A flexible distribution class for count data

Sellers, Kimberly F.; Swift, Andrew W.; Weems, Kimberly S.

doi:10.1186/s40488-017-0077-0

Cited by 15 publications

(16 citation statements)

References 24 publications

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“…As is often the case with count data (Manté et al ., ; Sellers et al ., ), the post‐trial louse abundance data from the leaping experiment were over‐dispersed (Figure ), demonstrating greater than expected variation relative to a Poisson distribution (mean motile abundance = 1.87, variance = 2.46).…”

Section: Methodsmentioning

confidence: 99%

Oust the louse: leaping behaviour removes sea lice from wild juvenile sockeye salmon Oncorhynchus nerka

Atkinson

Bateman

Dill

et al. 2018

Journal of Fish Biology

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We conducted a manipulative field experiment to determine whether the leaping behaviour of wild juvenile sockeye salmon Oncorhynchus nerka dislodges ectoparasitic sea lice Caligus clemensi and Lepeophtheirus salmonis by comparing sea-lice abundances between O. nerka juveniles prevented from leaping and juveniles allowed to leap at a natural frequency. Juvenile O. nerka allowed to leap had consistently fewer sea lice after the experiment than fish that were prevented from leaping. Combined with past research, these results imply potential costs due to parasitism and indicate that the leaping behaviour of juvenile O. nerka does, in fact, dislodge sea lice.

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Section: Methodsmentioning

confidence: 99%

Oust the louse: leaping behaviour removes sea lice from wild juvenile sockeye salmon Oncorhynchus nerka

Atkinson

Bateman

Dill

et al. 2018

Journal of Fish Biology

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“…The sCMP(

λ, ν, n

) distribution encompasses the Poisson distribution with rate parameter

n λ

(for

ν = 1

), negative binomial(

n, 1 - λ

) distribution (for

ν = 0

and

λ < 1

), and Binomial(

n, p

) distribution

((), as ν \to \infty with success probability p = \frac{λ}{λ + 1})

as special cases. Furthermore, for

n = 1

, the sCMP(

λ, ν, n = 1

) is simply the CMP(

λ, ν

) distribution Sellers et al ().…”

Section: Motivating Distributionsmentioning

confidence: 99%

“…Along with a sCMP, Sellers et al () introduce a generalized Conway–Maxwell–Binomial (gCMB) distribution. Let

Y_{*}

be a gCMB(

p, ν, s, n_{1}, n_{2}

) random variable; accordingly, it has the pmf

P false(Y_{*} = y false) = \frac{{((), \binom{s}{y})}^{ν} p^{y} {false(1 - p false)}^{s - y} ([], true \sum_{a_{1}, \dots, a_{n_{1}}}^{y} {((), \binom{y}{a_{1}, \dots, a_{n_{1}}})}^{ν}) ([], true \sum_{b_{1}, \dots, b_{n_{2}}}^{s - y} {((), \binom{s - y}{b_{1}, \dots, b_{n_{2}}})}^{ν})}{G ((), p, ν, s, n_{1}, n_{2})}

where

\begin{matrix} eqnarrayleft center right \\ eqnarray-1 G (p, ν, s, n_{1}, n_{2}) = {falsefalse}_{\sum}^{y = 0} {(\frac{0}{s})}^{ν} p^{y} (1 - p \end{matrix}

…”

Section: Motivating Distributionsmentioning

confidence: 99%

“…Finally, Sellers et al () determine the pgf of the gCMB distribution to be

\begin{matrix} eqnarrayleft center right \\ eqnarray-1 φ_{Y_{*}} (t) & eqnarray-2 = & eqnarray-3 \frac{{falsefalse}_{false\sum}^{y = 0} {(\frac{0}{s})}^{ν} {(t p)}^{y} {(1 - p)}^{s - y} [{falsefalse}_{false\sum}^{a_{1}, \dots, a_{n_{1}}} {(\frac{0}{y})}^{ν}] [{falsefalse}_{false\sum}^{b_{1}, \dots, b_{n_{2}}} {(\frac{0}{s - y})}^{ν}]}{G (p, ν, s, n_{1}, n_{2})} \\ eqnarray-1 & eqnarray-2 = & eqnarray-3 \frac{H (\frac{t p}{1 - p}, ν, s, n_{1}, n_{2})}{H (\frac{p}{1 - p}, ν, s, n_{1}, n_{2})}, \end{matrix}

where

…”

Section: Motivating Distributionsmentioning

confidence: 99%

“…The following is a flexible INAR(1) model to effectively represent count data that express over‐ or under‐dispersion. Based on the sCMP and gCMB distributions respectively (as described in Section 2 with more detail available in Sellers et al ()), we use the sCMP distribution to model the marginals of the INAR(1) process as

X_{t} = C_{t} false(X_{t - 1} false) + ϵ_{t} t = 1, 2, \dots,

where (i)

ϵ_{t} \sim sCMP false(λ, ν, n_{2} false)

; and (ii)

({}, C_{t} false(• false) : t = 1, 2, \dots)

is a sequence of independent gCMB

((), \frac{1}{2}, ν, •, n_{1}, n_{2})

operators, independent of

false{ϵ_{t} false}

. …”

Section: Introducing the Scmpar(1) Modelmentioning

confidence: 99%

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A Flexible Univariate Autoregressive Time‐Series Model for Dispersed Count Data

Sellers

Peng

Arab

2019

Journal Time Series Analysis

Self Cite

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Integer‐valued time series data have an ever‐increasing presence in various applications (e.g., the number of purchases made in response to a marketing strategy, or the number of employees at a business) and need to be analyzed properly. While a Poisson autoregressive (PAR) model would seem like a natural choice to model such data, it is constrained by the equi‐dispersion assumption (i.e., that the variance and the mean equal). Hence, data that are over‐ or under‐dispersed (i.e., have the variance greater or less than the mean respectively) are improperly modeled, resulting in biased estimates and inaccurate forecasts. This work instead develops a flexible integer‐valued autoregressive model for count data that contain over‐ or under‐dispersion. Using the Conway–Maxwell–Poisson (CMP) distribution and related distributions as motivation, we develop a first‐order sum‐of‐CMP's autoregressive (SCMPAR(1)) model that will instead offer a generalizable construct that captures the PAR, and versions of what we refer to as a negative binomial AR model, and binomial AR model respectively as special cases, and serve as an overarching representation connecting these three special cases through the dispersion parameter. We illustrate the SCMPAR model's flexibility and ability to effectively model count time series data containing data dispersion through simulated and real data examples.

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Stationary count time series models

Weiß

2020

WIREs Computational Stats

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During the last 20-30 years, there was a remarkable growth in interest on approaches for stationary count time series. We consider popular classes of models for such time series, including thinning-based models, conditional regression models, and Hidden-Markov models. We review and compare important members of these model families, having regard to stochastic properties such as the dispersion and autocorrelation structure. Our survey covers univariate and multivariate count data, as well as unbounded and bounded counts. We also discuss an illustrative data example. Besides this critical presentation of the current state-of-the-art, some existing challenges and opportunities for future research are identified.

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A flexible distribution class for count data

Cited by 15 publications

References 24 publications

Oust the louse: leaping behaviour removes sea lice from wild juvenile sockeye salmon Oncorhynchus nerka

Oust the louse: leaping behaviour removes sea lice from wild juvenile sockeye salmon Oncorhynchus nerka

A Flexible Univariate Autoregressive Time‐Series Model for Dispersed Count Data

Stationary count time series models

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