The first-order autoregressive Mittag–Leffler process

Jayakumar, K.; Pillai, R. N.

doi:10.2307/3214855

Cited by 32 publications

(19 citation statements)

References 3 publications

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“…The fundamental paper in autoregressive modeling with non-Gaussian marginal distribution is by Gaver and Lewis (1980). Subsequently, Lewis (1981, 1985), Dewald and Lewis (1985), Anderson and Arnold (1993) and Jayakumar and Pillai (1993) developed autoregressive models with different marginal distributions.…”

Section: Asymmetric Laplace Processesmentioning

confidence: 99%

A time-series model using asymmetric Laplace distribution

Jayakumar

Kuttykrishnan²

2007

Statistics & Probability Letters

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Section: Asymmetric Laplace Processesmentioning

confidence: 99%

A time-series model using asymmetric Laplace distribution

Jayakumar

Kuttykrishnan²

2007

Statistics & Probability Letters

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“…Here, we describe a different way to compute the survival cure rate model

S_{ϕ} (t)

based on the Mittag‐Leffler distribution of order ϕ and scale parameter λ that is an extension of the exponential distribution (Cahoy et al., ; Laskin, ; Mauro et al., ; Jayakumar and Pillai, ), and is given by

Ψ (t) = 1 - E_{ϕ} (- λ t^{ϕ}), t > 0,

for

0 < ϕ \leq 1

and

λ > 0

. The idea for computing the proposed relaxed cure rate model can be described as follows.…”

Section: Simulation Studymentioning

confidence: 99%

Relaxed Poisson cure rate models

2015

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The purpose of this article is to make the standard promotion cure rate model (Yakovlev and Tsodikov, ) more flexible by assuming that the number of lesions or altered cells after a treatment follows a fractional Poisson distribution (Laskin, ). It is proved that the well-known Mittag-Leffler relaxation function (Berberan-Santos, ) is a simple way to obtain a new cure rate model that is a compromise between the promotion and geometric cure rate models allowing for superdispersion. So, the relaxed cure rate model developed here can be considered as a natural and less restrictive extension of the popular Poisson cure rate model at the cost of an additional parameter, but a competitor to negative-binomial cure rate models (Rodrigues et al., ). Some mathematical properties of a proper relaxed Poisson density are explored. A simulation study and an illustration of the proposed cure rate model from the Bayesian point of view are finally presented.

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“…It may be denoted by ML(α, λ ). Jayakumar and Pillai (1993) considered a more general class called semi-Mittag-Leffler distribution which included the Mittag-Leffler distribution as a special case. A random variable x with positive support is said to have a semi-Mittag-Leffler distribution if its Laplace transform is given by…”

Section: Mittag-leffler Distributionmentioning

confidence: 99%

“…When α = 1, this corresponds to the exponential distribution with unit mean. Jayakumar and Pillai (1993) considered the semi-Mittag-Leffler distribution with exponent α. Its Laplace transform is of the form 1 1+η(t) where η(t) satisfies the functional equation…”

Section: Mittag-leffler Autoregressive Structurementioning

confidence: 99%

Applications to Stochastic Process and Time Series

2008

Special Functions for Applied Scientists

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The first-order autoregressive Mittag–Leffler process

Abstract: The first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process is introduced and its properties are studied. As an illustration, we discuss the special case of the first-order autoregressive Mittag-Leffler (MLAR(1)) process.

Cited by 32 publications

References 3 publications

A time-series model using asymmetric Laplace distribution

A time-series model using asymmetric Laplace distribution

Relaxed Poisson cure rate models

Applications to Stochastic Process and Time Series

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