The Touchard distribution

Matsushita, Raul; Pianto, Donald M.; Andrade, Bernardo Borba de; Cançado, André Luiz Fernandes; Silva, Sergio Da

doi:10.1080/03610926.2018.1444177

Cited by 3 publications

(7 citation statements)

References 20 publications

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“…Touchard probabilities can be written recursively as

\begin{matrix} true \\ right p_{k + 1} & left = \frac{λ}{k + 1} {(\frac{k + a + 1}{k + a})}^{δ} p_{k} . \end{matrix}

The above property and other recursion formulas similar to those in section 5 in Matsushita et al 30 . facilitate several computations related to moments and random number generation.…”

Section: About the Touchard Distributionmentioning

confidence: 93%

“…The concept of weighting distributions goes back to Fisher's 25 method of ascertainment and it was later revisited and expanded by Rao, Patil, and collaborators 23,24,26 . Recent advances include theoretical aspects of WP models and applications 27–31 but no exploration has been made within statistical quality control.…”

Section: About the Touchard Distributionmentioning

confidence: 99%

“…A family within the WP class with important theoretical properties is obtained with

w (x) = \exp [δ t (x)]

, where

t (\cdot)

is a convex function (depending or not on

λ

) and

δ \in R

provides overdispersion (

δ < 0

), Poissoness (

δ = 0

) or underdispersion (

δ > 0

) 28 . The special case with

t (x) = \log (x + a)

a \geq 0

has been studied by del Castillo and Pérez 27,33 and later labelled the Touchard model 30,31 …”

Section: About the Touchard Distributionmentioning

confidence: 99%

“…This type of weight function has been studied by different authors 27,28,30,31 and has desirable properties, yielding a model in the regular exponential family, it belongs to the family of two‐parameter power series distributions, 34 and has pointwise duality in terms of over/underdispersion 28 . Following recent studies 30,31 we will call the resulting model the Touchard distribution , whose main properties are described next.…”

Section: About the Touchard Distributionmentioning

confidence: 99%

“…The value of

τ

is obtained numerically by truncation of the corresponding series typically at

y_{normalmax} = false[5 λ false]

. Automated choices for

y_{\max}

based on a given relative precision can be used by means of recursive properties 30 . Note that, in general, the normalizing constant is also a function of

a

but we omit this from notation keeping in mind that

a

is considered predetermined.…”

Section: About the Touchard Distributionmentioning

confidence: 99%

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Monitoring count data with Shewhart control charts based on the Touchard model

Andrade

Bourguignon

et al. 2021

Quality & Reliability Eng

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The most common control chart used to monitor count data is based on Poisson distribution, which presents a strong restriction: The mean is equal to the variance. To deal with under‐ or overdispersion, control charts considering other count distributions as Negative Binomial (NB) distribution, hyper‐Poisson, generalized Poisson distribution (GPD), Conway–Maxwell–Poisson (COM‐Poisson), Poisson–Lindley, new generalized Poisson–Lindley (NGPL) have been developed and can be found in the literature. In this paper we also present a Shewhart control chart to monitor count data developed on Touchard distribution, which is a three‐parameter extension of the Poisson distribution (Poisson distribution is a particular case) and in the family of weighted Poisson models. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under‐ or overdispersion and concentration of zeros that are frequently found in non‐Poisson count data. Consequences in terms of speed to signal departures of stability of the parameters are obtained when incorrect control limits based on non‐Touchard distribution (like Poisson, NB or COM‐Poisson) are used to monitor count data generated by a Touchard distribution. Numerical examples illustrate the current proposal.

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“…Touchard probabilities can be written recursively as

\begin{matrix} true \\ right p_{k + 1} & left = \frac{λ}{k + 1} {(\frac{k + a + 1}{k + a})}^{δ} p_{k} . \end{matrix}

The above property and other recursion formulas similar to those in section 5 in Matsushita et al 30 . facilitate several computations related to moments and random number generation.…”

Section: About the Touchard Distributionmentioning

confidence: 93%

Section: About the Touchard Distributionmentioning

confidence: 99%

“…A family within the WP class with important theoretical properties is obtained with

w (x) = \exp [δ t (x)]

, where

t (\cdot)

is a convex function (depending or not on

λ

) and

δ \in R

provides overdispersion (

δ < 0

), Poissoness (

δ = 0

) or underdispersion (

δ > 0

) 28 . The special case with

t (x) = \log (x + a)

a \geq 0

has been studied by del Castillo and Pérez 27,33 and later labelled the Touchard model 30,31 …”

Section: About the Touchard Distributionmentioning

confidence: 99%

Section: About the Touchard Distributionmentioning

confidence: 99%

“…The value of

τ

is obtained numerically by truncation of the corresponding series typically at

y_{normalmax} = false[5 λ false]

. Automated choices for

y_{\max}

based on a given relative precision can be used by means of recursive properties 30 . Note that, in general, the normalizing constant is also a function of

a

but we omit this from notation keeping in mind that

a

is considered predetermined.…”

Section: About the Touchard Distributionmentioning

confidence: 99%

See 3 more Smart Citations

Monitoring count data with Shewhart control charts based on the Touchard model

Andrade

Bourguignon

et al. 2021

Quality & Reliability Eng

View full text Add to dashboard Cite

show abstract

A data transformation to deal with constant under/over-dispersion in Poisson and binomial regression models

Vanegas

Rondón

2020

Journal of Statistical Computation and Simulation

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A distribuição Touchard e suas aplicações

Oliveira¹

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The Poisson distribution, one of the most important distributions in probability theory, has been widely used to model count data. The Poisson distribution depends on a single parameter λ. The expected value and variance of a Poissondistributed random variable are both equal to λ, so using standard Poisson model with under or overdispersed data may result in lack-of-fit. This dissertation presents a two-parameter extension of the Poisson distribution: the Touchard distribution. It is a flexible distribution that can account for both under-or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. Touchard regression and three-parameter extension of the Poisson distribution will also be shown in this work.Several applications will illustrate the capabilities of this approach to be a useful model for assessing non-Poisson data.

show abstract

The Touchard distribution

Cited by 3 publications

References 20 publications

Monitoring count data with Shewhart control charts based on the Touchard model

Monitoring count data with Shewhart control charts based on the Touchard model

A data transformation to deal with constant under/over-dispersion in Poisson and binomial regression models

A distribuição Touchard e suas aplicações

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