CharFun (version 0.1.0)

cfE_Empirical: Characteristic function of Empirical distribution

Description

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).

Usage

cfE_Empirical(t, data, cfX)

Arguments

t

numerical values (number, vector...)

data

set of observed data

cfX

function

Value

characteristic function cf(t) of the Empirical distribution, based on the observed data

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Empirical_characteristic_function

Examples

# 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)