Bayesian Modelling of Integer Data Using the Generalised Poisson Difference Distribution

Shahtahmassebi, Golnaz; Moyeed, Rana

doi:10.5539/ijsp.v3n1p35

Cited by 7 publications

(11 citation statements)

References 29 publications

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“…Let us assume X and Y are two non‐negative integer random variables. The random variable Z = X − Y follows a GPDD (SHAHTAHMASSEBI and MOYEED, ) if its probability density function has the following form

f_{normalGPDD} (Z = X - Y = z) = e^{- λ_{1} - λ_{2} - θ_{1} z} \sum_{y = 0}^{\infty} {((), λ_{1}, θ_{1})}_{z + y} {((), λ_{2}, θ_{2})}_{y} e^{- (θ_{1} + θ_{2}) y},

for any value of

z \in double-struckZ

, where

{(λ, θ)}_{x} = \frac{λ {(λ + xθ)}^{x - 1}}{x!} .

Lower limits for θ 1 and θ 2 have been set to ensure that there are at least five classes of non‐zero probabilities at both tails when θ 1 <0 or θ 2 <0:

\begin{matrix} \max (- 1, - λ_{1} / m_{1}) & < θ_{1} < 1, \\ \max (- 1, - λ_{2} / m_{2}) & < θ_{2} < 1, \end{matrix}

where m 1 , m 2 ≥4 are the largest positive integers in which λ 1 + m 1 θ 1 >0 and λ 2 + m 2 θ 2 >0. Therefore, for any z > m 1 when θ 1 <0, or z <− m 2 when θ 2 <0,

f_{GPDD} (z | λ_{1 <...>})

…”

Section: Generalized Poisson Difference Distributionmentioning

confidence: 99%

“…The cumulants of the probability distribution of the random variable Z can be derived using the following recurrence relation ((SHAHTAHMASSEBI and MOYEED, ))

\begin{matrix} (1 - θ_{1}) (1 - θ_{2}) L_{k + 1} & = (2 - θ_{2}) λ_{1} θ_{1} \frac{\partial^{2} L_{k}}{\partial θ_{1} \partial λ_{1}} + (2 - θ_{2}) λ_{1} \frac{\partial L_{k}}{\partial λ_{1}} + θ_{1} θ_{2} λ_{2} \frac{\partial^{2} L_{k}}{\partial θ_{2} \partial λ_{2}} \\ + θ_{1} λ_{2} \frac{\partial L_{k}}{\partial λ_{2}} - θ_{1} \frac{\partial L_{k}}{\partial θ_{1}} - θ_{2} \frac{\partial L_{k}}{\partial θ_{2}} - L_{k}, \end{matrix}

and the expression for the first four cumulants of the GPDD are obtained as follows:

L_{1} = \frac{λ_{1}}{((), 1 - θ_{1})} - \frac{λ_{2}}{((), 1 - θ_{2})},

…”

Section: Generalized Poisson Difference Distributionmentioning

confidence: 99%

“…That is when θ 1 >0 and/or θ 2 >0, the GPDD will have a longer left and/or right tail than a standard PD distribution. On the other hand, when θ 1 <0 and/or θ 2 <0, the GPDD will have a shorter left and/or right tail than a standard PD distribution, a property that had not been addressed by many alternative distributions to the PD distribution (for further details see (SHAHTAHMASSEBI and MOYEED, )).…”

Section: Generalized Poisson Difference Distributionmentioning

confidence: 99%

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An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores

Shahtahmassebi

Moyeed

2016

Statistica Neerlandica

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The analysis of sports data, in particular football match outcomes, has always produced an immense interest among the statisticians. In this paper, we adopt the generalized Poisson difference distribution (GPDD) to model the goal difference of football matches. We discuss the advantages of the proposed model over the Poisson difference (PD) model, which was also used for the same purpose. The GPDD model, like the PD model, is based on the goal difference in each game that allows us to account for the correlation without explicitly modelling it. The main advantage of the GPDD model is its flexibility in the tails by considering shorter as well as longer tails than the PD distribution. We carry out the analysis in a Bayesian framework in order to incorporate external information, such as historical knowledge or data, through the prior distributions. We model both the mean and the variance of the goal difference and show that such a model performs considerably better than a model with a fixed variance. Finally, the proposed model is fitted to the 2012-2013 Italian Serie A football data, and various model diagnostics are carried out to evaluate the performance of the model.

show abstract

f_{normalGPDD} (Z = X - Y = z) = e^{- λ_{1} - λ_{2} - θ_{1} z} \sum_{y = 0}^{\infty} {((), λ_{1}, θ_{1})}_{z + y} {((), λ_{2}, θ_{2})}_{y} e^{- (θ_{1} + θ_{2}) y},

for any value of

z \in double-struckZ

, where

{(λ, θ)}_{x} = \frac{λ {(λ + xθ)}^{x - 1}}{x!} .

Lower limits for θ 1 and θ 2 have been set to ensure that there are at least five classes of non‐zero probabilities at both tails when θ 1 <0 or θ 2 <0:

\begin{matrix} \max (- 1, - λ_{1} / m_{1}) & < θ_{1} < 1, \\ \max (- 1, - λ_{2} / m_{2}) & < θ_{2} < 1, \end{matrix}

where m 1 , m 2 ≥4 are the largest positive integers in which λ 1 + m 1 θ 1 >0 and λ 2 + m 2 θ 2 >0. Therefore, for any z > m 1 when θ 1 <0, or z <− m 2 when θ 2 <0,

f_{GPDD} (z | λ_{1 <...>})

…”

Section: Generalized Poisson Difference Distributionmentioning

confidence: 99%

“…The cumulants of the probability distribution of the random variable Z can be derived using the following recurrence relation ((SHAHTAHMASSEBI and MOYEED, ))

\begin{matrix} (1 - θ_{1}) (1 - θ_{2}) L_{k + 1} & = (2 - θ_{2}) λ_{1} θ_{1} \frac{\partial^{2} L_{k}}{\partial θ_{1} \partial λ_{1}} + (2 - θ_{2}) λ_{1} \frac{\partial L_{k}}{\partial λ_{1}} + θ_{1} θ_{2} λ_{2} \frac{\partial^{2} L_{k}}{\partial θ_{2} \partial λ_{2}} \\ + θ_{1} λ_{2} \frac{\partial L_{k}}{\partial λ_{2}} - θ_{1} \frac{\partial L_{k}}{\partial θ_{1}} - θ_{2} \frac{\partial L_{k}}{\partial θ_{2}} - L_{k}, \end{matrix}

and the expression for the first four cumulants of the GPDD are obtained as follows:

L_{1} = \frac{λ_{1}}{((), 1 - θ_{1})} - \frac{λ_{2}}{((), 1 - θ_{2})},

…”

Section: Generalized Poisson Difference Distributionmentioning

confidence: 99%

Section: Generalized Poisson Difference Distributionmentioning

confidence: 99%

See 1 more Smart Citation

An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores

Shahtahmassebi

Moyeed

2016

Statistica Neerlandica

Self Cite

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show abstract

“…However, a major drawback of these models in our current context is their lack of flexibility to incorporate missing observations and to allow for a time-varying variance process. Most related to our work is the contribution by Shahtahmassebi (2011) and Shahtahmassebi and Moyeed (2014) who adopted the Skellam distribution to analyze time series data in Z within a Bayesian framework, whereas we use simulated maximum likelihood methods. However, their work does not treat the specific features of intraday financial price changes such as intraday seasonality, long stretches of missing values, and the time-varying modifications for the Skellam distribution.…”

Section: Introductionmentioning

confidence: 99%

Intraday Stochastic Volatility in Discrete Price Changes: The Dynamic Skellam Model

Koopman

Lit

Lucas

2017

Journal of the American Statistical Association

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We study intraday stochastic volatility for four liquid stocks traded on the New York Stock Exchange using a new dynamic Skellam model for high-frequency tick-by-tick discrete price changes. Since the likelihood function is analytically intractable, we rely on numerical methods for its evaluation. Given the high number of observations per series per day (1000 to 10,000), we adopt computationally efficient methods including Monte Carlo integration. The intraday dynamics of volatility and the high number of trades without price impact require nontrivial adjustments to the basic dynamic Skellam model. In-sample residual diagnostics and goodness-of-fit statistics show that the final model provides a good fit to the data. An extensive dayto-day forecasting study of intraday volatility shows that the dynamic modified Skellam model provides accurate forecasts compared to alternative modeling approaches. Supplementary materials for this article are available online.

show abstract

“…However, a major drawback of these models in our current context is their lack of flexibility to incorporate missing observations and to allow for a time-varying variance process. Most related to our work is the contribution of Shahtahmassebi (2011) and Shahtahmassebi and Moyeed (2014) who adopt the Skellam distribution to analyze time series data in Z within a Bayesian framework, whereas we use simulated maximum likelihood methods. However, their work does not treat the specific features of intraday financial price changes such as intraday seasonality, long stretches of missing values, and the time-varying modifications for the Skellam distribution.…”

Section: Introductionmentioning

confidence: 99%

The Dynamic Skellam Model with Applications

Koopman

Lit²

2014

SSRN Journal

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Bayesian Modelling of Integer Data Using the Generalised Poisson Difference Distribution

Cited by 7 publications

References 29 publications

An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores

An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores

Intraday Stochastic Volatility in Discrete Price Changes: The Dynamic Skellam Model

The Dynamic Skellam Model with Applications

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