Corrigendum: Degenerate Beta autoregressive model for proportion time-series with zeros or ones: an application to antimicrobial resistance rate using R shiny app

A corrigendum on
Degenerate Beta autoregressive model for proportion time-series with zeros or ones: an application to antimicrobial resistance rate using R shiny app

by Lobo, J., Kamath, A., and Kalwaje Eshwara, V. (2023). Front. Public Health 10:969777. doi: 10.3389/fpubh.2022.969777

In the published article reference 14 was not cited in the article and an additional citation for reference 11 was missed. The citations have now been inserted in Material and Methods, Degenerate Beta Autoregressive (DeβAR) model, Parameter estimation and should read:

“Here, let ${\text{x}}_t^{*}=log(\frac{x_t}{1-x_t})$ if x_tϵ(0, 1) else ${\text{x}}_t^{*}=0$ (11, 14) and $\psi ({\mu }_t\zeta )-\psi ((1-{\mu }_t)\zeta )={\mu }_t^{*}$, where ψ(.) is a digamma function.”

In the published article, there was an error. Inbetween steps of likelihood derivation was missed.

A correction has been made to Material and Methods, Degenerate Beta Autoregressive (DeβAR) model, Parameter estimation. “This sentence previously stated:”

$L(\theta ;x)={\displaystyle \prod _{t=p+1}^n}{f}_{X_t}(x_t|F_{t-1})$

The likelihood function for the parameters of Degenerate Beta AR model is given by,

$L(\theta ;x)={\displaystyle \prod _{t=p+1}^n}\left\{\omega I_{x_t=c}+I_{x_tϵ(0,1)}(1-\omega )\frac{\Gamma (\zeta )}{\Gamma ({\mu }_t\zeta )\Gamma ((1-{\mu }_t)\zeta )}{x}_t^{{\mu }_t\zeta -1}{(1-x_t)}^{(1-{\mu }_t)\zeta -1}\right\}$

“The corrected sentence appears below:”

$\begin{array}{l}L(\theta ;x_t)={\displaystyle \prod _{t=p+1}^n}{f}_{X_t}(x_t|F_{t-1})\end{array}$

$\begin{array}{l}L(\theta ;x_t)={\displaystyle \prod _{t=p+1}^n}b{i}_c(z_t;\omega ,{\mu }_t,\zeta )=L_1(\omega )L_2({\mu }_t,\zeta )\end{array}$

where,

$\begin{array}{l}{L}_1(\omega )={\displaystyle \prod _{t=p+1}^n}{\omega }^{I_{x_t=c}}{(1-\omega )}^{I_{x_tϵ(0,1)}}\end{array}$

$\begin{array}{r}{L}_2({\mu }_t,\zeta )={\displaystyle \prod _{t=p+1}^n\{}\frac{\Gamma (\zeta )}{\Gamma ({\mu }_t\zeta )\Gamma ((1-{\mu }_t)\zeta )}{x}_t^{{\mu }_t\zeta -1}(1-x_t)\\ {}^{(1-{\mu }_t)\zeta -1}{\}}^{{ℐ}_{x_tϵ(0,1)}}\end{array}$

The likelihood function for the parameters of Degenerate Beta AR model is given by,

$\begin{array}{r}L(\theta ;x_t)={\displaystyle \prod _{t=p+1}^n\{\omega {ℐ}_{x_t=c}+{ℐ}_{x_tϵ(0,1)}(}1-\omega )\\ \frac{\Gamma (\zeta )}{\Gamma ({\mu }_t\zeta )\Gamma ((1-{\mu }_t)\zeta )}{x}_t^{{\mu }_t\zeta -1}(1-x_t{)}^{(1-{\mu }_t)\zeta -1}\}\end{array}$

In the published article, there was an error. Limitation of the model has been added and reference 19 has been added.

A correction has been made to Material and Methods, Degenerate Beta Autoregressive (DeβAR) model, Parameter estimation. “This sentence previously stated:”

Large sample inference: If the model specified by Equation (5) follows the regularity condition of maximum likelihood estimation (MLE) then, MLEs of θ and J(θ) (Fisher information matrix) are consistent. Assuming that $I(\theta )=\underset{n\to \infty }{lim}\left\{n^{-1}J(\theta )\right\}$ exists and is non-singular, we have $\sqrt{n}(\widehat{\theta }-\theta )$ converges in distribution to N(0, I(θ)⁻¹).

“The corrected sentence appears below:”

Large sample inference: If the model specified by Equation (5) follows the regularity condition of maximum likelihood estimation (MLE) then, MLE of θ and J(θ) (Fisher information matrix) are consistent. Assuming that $I(\theta )=\underset{n\to \infty }{lim}\left\{n^{-1}J(\theta )\right\}$ exists and is nonsingular, we have $\sqrt{n}(\widehat{\theta }-\theta )$ converges in distribution to N(0, I(θ)⁻¹).

Note: The proposed DeβAR model is applicable when $x_t^{*}$ is converted to 0 as mentioned above. To overcome with this limitation Bayer et al. (19) proposed Inflated beta autoregressive moving average models, which are more suitable when interval data includes 0 or 1.

The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.

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References

11. Ospina R, Ferrari SL. A general class of zero-or-one inflated beta regression models. Comput Stat Data Anal. (2012) 56:1609–23. doi: 10.1016/j.csda.2011.10.005

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14. Benjamin MA, Rigby RA, Stasinopoulos MD. Fitting non-Gaussian time series models. In: InCOMPSTAT: Proceedings in Computational Statistics 13th Symposium held in Bristol, Great Britain, 1998. (1998). p. 191–6.

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19. Bayer FM, Pumi G, Pereira TL, Souza TC. Inflated beta autoregressive moving average models. Comput Appl Math. (2023) 42:183. doi: 10.1007/s40314-023-02322-w

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