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 Quantlet: fracbrown Description: calculates the singular value decomposition of the covariance matrix of a fractional Brownian motion.

Reference(s):
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, London Hall, P., Härdle, W., Kleinow, T., Schmidt, P. (2000). Semiparametric Bootstrap Approach to Hypothesis Tests and Confidence Intervals For The Hurst Coefficient, Statistical Inference for Stochastic Processes, Vol. 3

 Usage: sv=fracbrown(p,nu,alpha) Input: p scalar, temporal scale factor nu scalar, the resulting matrix is a (2*p*nu+1) matrix alpha scalar, coefficient of the fractional Brownian motion Output: sv (2*p*nu+1)x(2*p*nu+1) matrix, singular value decomposition of the covariance matrix of a fractional Brownian motion with coefficient alpha

Note:
sv*normal(2*p*nu+1) simulates the path of a fractional Brownian motion with respect to alpha=2*H ( H = Hurst coefficient) in the time interval [-p, p] and time step size 1/nu. The covariance matrix v of the vector sv*normal(p*nu+1) has the following structure:

v = sv*sv' ;

v[i,j] = (1/2)*(| s[i] |^alpha + | s[j] |^alpha - | s[i]-s[j] |^alpha), s=(-p*nu, ..., p*nu)

Example:
```library("times")
randomize(234)
p=1
nu = 50
alpha = 1.5
// simulate frac. BM
x=fracbrown(p,nu,alpha)*normal(2*p*nu+1)
dim(x)			// 2*p*nu+1
h=hurst(x,12)
2*h.b[2]		// estimate alpha

```
Result:
```Contents of dim
[1,]      101
Contents of _tmp
[1,]   1.5237
```

Author: W. Haerdle, T. Kleinow, 20010503 license MD*Tech
(C) MD*TECH Method and Data Technologies, 05.02.2006