Description Usage Arguments Value See Also Examples
cfE_Empirical(t, data, cfX) evaluates the characteristic function cf(t) of the Empirical distribution, based on the observed data. In particular, cf(t) = cfE_Empirical(t, data) =
(1/n) * sum_{j=1}^n cf_Dirac(data(j)*t)
, where cfX is function handle of the characteristic function cfX(t) of the random variable X (as e.g. another empirical CF based on observed data of X).
1 | cfE_Empirical(t, data, cfX)
|
t |
numerical values (number, vector...) |
data |
set of observed data |
cfX |
function |
characteristic function cf(t) of the Empirical distribution, based on the observed data
For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Empirical_characteristic_function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | # EXAMPLE1 (Empirical CF - a weighted mixture of independent Dirac variables)
set.seed(101)
n <- 1000
data <- c(rnorm(3 * n, 5, 0.2), rt(n, 3), rchisq(n, 1))
t <- seq(-50, 50, length.out = 2 ^ 10)
plotGraf(function(t)
cfE_Empirical(t, data), t, title = "Empirical CF - CF of the mixture of Dirac random variables")
# EXAMPLE2 (Convolution of the ECF and the Gaussian kernel)
set.seed(101)
n <- 1000
data <- c(rnorm(3 * n, 5, 0.2), rt(n, 3), rchisq(n, 1))
bandwidth <- 0.25
cf_DATA <- function(t)
cfE_Empirical(t, data)
cf_KERNEL <- function(t)
exp(-(bandwidth * t) ^ 2 / 2)
cf <- function(t)
cf_DATA(t) * cf_KERNEL(t)
t <- seq(-50, 50, length.out = 2 ^ 10)
plotGraf(cf, t, title = "Smoothed Empirical CF")
result <- cf2DistGP(cf)
# (PDF/CDF of the compound Empirical-Empirical distribution)
set.seed(101)
lambda <- 25
nN <- 10
Ndata <- rpois(nN, lambda)
mu <- 0.1
sigma <- 2
nX <- 1500
Xdata <- rlnorm(nX, mu, sigma)
cfX <- function(t)
cfE_Empirical(t, Xdata)
cf <- function(t)
cfE_Empirical(t, Ndata, cfX)
t <- seq(-0.2, 0.2, length.out = 2 ^ 10)
plotGraf(cf, t, title = "Compound Empirical CF")
x <- seq(0, 1000, length.out = 501)
prob <- c(0.9, 0.95)
result <- cf2DistGP(cf, x, prob, N = 2 ^ 10, SixSigmaRule = 10)
|
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.