vstdct (version 0.2)

Data Examples: Data Examples

Description

example1, example2 and example3 generate i.i.d. vectors from a given distribution with different Toeplitz covariance matrices. The covariance function \(\sigma\) of the Toeplitz covariance matrix of

• example1: has a polynomial decay, \(\sigma(\tau)= sd^2(1+|\tau|)^{-gamma}\),

• example2: follows an \(ARMA(2,2)\) model with coefficients \((0.7,-0.4,-0.2,0.2)\) and innovations variance \(sd^2\),

• example3: yields a Lipschitz continuous spectral density \(f\) that is not differentiable, i.e. \(f(x)= sd^2({|\sin(x+0.5\pi)|^{gamma}+0.45})\)

Usage

example1(p, n, sd, gamma, family = "Gaussian")
example2(p, n, sd, family = "Gaussian")
example3(p, n, sd, gamma, family = "Gaussian")

Value

A list containing the following elements:

• Y: pxn dimensional data matrix

• sdf: true spectral density function

• acf: true covariance function

Arguments

p

vector length

n

sample size

sd

standard deviation

gamma

polynomial decay of covariance function for example1 resp. exponent for example3

family

distribution of the simulated data. Available distributions are "Gaussian", "Gamma", "Uniform". The default is "Gaussian".

Examples

example1(p=10, n=1, sd=1, gamma=1.2, family="Gaussian")
example2(p=10,n=1,sd=1,family="Gaussian")
example3(p=10, n=1, sd=1, gamma=2,family="Gaussian")