View source: R/cf_LogRV_ChiSquareNC.R
cf_LogRV_ChiSquareNC | R Documentation |
cf_LogRV_ChiSquareNC(t, df, delta, coef, niid, tol)
evaluates characteristic function
of a linear combination (resp. convolution) of independent LOG-TRANSFORMED non-central ChiSquare random variables,
with distributions ChiSquare(df_i,\delta_i)
.
That is, cf_LogRV_ChiSquareNC
evaluates the characteristic function cf(t)
of Y = coef_i*log(X_1) +...+ coef_N*log(X_N)
, where X_i ~ ChiSquare(df_i,\delta_i)
are inedependent RVs, with df_i
degrees of freedom and the noncentrality parameters \delta_i >0
, for i = 1,...,N
.
The characteristic function of Y = log(X)
with X ~ ChiSquare(df,\delta)
is Poisson mixture
of CFs of the central log-transformed ChiSquare RVs of the form
cf(t) = cf_LogRV_ChiSquareNC(t,df,\delta) = exp(-\delta/2) sum_{j=1}^Inf (\delta/2)^j/j! * cf_LogRV_ChiSquare(df+2*j),
where cf_LogRV_ChiSquare(df+2+j)
are the CFs of central log-transformed ChiSquare RVs with df+2+j
degrees of freedom.
Alternatively,
cf(t) = cf_LogRV_ChiSquareNC(t,df,\delta) = cf_LogRV_ChiSquare(t,df) * 1F1(-1i*t;df/2;-\delta/2),
where 1F1(a;b;z)
is hypergeometric function.
Hence, the characteristic function of Y = coef(1)*Y1 + ... + coef(N)*YN
cf_Y(t) = cf_Y1(coef(1)*t) * ... * cf_YN(coef(N)*t),
where cf_Yi(t)
is evaluated with the parameters df_i
and delta_i
.
cf_LogRV_ChiSquareNC(t, df, delta, coef, niid, tol)
t |
vector or array of real values, where the CF is evaluated. |
df |
vector of the degrees of freedom |
delta |
vector of the non-centrality parameters |
coef |
vector of the coefficients of the linear combination of the Beta distributed random variables.
If coef is scalar, it is assumed that all coefficients are equal. If empty, default value is |
niid |
scalar convolution coeficient |
tol |
tolerance factor for selecting the Poisson weights, i.e. such that |
Characteristic function cf(t)
of a linear combination of independent
LOG-TRANSFORMED non-central ChiSquare random variables.
Ver.: 20-Sep-2018 19:53:05 (consistent with Matlab CharFunTool v1.3.0, 10-Aug-2018 22:25:01).
For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution.
Other Continuous Probability Distribution:
cfS_Arcsine()
,
cfS_Beta()
,
cfS_Gaussian()
,
cfS_Laplace()
,
cfS_Rectangular()
,
cfS_Student()
,
cfS_TSP()
,
cfS_Trapezoidal()
,
cfS_Triangular()
,
cfS_Wigner()
,
cfX_ChiSquare()
,
cfX_Exponential()
,
cfX_FisherSnedecor()
,
cfX_Gamma()
,
cfX_InverseGamma()
,
cfX_LogNormal()
,
cf_ArcsineSymmetric()
,
cf_BetaNC()
,
cf_BetaSymmetric()
,
cf_Beta()
,
cf_ChiSquare()
,
cf_Exponential()
,
cf_FisherSnedecorNC()
,
cf_FisherSnedecor()
,
cf_Gamma()
,
cf_InverseGamma()
,
cf_Laplace()
,
cf_LogRV_BetaNC()
,
cf_LogRV_Beta()
,
cf_LogRV_ChiSquare()
,
cf_LogRV_FisherSnedecorNC()
,
cf_LogRV_FisherSnedecor()
,
cf_LogRV_MeansRatioW()
,
cf_LogRV_MeansRatio()
,
cf_LogRV_WilksLambdaNC()
,
cf_LogRV_WilksLambda()
,
cf_Normal()
,
cf_RectangularSymmetric()
,
cf_Student()
,
cf_TSPSymmetric()
,
cf_TrapezoidalSymmetric()
,
cf_TriangularSymmetric()
,
cf_vonMises()
Other Non-central Probability Distribution:
cf_BetaNC()
,
cf_FisherSnedecorNC()
,
cf_LogRV_BetaNC()
,
cf_LogRV_FisherSnedecorNC()
## EXAMPLE 1
# CF of the minus log-transformed non-central ChiSquare RV with delta = 1
df <- 3
delta <- 1
coef <- -1
t <- seq(from = -10,
to = 10,
length.out = 201)
plotReIm(function(t)
cf_LogRV_ChiSquareNC(t, df, delta, coef),
t,
title = 'CF of minus log-transformed ChiSquare RV')
## EXAMPLE 2
# CDF/PDF of the minus log-transformed non-central ChiSquare RV
df <- 3
delta <- 1
coef <- -1
cf <- function(t)
cf_LogRV_ChiSquareNC(t, df, delta, coef)
options <- list()
options$N <- 2 ^ 10
result <- cf2DistGP(cf = cf, options = options)
## EXAMPLE 3
# CDF/PDF of the linear combination of the minus log-transformed non-central ChiSquare RVs
df <- c(3, 4, 5)
delta <- c(0, 1, 2)
coef <- c(-1,-1,-1) / 3
cf <- function(t)
cf_LogRV_ChiSquareNC(t, df, delta, coef)
options <- list()
options$N <- 2 ^ 10
result <- cf2DistGP(cf = cf, options = options)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.