Mixed Tempered Stable Distribution

Description

This package provides detailed functions for univariate Mixed Tempered Stable distribution distribution with Gamma density. This distribution encompasses, Variance Gamma and Symmetric Geo-Stable as special cases. The package contains routine for mle estimation, for the computation of density, probability, quantile and random numbers

Details

Author(s)

Lorenzo Mercuri, Edit Rroji

Maintainer: Lorenzo Mercuri <lorenzo.mercuri@unimi.it>

References

Barndorff-Nielsen,O.E., Kent,J. and Sorensen, M. (1982): Normal variance-mean mixtures and z-distributions, International Statistical Review, 50, 145-159.

Kuchler, U. and Tappe, S. (2014): Exponential stockmodels driven by tempered stable processes. Journal of Econometrics,181 (1), 53-63.

Madan, D.B. and Seneta E. (1990): The variance gamma (V.G.) model for share market returns, Journal of Business, 63, 511-524

Rroji, E and Mercuri, L.(2014): Mixed Tempered Stable distribution UNIMI-Research Papers in Economics, Business, and Statistics, 64.

"MixedTS": A class for informations about Mixed Tempered Stable

Description

Mathematical description of the Mixed Tempered Stable distribution.

This class inherits from the class param.MixedTS and is a superclass for MixedTS.qmle-class.

Objects from the Class

This object is built by the following methods:

dMixedTS, pMixedTS, qMixedTS, rMixedTS.

Slots

Data:

Object of class "numeric" containing a random number. This slot is filled when the method rMixedTS is used.

dens:

Object of class "numeric" that contains the density of the MixedTS. This slot is filled by dMixedTS.

prob:

Object of class "numeric" that contains the probability of the MixedTS. This slot is filled by pMixedTS and pMixedTS.

xMixedTS:

Object of class "numeric" that contains the support for the density and probability.

quantile:

Object of class "logical". If TRUE the object is built by the method qMixedTS. If FALSE the object is built by the method qMixedTS.

mu0:

Object of class "numeric". See param.MixedTS.

mu:

Object of class "numeric". See param.MixedTS.

sigma:

Object of class "numeric". See param.MixedTS.

a:

Object of class "vector". See param.MixedTS.

alpha:

Object of class "numeric". See param.MixedTS.

lambda_p:

Object of class "numeric". See param.MixedTS.

lambda_m:

Object of class "numeric". See param.MixedTS.

Mixing:

Object of class "character". See param.MixedTS.

paramMixing:

Object of class "list". See param.MixedTS.

MixingLogMGF:

Object of class "OptionalFunction". See param.MixedTS.

Extends

Class "param.MixedTS", directly.

Methods

plot

signature(x = "MixedTS", ...)

MixedTS.qmle: a class for Maximum Likelihood of Mixed Tempered Stable

Description

This class is constructed by function MixedTS.qmle. It is a subclass for the MixedTS-class

Objects from the Class

Objects can be created by function MixedTS.qmle.

Slots

time:

Object of class "numeric". Computational Time.

coef:

Object of class "numeric". Estimated parameters.

vcov:

Object of class "matrix". Approximate variance-covariance matrix.

min:

Object of class "numeric". Minimum value of objective function.

details:

Object of class "list". A list as returned from constrOptim

nobs:

Object of class "integer". Number of observation.

method:

Object of class "character". The optimization method used.

Data:

Object of class "numeric". See MixedTS-class.

dens:

Object of class "numeric". See MixedTS-class.

prob:

Object of class "numeric". See MixedTS-class.

xMixedTS:

Object of class "numeric" . See MixedTS-class.

quantile:

Object of class "logical". See MixedTS-class.

mu0:

Object of class "numeric". See MixedTS-class.

mu:

Object of class "numeric". See MixedTS-class.

sigma:

Object of class "numeric". See MixedTS-class.

a:

Object of class "vector". See MixedTS-class.

alpha:

Object of class "numeric". See MixedTS-class.

lambda_p:

Object of class "numeric". See MixedTS-class.

lambda_m:

Object of class "numeric". See MixedTS-class.

Mixing:

Object of class "character". See MixedTS-class.

paramMixing:

Object of class "list" . See MixedTS-class.

MixingLogMGF:

Object of class "OptionalFunction". See MixedTS-class.

Extends

Class "MixedTS", directly. Class "param.MixedTS", by class "MixedTS", distance 2.

Methods

summary

signature(.Object = "MixedTS.qmle")

coef

signature(.Object = "MixedTS.qmle")

vcov

signature(.Object = "MixedTS.qmle")

logLik

signature(.Object = "MixedTS.qmle")

BIC

signature(.Object = "MixedTS.qmle")

AIC

signature(.Object = "MixedTS.qmle")

Density of Mixed Tempered Stable distribution

Description

This Method returns the density of a Mixed Tempered Stable

Methods

signature(object = "param.MixedTS",x = numeric(), setSup=NULL,setInf=NULL,N=2^10)

This method returns an object of class MixedTS where the slot dens contains the value of the density evaluated on the x. setSup and setInf are used to choose + infinity and - infinty. N is the number of point used for discretization in fft algorithm.

Examples

# First Example

# Density of MixedTS with Gamma

ParamEx1<-setMixedTS.param(mu0=0, mu=0, sigma=0.4, a=1.5,
                           alpha=0.8, lambda_p=4, lambda_m=1, 
                           Mixing="Gamma")

# support

x<-seq(-3,1,length=100)

dens1<-dMixedTS(x=x,object=ParamEx1,setSup=10,setInf=-10,N=2^7)

plot(dens1)

# Density of MixedTS with IG

Mix<-"User"

logmgf<-("lamb/mu1*(1-sqrt(1-2*mu1^2/lamb*u))")

parMix<-list(lamb=1,mu1=1)

ParamEx2<-setMixedTS.param(mu0=0, mu=0, sigma=0.4, a=logmgf,
                           alpha=0.8, lambda_p=4, lambda_m=1,
                           Mixing=Mix,paramMixing=parMix)

x<-seq(-3,1,length=100)

dens2<-dMixedTS(x=x,object=ParamEx2,setSup=10,setInf=-10,N=2^7)

plot(dens2)

Maximum Likelihood Estimation for MixedTS distribution

Description

Estimate MixedTS parameters using the Maximum Likelihood Estimation procedure.

Usage

mle.MixedTS(object, start = list(), Data = NULL, 
          method = "L-BFGS-B", fixed.param = NULL, 
          lower.param = NULL, upper.param = NULL, 
          setSup = NULL, setInf = NULL, N = 2^10)

Arguments

Value

The function returns an object of class MixedTS.qmle.

Examples

# First Example:
# We define the Mixed Tempered Stable using the function setMixedTS.param


ParamEx1<-setMixedTS.param(mu0=0, mu=0, sigma=0.4, a=1.5,
                           alpha=0.8, lambda_p=4, lambda_m=1, Mixing="Gamma")

# We generate a sample using the rMixedTS method
set.seed(100)
Rand1 <- rMixedTS(x=5000,object=ParamEx1, setSup=10,setInf=-10,N=2^9)

# Estimate procedure
## Not run: 
est1<-mle.MixedTS(object=Rand1 , setSup=10,setInf=-10,N=2^9)
# Show results

summary(est1)

## End(Not run)

Probability of Mixed Tempered Stable distribution

Description

This Method returns the cdf of a Mixed Tempered Stable

Methods

signature(object = "param.MixedTS",x = numeric(), setSup=NULL,setInf=NULL,N=2^10)

This method returns an object of class MixedTS where the slot prob contains the value of the probability evaluated on the x. setSup and setInf are used to choose + infinity and - infinty. N is the number of point used for discretization in fft algorithm.

Examples

# First Example

# Density of MixedTS with Gamma

ParamEx1<-setMixedTS.param(mu0=0, mu=0, sigma=0.4, a=1.5,
                           alpha=0.8, lambda_p=4, lambda_m=1, 
                           Mixing="Gamma")
# support

x<-seq(-3,1,length=100)

prob1<-pMixedTS(x=x,object=ParamEx1,setSup=10,setInf=-10,N=2^7)

plot(prob1)

# Prob of MixedTS with IG

Mix<-"User"

parMix<-list(lamb=1,mu1=1)

logmgf<-("lamb/mu1*(1-sqrt(1-2*mu1^2/lamb*u))")

ParamEx2<-setMixedTS.param(mu0=0, mu=0, sigma=0.4, a=logmgf,
                           alpha=0.8, lambda_p=4, lambda_m=1,
                           Mixing=Mix,paramMixing=parMix)

x<-seq(-3,1,length=100)

prob2<-pMixedTS(x=x,object=ParamEx2,setSup=10,setInf=-10,N=2^7)
plot(prob2)

"param.MixedTS": A mathematical Description of the Mixed Tempered Stable

Description

Main class of the package MixedTS.

Objects from the Class

Objects can be created by calls of the form setMixedTS.

Slots

mu0:

a numeric object. mu0 parameter belongs to the real axis.

mu:

a numeric object. mu parameter belongs to the real axis

sigma

a numeric object. sigma parameter assumes value from zero to infinity.

a

a vector object. If numeric, the mixing density V is a Gamma and a is the value of the shape parameter. If string, a is the log of the moment generating function of the mixing density V.

alpha

a numeric object that takes value from 0 to 2. If alpha is fixed to 2, the Mixed Tempered Stable becomes the Normal Variance Mean mixture.

lambda_p

a positive numeric object. It is the right tempering parameter of the random variable X.

lambda_m

a positive numeric object. It is the left tempering parameter of the random variable X

Mixing

a string object indicating the nature of the mixing density V. If Mixing="Gamma" (default value), the V randm variable is a Gamma. If Mixing="Gamma", the user have to specify the log of the moment generating function of the V random variable.

paramMixing

a list object. It is an empty list when Mixing="Gamma". If Mixing="User", it is used to pass the values of the Mixing density parameters defined by the User through slot a.

MixingLogMGF:

This slot contains a function that returns the logarithm of mgf for the Mixing density. The function is built internally using the information contains into the slots a, paramMixing.

Parametrization:

String that indicates the parametrization used by user for the MixedTS

Methods

dMixedTS

signature(object = "param.MixedTS"): Method for computing density of MixedTS. See "dMixedTS-methods" for more details.

pMixedTS

signature(object = "param.MixedTS"): Method for computing probability of MixedTS. See "pMixedTS-methods" for more details.

qMixedTS

signature(object = "param.MixedTS"): Method for computing quantile of MixedTS. See "qMixedTS-methods" for more details.

rMixedTS

signature(object = "param.MixedTS"): Method for computing random numbers of MixedTS. See "rMixedTS-methods" for more details.

initialize

signature(object = "param.MixedTS").

Qparam.MixedTS

signature(object = "param.MixedTS").

Quantile of Mixed Tempered Stable distribution

Description

This Method returns the quantile of a Mixed Tempered Stable.

Methods

signature(object = "param.MixedTS",x = numeric(), setSup=NULL,setInf=NULL,N=2^10)

This method returns an object of class MixedTS where the slot prob contains the value of the quantile evaluated on the x (x is the probability). setSup and setInf are used to choose + infinity and - infinty. N is the number of point used for discretization in fft algorithm.

Random number of Mixed Tempered Stable distribution

Description

This Method returns the quantile of a Mixed Tempered Stable.

Methods

signature(object = "param.MixedTS",x = numeric(), setSup=NULL,setInf=NULL,N=2^10)

This method returns an object of class MixedTS where the slot Data contains a set of size x of random numbers. setSup and setInf are used to choose + infinity and - infinty. N is the number of point used for discretization in fft algorithm.

Mixed Tempered Stable distribution

Description

setMixedTS describes the Mixed Tempered Stable distribution introduced in Rroji and Mercuri (2014):

Definition

We say that a continuous random variable Y follows a Mixed Tempered Stable distribution if:

Y= mu0+ mu*V + sigma*sqrt{V}*Z

The conditional distribution of random variable given V=v is a standardized Tempered Stable with parameters (alpha, lambda_p*sqrt{v}, lambda_m) (see Kuchler, U. and Tappe, S. 2014). The distribution of V is infinitely divisible defined on the positive axis.

Usage

setMixedTS.param(mu0 = numeric(), mu = numeric(), 
  sigma = numeric(), a, alpha = numeric(), 
  lambda_p = numeric(), lambda_m = numeric(), 
  param = numeric(), Mixing = "Gamma", paramMixing = list(), Parametrization = "A")

Arguments

where V is distributed as a Gamma(a, 1).

Details

For particular choices of the tempering parameters the tails of the MixedTS distribution can be heavy or semi-heavy. In particular if the Mixing density is a Gamma, we get the Variance Gamma (Madan and Seneta 1990) and the symmetric Geo-Stable distribution as special cases.

Value

This function returns an object of class "param.MixedTS".

Note

This class of distributions has the Normal Variance Mean Mixture (Barndorff-Nielsen et al. 1982) as special case.

References

Barndorff-Nielsen,O.E., Kent,J. and Sorensen, M. (1982): Normal variance-mean mixtures and z-distributions, International Statistical Review, 50, 145-159.

Kuchler, U. and Tappe, S. (2014): Exponential stockmodels driven by tempered stable processes. Journal of Econometrics,181 (1), 53-63.

Madan, D.B. and Seneta E. (1990): The variance gamma (V.G.) model for share market returns, Journal of Business, 63, 511-524

Rroji, E and Mercuri, L.(2014): Mixed Tempered Stable distribution UNIMI-Research Papers in Economics, Business, and Statistics, 64.

Examples

# Mixed Tempered Stable with Gamma Mixing density.

ParamEx1<-setMixedTS.param(mu0=0, mu=0, sigma=0.4, a=1.5,
                           alpha=0.8, lambda_p=4, lambda_m=1)


# Mixed Tempered Stable with Inverse Gaussian Mixing density.
## As first step we set the "a" parameter 
## equal to the log mgf of the inverse gaussian random variable
# The log mgf of an Ig with parameter (lamb, mu1) is defined as:

logmgf<-("lamb/mu1*(1-sqrt(1-2*mu1^2/lamb*u))")
Mix<-"User"

# The parameters of the mixing density are set by the following command
# line:

parMix<-list(lamb=1,mu1=1)

ParamEx2<-setMixedTS.param(mu0=0, mu=0, sigma=0.4, a=logmgf,
                           alpha=0.8, lambda_p=4, lambda_m=1,
                           Mixing=Mix,paramMixing=parMix)