# PCvM.statistic

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

- Keywords
- htest

##### 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.

##### 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).

##### Value

For `PCvM.statistic`

, the value of the statistic. For `Adot`

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

.

##### 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. 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

##### See Also

##### Examples

```
# 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)
# }
```

*Documentation reproduced from package fda.usc, version 2.0.1, License: GPL-2*