A Gamma Regression for Bounded Continuous Variables

Mousa, Amany; El-Sheikh, Ahmed A.; Abdelfattah, Mahmoud

doi:10.17654/as049040305

Cited by 24 publications

(19 citation statements)

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“…Let y 1 , … , y n be independent random variables such that y t , for t = 1, … , n , is unit gamma distributed with parameters

μ_{t}

(mean) and

ϕ_{t}

(precision). The unit gamma regression model proposed by Mousa et al (2016) is

\begin{align} g false(μ_{t} false) = true \sum_{i = 1}^{p} x_{t i} β_{i} = η_{1 t} and h false(ϕ_{t} false) = true \sum_{j = 1}^{q} z_{t j} γ_{j} = η_{2 t}, \end{align}

where

β = {(β_{1}, \dots, β_{p})}^{⊤} \in ℝ^{p}

and

γ = {(γ_{1}, \dots, γ_{q})}^{⊤} \in ℝ^{q}

are unknown parameter vectors ( p + q < n ), x t 1 , … , x tp and z t 1 , … , z tq are observations on the mean and precision covariates, respectively. Also,

g : (0, 1) \mapsto ℝ

and

h : ℝ_{+} \mapsto ℝ

are strictly monotone and twice‐differentiable link functions.…”

Section: Unit Gamma Modelmentioning

confidence: 99%

“…Regression analysis is used to model the dependence of a variable of interest (known as dependent variable or response) on a set of other variables (known as independent variables, covariates or regressors). The unit gamma regression model was proposed by Mousa et al (2016) based on the unit gamma distribution (Grassia, 1977). It is useful for regression analyses in which the response assumes values in the standard unit interval, (0, 1), such as rates, proportions and concentration indices.…”

Section: Introductionmentioning

confidence: 99%

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Residual and local influence analyses for unit gamma regressions

Rocha

Espinheira

Cribari‐Neto

2020

Statistica Neerlandica

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We obtain local influence measures and residuals for the unit gamma regression model. In particular, we introduce four residuals that are based on Fisher's iterative scoring parameter estimation algorithm and develop local influence analysis based on several different perturbation schemes: cases weighting, response additive perturbation, and covariate(s) additive perturbation. An empirical application in which variables related to education and investment in research and development are used to explain the proportion of nonpoor people in a set of countries is presented and discussed. Residual and local influence analyses show that the unit gamma regression model yields a good fit to the data, even outperforming the beta regression model. The diagnostic analysis singles out countries whose data are worthy of further investigation. Our results reveal that lower poverty levels are associated with higher shares of investment in high technology. The statistical significance of such a relationship is not sensitive to atypical data points.

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“…Let y 1 , … , y n be independent random variables such that y t , for t = 1, … , n , is unit gamma distributed with parameters

μ_{t}

(mean) and

ϕ_{t}

(precision). The unit gamma regression model proposed by Mousa et al (2016) is

\begin{align} g false(μ_{t} false) = true \sum_{i = 1}^{p} x_{t i} β_{i} = η_{1 t} and h false(ϕ_{t} false) = true \sum_{j = 1}^{q} z_{t j} γ_{j} = η_{2 t}, \end{align}

where

β = {(β_{1}, \dots, β_{p})}^{⊤} \in ℝ^{p}

and

γ = {(γ_{1}, \dots, γ_{q})}^{⊤} \in ℝ^{q}

are unknown parameter vectors ( p + q < n ), x t 1 , … , x tp and z t 1 , … , z tq are observations on the mean and precision covariates, respectively. Also,

g : (0, 1) \mapsto ℝ

and

h : ℝ_{+} \mapsto ℝ

are strictly monotone and twice‐differentiable link functions.…”

Section: Unit Gamma Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Residual and local influence analyses for unit gamma regressions

Rocha

Espinheira

Cribari‐Neto

2020

Statistica Neerlandica

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show abstract

“…Mousa et al considered a new parameterization for the Unit Gamma distribution. Let

θ = \frac{μ^{1 false/ τ}}{false(1 - μ^{1 false/ τ} false)}

, from this new parameterization, the Unit Gamma density can be written as

f false(y false| μ, τ false) = \frac{{([], \frac{μ^{1 false/ τ}}{1 - μ^{1 false/ τ}})}^{τ}}{normalΓ false(τ false)} y^{\frac{μ^{1 false/ τ}}{1 - μ^{1 false/ τ}} - 1} {([], \log ((), \frac{1}{y}))}^{τ - 1}, 0 < y < 1,

where 0 < μ < 1 and τ > 0.…”

Section: Some Distributions For Modelling Rates and Proportionsmentioning

confidence: 99%

“…Additionally, it is evident that the Beta and Unit gamma distributions presents the same fit to the current data. In Figure 3, we present the control chart for the complete set of the available data (Phase I: 1-20; Phase II: [21][22][23][24][25][26][27][28][29][30][31][32][33][34] using error type I = 0.0027 to yield ARL 0 = 370. The control chart in Figure 3 does not trigger an alarm during Phase I, so we obtain no contradiction against the models.…”

Section: Application To a Real Data Setmentioning

confidence: 99%

Control charts to monitor rates and proportions

Fernandes

Bourguignon

2018

Quality & Reliability Eng

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In this paper, we call attention for monitoring proportions and rates when they are not results of Bernoulli experiments. To deal with this problem, usually control charts based on Beta distributions are built as this distribution is a well‐known one. However, in practice, there are other distributions to be considered as Simplex and Unit Gamma distributions. The impact of the speed to signal a shift in the average proportion in terms of out‐of‐control average run length is measured when the control limits determined under a Beta distribution is equivocally used to monitor individual rates from Simplex or Unit Gamma distributions or vice versa.

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“…Ratnaparkhl and Mosimann (1990) studied the logarithmic and Tukey's lambda-type transformation on the unit-Gamma distribution. More recently, Mousa et al (2016) formulated the UG regression model while Mazucheli et al (2018) derived second order bias corrections for the parameters of UG distribution. Ho et al (2019) considered the UG distribution to construct control charts to monitor rates and proportions.…”

Section: Introductionmentioning

confidence: 99%

Comparison of estimation methods for unit-Gamma distribution

Dey¹,

Menezes²,

Mazucheli³

2021

Journal of Data Science

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In this study we have considered different methods of estimation of the unknown parameters of a two-parameter unit-Gamma (UG) distribution from the frequentists point of view. First, we briefly describe different frequentists approaches: maximum likelihood estimators, moments estimators, least squares estimators, maximum product of spacings estimators, method of Cramer-von-Mises, methods of Anderson-Darling and four variants of Anderson-Darling test and compare them using extensive numerical simulations. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. The performances of the estimators have been compared in terms of their bias and root mean squared error using simulated samples. Also, for each method of estimation, we consider the interval estimation using the bootstrap method and calculate the coverage probability and the average width of the bootstrap confidence intervals. The study reveals that the maximum product of spacing estimators and Anderson-Darling 2 (AD2) estimators are highly competitive with the maximum likelihood estimators in small and large samples. Finally, two real data sets have been analyzed for illustrative purposes.

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A Gamma Regression for Bounded Continuous Variables

Cited by 24 publications

References 0 publications

Residual and local influence analyses for unit gamma regressions

Residual and local influence analyses for unit gamma regressions

Control charts to monitor rates and proportions

Comparison of estimation methods for unit-Gamma distribution

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