# PCvM.statistic: PCvM statistic for the Functional Linear Model with scalar... In fda.usc: Functional Data Analysis and Utilities for Statistical Computing

 PCvM.statistic R Documentation

## 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=<X,β>+ε.

### Usage

```Adot(X, inpr)

```

### 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 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).

### 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.

### Author(s)

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

### 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

`flm.test`

### Examples

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

# Obtain the entire matrix Adot