A Trinomial Difference Distribution

Omair, Maha A.; Alzaid, Abdulhamid A.; Odhah, Omalsad Hamood

doi:10.15446/rce.v39n1.55131

Cited by 4 publications

(8 citation statements)

References 6 publications

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“…Proof By Proposition 1 in Omair et al (2016),

TD 1 false(n, α_{1}, α_{2} false)

can be rewritten as the difference of two independent random variables

X_{1}

and

X_{2}

, if

{false(1 - α_{1} - α_{2} false)}^{2} - 4 α_{1} α_{2} \geq 0

, where

X_{i} \sim Bin false(n, p_{i} false)

with

p_{1} = \frac{1 + α_{1} - α_{2} \pm \sqrt{{(1 - α_{1} - α_{2})}^{2} - 4 α_{1} α_{2}}}{2}, p_{2} = \frac{1 - α_{1} - α_{2} \pm \sqrt{{(1 - α_{1} - α_{2})}^{2} - 4 α_{1} α_{2}}}{2} .

Hence,

E false(X_{i}^{k} false)

exists and

E false(X_{i}^{k} false) = n false(n - 1 false) \dots false(n - k + 1 false) p_{i}^{k} = \frac{n!}{k!} p_{i}^{k}

\forall k \geq 1 .

…”

Section: Model Formulation and Stability Propertiesmentioning

confidence: 79%

“…By Proposition 2 in Omair et al (2016),

α \circ_{ν} X false| false(false| X false| = n false) \sim TD 2 false(n, α, ν false)

. Hence,

({, \begin{matrix} left \\ array E (α \circ_{ν} X | | X | = n) = n α, E ({(α \circ_{ν})}^{2} | | X | = n) = n^{2} α^{2} + n ν, \\ array Var (α \circ_{ν} X | | X | = n) = n ν, \\ array E ({(α \circ_{ν})}^{3} | | X | = n) = n {(n - 1)}^{2} α^{3} + n α (3 ν (n - 1) + 1), \\ array E ({(α \circ_{ν})}^{4} | | X | = n) = n (n - 1) [(n^{2} + n - 3) α^{4} + 3 (2 (n - 1) ν + 1) α^{2} + 3 ν^{2}] . \end{matrix})

…”

Section: Model Formulation and Stability Propertiesunclassified

“…Proposition 1. (Omair et al, 2016) If Z $ TD1ðn, α 1 , α 2 Þ, then Z can be written as the difference of two independent random variables X 1 $ Binðn, pÞ and X 2 $ Binðn, qÞ if and only if 4…”

Section: Trinomial Difference Distributionmentioning

confidence: 99%

“…The second one is that the likelihood functions for

normalℤ

‐valued AR models are cumbersome to work with such that the closed form of maximum likelihood estimator may not exist. To fill these gaps, we are inspired by the work of Omair et al (2016), and we first construct a new thinning operator (named as the trinomial difference thinning operator), which is a monumental contribution because its counting sequence retains i.i.d. and includes −1, that is, the counting sequence takes value in

false{- 1,0, 1 false}

, and the sum of its counting sequence follows a trinomial difference distribution, which makes the probability mass function (PMF) of the trinomial difference thinning operator available.…”

Section: Introductionmentioning

confidence: 99%

“…Proof. By Proposition 1 in Omair et al (2016), TD1ðn, α 1 , α 2 Þ can be rewritten as the difference of two independent random vari-…”

mentioning

confidence: 99%

See 4 more Smart Citations

A trinomial difference autoregressive model and its applications

Chen,

Zhang,

Zhu

2023

Stat

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This paper considers the autoregressive modelling problem of ‐valued time series of counts, whose counting sequence consists of −1, 0 and 1. Most existing methods are based on the signed binomial thinning operator and its some extensions, in which the counting sequence consists of 0 and 1, that is, the cases of −1 is ignored. To fill this gap, we first construct the trinomial difference thinning operator and then propose the trinomial difference Z‐valued autoregressive (TDZAR) model and give some stochastic properties. An attractive merit of the TDZAR model is that the incorporated trinomial difference thinning operator makes the conditional maximum likelihood estimate more wieldy and easy. Second, we discuss the two‐step conditional least squares estimate and the conditional maximum likelihood estimate and establish their asymptotic properties of the estimators. Third, the performances of these estimators are compared via simulation study. Finally, we apply the proposed model to a real data set.

show abstract

“…Proof By Proposition 1 in Omair et al (2016),

TD 1 false(n, α_{1}, α_{2} false)

can be rewritten as the difference of two independent random variables

X_{1}

and

X_{2}

, if

{false(1 - α_{1} - α_{2} false)}^{2} - 4 α_{1} α_{2} \geq 0

, where

X_{i} \sim Bin false(n, p_{i} false)

with

p_{1} = \frac{1 + α_{1} - α_{2} \pm \sqrt{{(1 - α_{1} - α_{2})}^{2} - 4 α_{1} α_{2}}}{2}, p_{2} = \frac{1 - α_{1} - α_{2} \pm \sqrt{{(1 - α_{1} - α_{2})}^{2} - 4 α_{1} α_{2}}}{2} .

Hence,

E false(X_{i}^{k} false)

exists and

E false(X_{i}^{k} false) = n false(n - 1 false) \dots false(n - k + 1 false) p_{i}^{k} = \frac{n!}{k!} p_{i}^{k}

\forall k \geq 1 .

…”

Section: Model Formulation and Stability Propertiesmentioning

confidence: 79%

“…By Proposition 2 in Omair et al (2016),

α \circ_{ν} X false| false(false| X false| = n false) \sim TD 2 false(n, α, ν false)

. Hence,

({, \begin{matrix} left \\ array E (α \circ_{ν} X | | X | = n) = n α, E ({(α \circ_{ν})}^{2} | | X | = n) = n^{2} α^{2} + n ν, \\ array Var (α \circ_{ν} X | | X | = n) = n ν, \\ array E ({(α \circ_{ν})}^{3} | | X | = n) = n {(n - 1)}^{2} α^{3} + n α (3 ν (n - 1) + 1), \\ array E ({(α \circ_{ν})}^{4} | | X | = n) = n (n - 1) [(n^{2} + n - 3) α^{4} + 3 (2 (n - 1) ν + 1) α^{2} + 3 ν^{2}] . \end{matrix})

…”

Section: Model Formulation and Stability Propertiesunclassified

Section: Trinomial Difference Distributionmentioning

confidence: 99%

“…The second one is that the likelihood functions for

normalℤ

false{- 1,0, 1 false}

, and the sum of its counting sequence follows a trinomial difference distribution, which makes the probability mass function (PMF) of the trinomial difference thinning operator available.…”

Section: Introductionmentioning

confidence: 99%

“…Proof. By Proposition 1 in Omair et al (2016), TD1ðn, α 1 , α 2 Þ can be rewritten as the difference of two independent random vari-…”

mentioning

confidence: 99%

See 3 more Smart Citations

A trinomial difference autoregressive model and its applications

Chen,

Zhang,

Zhu

2023

Stat

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show abstract

Bivariate Distributions on Z2

Omair

Alomani²,

Alzaid³

2022

Bull. Malays. Math. Sci. Soc.

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A trinomial difference autoregressive process for the bounded ℤ‐valued time series

Chen,

Han,

Zhu

2024

Journal Time Series Analysis

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This article tackles the modeling challenge of bounded ‐valued time series by proposing a novel trinomial difference autoregressive process. This process not only maintains the autocorrelation structure presenting in the classical binomial GARCH model, but also facilitates the analysis of bounded ‐valued time series with negative or positive correlation. We verify the stationarity and ergodicity of the couple process (comprising both the observed process and its conditional mean process) while also presenting several stochastic properties. We further discuss the conditional maximum likelihood estimation and establish their asymptotic properties. The effectiveness of these estimators is assessed through simulation studies, followed by the application of the proposed models to two real datasets.

show abstract

A Trinomial Difference Distribution

Cited by 4 publications

References 6 publications

A trinomial difference autoregressive model and its applications

A trinomial difference autoregressive model and its applications

Bivariate Distributions on Z2

A trinomial difference autoregressive process for the bounded ℤ‐valued time series

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