23.34407
#>  [4,] 22.85707 23.32780
#>  [5,] 23.26055 23.06404
#>  [6,] 22.92071 23.78524
#>  [7,] 22.78013 24.01663
#>  [8,] 22.67189 23.20940
#>  [9,] 34.55752 23.34617
#> [10,] 23.34617 34.55752
# Statistic
PCvM.statistic(X,residuals=rnorm(10),p=5)
#> [1] 4.21528