Lévy processes time-changed by the first-exit time of the inverse Gaussian subordinator

Mselmi, Farouk

doi:10.2298/fil1807545m

Cited by 2 publications

(8 citation statements)

References 21 publications

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“…The variance function of the natural exponential family F ( η 1 ) is given, for all

m \in M_{F false(η_{1} false)} ∖ false{0 false}

, by

V_{F false(η_{1} false)} false(m false) = {()(, K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false)))}^{2} V_{F false(ρ_{1} false)} ()(, \frac{m}{K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false))}) + \frac{m V_{F false(Q_{1} false)} ()(, K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false)))}{K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false))},

where

V_{F false(ρ_{1} false)} false(x false) = \frac{4 x^{3} \sqrt{1 + 8 x}}{false(2 x + 1 false) \sqrt{1 + 8 x} + 6 x + 1}

, for all x ∈ (0, + ∞ ) (see Theorem 3.1 and Theorem 4.4. of Mselmi, 2018a). In the rest of the paper, we denote by

μ : = μ_{1}

ρ : = ρ_{1}

and

η : = η_{1}

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Its first-exit time is defined by YðtÞ ¼ inffs > 0; XðsÞ > tg , where X(t) is inverse Gaussian distributed (see Meerschaert & Scheffler, 2008;Meerschaert, Nane, & Vellaisamy, 2011, 2018a, 2018a, 2018b, 2018b. Using the same notations of Mselmi (2018a), we denote by Q y (dz), ρ t (dy),…”

Section: Natural Exponential Familymentioning

confidence: 99%

“…þ6xþ1 , for all x (0, + ∞) (see Theorem 3.1 and Theorem 4.4. of Mselmi, 2018a). In the rest of the paper, we denote by μ :¼ μ 1 , ρ :¼ ρ 1 and η :¼ η 1 .…”

Section: Natural Exponential Familymentioning

confidence: 99%

“…It is worth mentioning that the inverse Gaussian subordinator represents a particular case of the α ‐stable one with

α = 0.5

(see Louati, Masmoudi, & Mselmi, 2015b; 2017,2020; 2018c). Its first‐exit time is defined by

Y false(t false) = \inf false{s > 0; X false(s false) > t false}

, where X ( t ) is inverse Gaussian distributed (see Meerschaert & Scheffler, 2008; Meerschaert, Nane, & Vellaisamy, 2011, 2018a, 2018a, 2018b, 2018b). Using the same notations of Mselmi (2018a), we denote by Q y ( dz ), ρ t ( dy ),

μ_{t} false(d y, d z false) = Q_{y} false(d z false) ρ_{t} false(d y false)

and

η_{t} false(d z false) = \int_{0}^{+ \infty} Q_{y} false(d z false) ρ_{t} false(d y false)

the distributions of G ( y ), Y ( t ), ( Y ( t ), Z ( t )) and Z ( t ) respectively.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recently, Mselmi (2018a) has characterized the natural exponential family generated by the distribution of the class of time‐changed Lévy processes by the first‐exit time of the inverse Gaussian subordinator by the form of its variance function. Nevertheless it depends to its unknown link function.…”

Section: Introductionmentioning

confidence: 99%

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Approximation of the quasi‐deviance function for the time‐changed Lévy processes by the first‐exit time of the inverse Gaussian subordinator

Mselmi

2021

Stat

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“…The variance function of the natural exponential family F ( η 1 ) is given, for all

m \in M_{F false(η_{1} false)} ∖ false{0 false}

, by

V_{F false(η_{1} false)} false(m false) = {()(, K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false)))}^{2} V_{F false(ρ_{1} false)} ()(, \frac{m}{K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false))}) + \frac{m V_{F false(Q_{1} false)} ()(, K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false)))}{K_{Q_{1}}^{'} ()(, ψ_{η_{1}} false(m false))},

where

V_{F false(ρ_{1} false)} false(x false) = \frac{4 x^{3} \sqrt{1 + 8 x}}{false(2 x + 1 false) \sqrt{1 + 8 x} + 6 x + 1}

, for all x ∈ (0, + ∞ ) (see Theorem 3.1 and Theorem 4.4. of Mselmi, 2018a). In the rest of the paper, we denote by

μ : = μ_{1}

ρ : = ρ_{1}

and

η : = η_{1}

.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Natural Exponential Familymentioning

confidence: 99%

“…þ6xþ1 , for all x (0, + ∞) (see Theorem 3.1 and Theorem 4.4. of Mselmi, 2018a). In the rest of the paper, we denote by μ :¼ μ 1 , ρ :¼ ρ 1 and η :¼ η 1 .…”

Section: Natural Exponential Familymentioning

confidence: 99%

“…It is worth mentioning that the inverse Gaussian subordinator represents a particular case of the α ‐stable one with

α = 0.5

(see Louati, Masmoudi, & Mselmi, 2015b; 2017,2020; 2018c). Its first‐exit time is defined by

Y false(t false) = \inf false{s > 0; X false(s false) > t false}

μ_{t} false(d y, d z false) = Q_{y} false(d z false) ρ_{t} false(d y false)

and

η_{t} false(d z false) = \int_{0}^{+ \infty} Q_{y} false(d z false) ρ_{t} false(d y false)

the distributions of G ( y ), Y ( t ), ( Y ( t ), Z ( t )) and Z ( t ) respectively.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximation of the quasi‐deviance function for the time‐changed Lévy processes by the first‐exit time of the inverse Gaussian subordinator

Mselmi

2021

Stat

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Generalized linear model for subordinated Lévy processes

Mselmi

2021

Scandinavian J Statistics

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Generalized linear models, introduced by Nelder andWedderburn, allowed to model the regression of normal and nonnormal data. While doing so, the analysis of these models could not be obtained without the explicit form of the variance function. In this paper, we determine the link and variance functions of the natural exponential family generated by the class of subordinated Lévy processes. In this framework, we introduce a class of variance functions that depends on the Lambert function. In this regard, we call it the Lambert class, which covers the variance functions of the natural exponential families generated by the subordinated gamma processes and the subordinated Lévy processes by the Poisson subordinator. Notice that the gamma process subordinated by the Poisson one is excluded from this class. The concept of reciprocity in natural exponential families was given in order to obtain an exponential family from another one. In this context, we get the reciprocal class of the natural exponential family generated by the class of subordinated Lévy processes. It is well known that the variance function represents an essential element for the determination of the quasi-likelihood and deviance functions. Then, we use the expression of our variance function in order to maintain them. This leads us to analyze the proposed generalized linear model. We illustrate some of our

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Lévy processes time-changed by the first-exit time of the inverse Gaussian subordinator

Cited by 2 publications

References 21 publications

Approximation of the quasi‐deviance function for the time‐changed Lévy processes by the first‐exit time of the inverse Gaussian subordinator

Approximation of the quasi‐deviance function for the time‐changed Lévy processes by the first‐exit time of the inverse Gaussian subordinator

Generalized linear model for subordinated Lévy processes

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