# Bootstrap independent and identically distributed functional data or functional time series

### Description

Computes bootstrap or smoothed bootstrap samples based on either independent and identically distributed functional data or functional time series.

### Usage

1 2 | ```
pcscorebootstrapdata(dat, bootrep, statistic, bootmethod = c("st", "sm",
"mvn", "stiefel", "meboot"), smo)
``` |

### Arguments

`dat` |
An object of class |

`bootrep` |
Number of bootstrap samples. |

`statistic` |
Summary statistics. |

`bootmethod` |
Bootstrap method. When |

`smo` |
Smoothing parameter. |

### Details

We will presume that each curve is observed on a grid of *T* points with *0≤q t_1<t_2…<t_T≤q τ*.
Thus, the raw data set *(X_1,X_2,…,X_n)* of *n* observations will consist of an *n* by *T* data matrix.
By applying the singular value decomposition, *X_1,X_2,…,X_n* can be decomposed into *X = ULR^{\top}*,
where the crossproduct of *U* and *R* is identity matrix.

Holding the mean and *L* and *R* fixed at their realized values, there are four re-sampling methods that differ mainly by the ways of re-sampling U.

(a) Obtain the re-sampled singular column matrix by randomly sampling with replacement from the original principal component scores.

(b) To avoid the appearance of repeated values in bootstrapped principal component scores, we adapt a smooth bootstrap procedure by adding a white noise component to the bootstrap.

(c) Because principal component scores follow a standard multivariate normal distribution asymptotically, we can randomly draw principal component scores from a multivariate normal distribution with mean vector and covariance matrix of original principal component scores.

(d) Because the crossproduct of U is identitiy matrix, U is considered as a point on the Stiefel manifold, that is the space of *n* orthogonal vectors, thus we can randomly draw principal component scores from the Stiefel manifold.

### Value

`bootdata` |
Bootstrap samples. If the original data matrix is |

`meanfunction` |
Bootstrap summary statistics. If the original data matrix is |

### Author(s)

Han Lin Shang

### References

H. D. Vinod (2004), "Ranking mutual funds using unconventional utility theory and stochastic dominance", *Journal of Empirical Finance*, **11**(3), 353-377.

A. Cuevas, M. Febrero, R. Fraiman (2006), "On the use of the bootstrap for estimating functions with functional data", *Computational Statistics and Data Analysis*, **51**(2), 1063-1074.

D. S. Poskitt and A. Sengarapillai (2013), "Description length and dimensionality reduction in functional data analysis", *Computational Statistics and Data Analysis*, **58**, 98-113.

H. L. Shang (2015), "Re-sampling techniques for estimating the distribution of descriptive statistics of functional data", *Communications in Statistics–Simulation and Computation*, **44**(3), 614-635.

### See Also

`fbootstrap`

### Examples

1 2 3 4 5 6 | ```
# Bootstrapping the distribution of a summary statistics of functional data.
boot1 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "st")
boot2 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "sm", smo = 0.05)
boot3 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "mvn")
boot4 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "stiefel")
boot5 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "meboot")
``` |