setMixedTS.param {MixedTS}

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

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

param

a numeric object containing the Mixed Tempered Stable parameters. It is not necessary if we use the previous inputs for defining the distribution. See documentation for more details.

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.

Parametrization

a character string. If Parametrization="A" the default, we use the following definition for MixedTS with gamma density

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

where V is distributed as a Gamma(a, sigma^2). Otherwise if Parametrization="B" we have:

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

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)