# A Study on Computational Algorithms in the Estimation of Parameters for a Class of Beta Regression Models

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## Abstract

**:**

`optim`function of

`R`, we study sets of parameters that are hard to be estimated. We detect that this function fails in most cases, but when it is successful, it is more accurate and faster than the others. The annealing algorithm obtains satisfactory estimates in viable time with few failures so that we recommend its use when the

`optim`function fails.

## 1. Introduction

`R`software (http://www.r-project.org accessed on 6 January 2022) allows the beta regression to be applied [15]. This is the

`betareg`package, whose details can be found in [16].

`DIRECT`,

`DIRECT_L`, evolutionary, genetic, memetic, particle swarm, self-adapted evolutionary, and simulated annealing methods. The performance of these algorithms is evaluated by using the Monte Carlo simulation method with the

`R`software [20], considering the quality and computational time of the solutions found and analyzing the behavior in different scenarios. The codes and simulation results are available in the following repository: https://github.com/Raydonal/MLE-BetaRegression-Optimization (accessed on 6 January 2022).

`optim`function of the

`betareg`package of

`R`for these algorithms. We present the conclusions of our study in Section 4.

## 2. Methodology

#### 2.1. Beta Models

#### 2.2. Optimization Algorithms

`optim`of

`R`from the

`betareg`package.

`R`language applied to 48 different optimization problems with known solutions. In our case and during the first step, we carry out a simulation study considering only the problem of maximizing the log-likelihood function defined in (6), with 10 different methods available in

`R`packages. In the second stage, we use the methods that provided better results in the previous stage and the

`optim`function implemented in the

`betareg`package, in order to make a comparison. The methods utilized are defined as follows:

- Genetic algorithm: This is a heuristic inspired by the basic principle of biological evolution and natural selection, simulating evolution so that the fittest individuals survive, imitating its mechanisms such as the processes of selection, crossing, and mutation. The
`ga`function that implements this algorithm is available in an`R`package named`GA`[44]. - Differential evolutionary algorithm: This method is similar to the genetic algorithm indicated to find the global optimum of real-valued functions with real-valued parameters as well [45]. Such an algorithm does not need the function to be optimized that is continuous or differentiable and is available by the
`DEoptim`command of an`R`package named`DEoptim`[46]. - Self-adapted evolutionary algorithm: This method proposed in [47] is a strategy of self-adaptation of the covariance matrix that is implemented in the
`cma_es`function of the`cmaes`package of`R`. This is also an evolutionary method, which uses a covariance matrix approximation to be more efficient in the generation of next generations. - Simulated annealing algorithm: This is a metaheuristic based on the thermodynamic annealing process that performs a probabilistic local search replacing the current solution with a solution in its vicinity, obtaining good solutions regardless of the chosen starting point. The
`GenSA`function is available in an`R`package named`GenSA`[48]. A general strategy to improve the simulated annealing (SANN) algorithm is to inject noise via the Markov chain Monte Carlo algorithms (noise-boosted) to sample high probability regions of the search space and to accept solutions that increase the search breadth [49,50,51]. Another approach is based on hybridized search gradient methods or genetic algorithms for cases of difficulties in the convergence with SANN [52,53]. - Controlled random search algorithm: This is a direct search heuristic [54] that tries to balance the fulfillment of constraints and convergence by storing possible trial points by the
`nloptr_crs`function. Such a function has implemented this algorithm and is available in an`R`package named`nloptr`[55]. `DIRECT`algorithm: This is a deterministic method based on the division of the search space into increasingly smaller hyperrectangles and was proposed in [56]. The`nloptr_d`function has implemented such an algorithm and is available in the`nloptr`package of`R`.`DIRECT_L`algorithm: This is a variation of`DIRECT`containing certain randomness and was proposed in [57]. The`nloptr_d_l`function is available in the`nloptr`package.- Evolutionary algorithm: A common practice in evolutionary methods is to apply a penalty function to bias the search for viable solutions. This method is a strategy improved by stochastic ranking that proposes a way to eliminate subjectivity in the configuration of penalty parameters and was proposed in [58]. The
`nloptr_i`function implements this algorithm and is available in the`nloptr`package of`R`. - Memetic algorithm: This technique does a search locally combining the evolutionary method with a local search algorithm. The
`alschains`function has implemented this algorithm and is available in the`malschains`package of`R`[59]. - Particle swarm algorithm: This is a heuristic that considers the movement of a swarm of particles through the search space, using formulas for position and velocity that depend on the state of other particles. The
`PSopt`function that implements this algorithm is available in an`R`package named`NMOF`[60,61,62,63]. - Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm: This is a quasi-Newton optimization method that applies analytical gradients for parameter estimation and uses as initial guesses values obtained through an auxiliary linear regression of the transformed response. The BFGS algorithm is employed in the
`optim`function of the`betareg`package [16].

## 3. Results and Discussion

#### 3.1. First Stage of the Simulation

`nloptr_d`and

`nloptr_d_l`methods, the other algorithms obtained similar results to each other.

`malschains`and

`cma_es`methods. In the second set, the

`nloptr_crs`and

`malschains`methods stand out from the rest. In the third set, there is a certain heterogeneity in the estimates, whereas for the fourth set, the

`malschains`method stands out again, with the

`nloptr_d`and

`nloptr_d_l`methods being quite different from the others. Although the

`malschains`method seems to perform well in some sets, it has some inconsistency with 33,900 failed executions, which are about $42\%$.

`nloptr_d`and

`nloptr_d_l`methods report large variations in the results, which is evident in the graphical plot of Set 3. Note that the

`ga`and

`cma_es`methods present a large number of outliers in most sets. The other methods seem to behave similarly to each other, obtaining estimates close to the expected value.

`nloptr_d`and

`nloptr_d_l`methods present large variations in the results and the

`ga`and

`cma_es`methods show a large number of outliers, while the rest of the methods obtained similar results, approaching the blue line. Again, an approximation of the estimates in relation to the blue line is observed as the sample size increases.

`ga`algorithm is the slowest one, taking more than 1.5 s per run. The

`nloptr_crs`method is much faster than all the others. The other methods have a similar speed performance to each other.

`cma_es`method, which was much slower when dealing with estimates for $\varphi =148$.

`DEoptim`,

`GenSA`,

`nloptr_i`, and

`nloptr_crs`methods seem to be the most suitable, as the

`malschains`,

`nloptr_d`,

`nloptr_d_l`,

`PSopt`,

`ga`, and

`cma_es`methods show some inconsistencies in the parameter estimation.

#### 3.2. Second Stage of the Simulation

`DE`,

`SA`,

`isres`,

`crs`) and the

`optim`function currently utilized in the

`betareg`package. Thus, it is possible to compare how the heuristics behave in relation to the currently most employed method, that is, the

`optim`function of the

`betareg`package. Furthermore, sets are investigated in which the methods are expected to have greater difficulty in estimating the parameters, diversifying the distribution of the covariate to the cases: Uniform(0,1); Normal(0,1); and Student-t(3). Note that the value of the precision parameter $\varphi $ drastically decreases as the variance increases. The configuration used for the generated samples is found in Table 3. Note that, in Table 4, for Sets 5 and 6, only the

`DEoptim`and

`GenSA`methods can consistently estimate the parameters based on the number of failures per method.

`DE`and

`SA`methods reported successes, whereas for Set 6, the

`isres`,

`crs`, and

`optim`methods had few successes. This result is consistent with the number of failures shown in Table 4. Furthermore, the estimates performed with the

`DE`and

`SA`methods are similar. For the same sets, in Figure 9, the box plots of the estimates of ${\beta}_{1}$ are displayed, where similar behavior is detected, with the notable difference that, in this case, the

`optim`function provides more heterogeneous results in the case of samples with small sizes. The estimates of $\varphi $ are shown in Figure 10, where a similar behavior is observed in relation to the estimates of ${\beta}_{0}$ and ${\beta}_{1}$.

`crs`and

`isres`methods failed in all attempts, while the

`optim`function failed in most executions. In addition, the

`DE`and

`SA`methods had few flaws/failures, being the most consistent algorithms. The estimates of ${\beta}_{0}$ shown in Figure 11 report that the behavior remains similar between the

`DE`and

`SA`methods. Note that the

`optim`function did not have any success in estimating the parameters for samples of size $n=120$. The same can be seen in the estimates of ${\beta}_{1}$ in Figure 12 and in the estimates of $\varphi $ in Figure 13.

`crs`and

`isres`methods are unable to provide estimates. In this case, the

`optim`function had a lower number of failures, although it is still a considerably significant number. The

`DE`method follows without failing, while the

`SA`method reports some few flaws. Figure 14, Figure 15 and Figure 16 show the estimates of ${\beta}_{0}$, ${\beta}_{1}$, and $\varphi $, respectively. In this case, the

`optim`function did not obtain estimates for Set 9. Furthermore, a similar behavior between the

`DE`and

`SA`methods is confirmed. Note that the estimates of ${\beta}_{1}$ were not very accurate for Set 9.

`optim`function is quite efficient when it comes to estimating the parameters. Although the

`DE`and

`SA`methods have obtained close estimates, the execution time of the

`SA`method is much smaller, so that its use is indicated in the studied cases. For Sets 7 and 8, as previously shown in Figure 18, it is confirmed that the execution time of the

`SA`method is less than for the

`DE`method, despite the similarity in the estimates. For Sets 9 and 10, as for the execution times shown in Figure 19, the previous behavior is once again detected, with the

`SA`method being more efficient than the

`DE`method.

## 4. Conclusions

`DIRECT`,

`DIRECT_L`, memetic with local search strings, and particle swarm algorithms. Four other methods provided satisfactory results, that is, the differential evolutionary, simulated annealing, controlled random search, and evolutionary with improved stochastic ranking algorithms. All of them obtained similar and satisfactory results in relation to the estimates in the evaluated scenarios. However, among them, the controlled random search algorithm had a superior computational speed performance in relation to the others. In the second stage of the simulations, sets of parameters were used to make the estimation process more difficult. In this case, we observed that the controlled random search and evolutionary with improved stochastic ranking algorithms presented a very large number of failures, in some cases not achieving any success. The

`optim`function used in the

`betareg`package of

`R`had many failures, but in cases where it was successful, it outperformed the other methods both in terms of accuracy of estimation and speed. The differential evolutionary and simulated annealing algorithms provided satisfactory estimates with few failures, with the simulated annealing algorithm being more efficient in terms of computational time.

`optim`function, implemented in the

`betareg`package to maximize the log-likelihood function of the beta regression model and to estimate its parameters, is accurate and efficient for most of the cases. Nevertheless, in some scenarios reported in our study, it presents some difficulties in estimating such parameters. In these scenarios, we recommend the use of the simulated annealing algorithm, which for the cases studied in this work showed greater consistency, providing satisfactory estimates in viable computational time.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Berggren, N.; Daunfeldt, S.O.; Hellström, J. Social trust and central-bank independence. Eur. J. Political Econ.
**2014**, 34, 425–439. [Google Scholar] [CrossRef][Green Version] - Buntaine, M.T. Does the Asian development bank respond to past environmental performance when allocating environmentally risky financing? World Dev.
**2011**, 39, 336–350. [Google Scholar] [CrossRef] - Castellani, M.; Pattitoni, P.; Scorcu, A.E. Visual artist price heterogeneity. Econ. Bus. Lett.
**2012**, 1, 16–22. [Google Scholar] [CrossRef][Green Version] - De Paola, M.; Scoppa, V.; Lombardo, R. Can gender quotas break down negative stereotypes? Evidence from changes in electoral rules. J. Public Econ.
**2010**, 94, 344–353. [Google Scholar] [CrossRef] - Huang, X.; Oosterlee, C. Generalized beta regression models for random loss-given-default. J. Credit. Risk
**2011**, 7, 1–27. [Google Scholar] [CrossRef][Green Version] - Figueroa-Zúniga, J.; Bayes, C.L.; Leiva, V.; Liu, S. Robust beta regression modeling with errors-in-variables: A Bayesian approach and numerical applications. Stat. Pap.
**2022**. [Google Scholar] [CrossRef] - Martinez-Florez, G.; Leiva, V.; Gomez-Deniz, E.; Marchant, C. A family of skew-normal distributions for modeling proportions and rates with zeros/ones excess. Symmetry
**2020**, 12, 1439. [Google Scholar] [CrossRef] - Mazucheli, J.; Bapat, S.R.; Menezes, A.F.B. A new one-parameter unit Lindley distribution. Chil. J. Stat.
**2019**, 11, 53–67. [Google Scholar] - Huerta, M.; Leiva, V.; Lillo, C.; Rodriguez, M. A beta partial least squares regression model: Diagnostics and application to mining industry data. Appl. Stoch. Model. Bus. Ind.
**2018**, 34, 305–321. [Google Scholar] [CrossRef] - Figueroa-Zúniga, J.; Niklitschek, S.; Leiva, V.; Liu, S. Modeling heavy-tailed bounded data by the trapezoidal beta distribution with applications. Revstat, 2022, in press.
- Smithson, M.; Verkuilen, J. A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychol. Methods
**2006**, 11, 54–71. [Google Scholar] [CrossRef][Green Version] - Cribari-Neto, F.; Souza, T.C. Religious belief and intelligence: Worldwide evidence. Intelligence
**2013**, 41, 482–489. [Google Scholar] [CrossRef] - Souza, T.C.; Cribari-Neto, F. Intelligence and religious disbelief in the united states. Intelligence
**2018**, 68, 48–57. [Google Scholar] [CrossRef] - Mazucheli, M.; Leiva, V.; Alves, B.; Menezes, A.F.B. A new quantile regression for modeling bounded data under a unit Birnbaum–Saunders distribution with applications in medicine and politics. Symmetry
**2021**, 13, 682. [Google Scholar] [CrossRef] - Ferrari, S.; Cribari-Neto, F. Beta regression or modeling rates and proportions. J. Appl. Stat.
**2004**, 31, 799–815. [Google Scholar] [CrossRef] - Cribari-Neto, F.; Zeileis, A. Beta regression in R. J. Stat. Softw.
**2010**, 34, 1–24. [Google Scholar] [CrossRef][Green Version] - Paolino, P. Maximum likelihood estimation of models with beta-distributed dependent variables. Political Anal.
**2001**, 9, 325–346. [Google Scholar] [CrossRef] - Cribari-Neto, F.; Vasconcellos, K. Nearly unbiased maximum likelihood estimation for the beta distribution. J. Stat. Comput. Simul.
**2002**, 72, 107–118. [Google Scholar] [CrossRef] - Ospina, R.; Cribari-Neto, F.; Vasconcellos, K. Improved point and interval estimation for a beta regression model. Comput. Stat. Data Anal.
**2006**, 51, 960–981. [Google Scholar] [CrossRef] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing, Vienna, Austria. 2021. Available online: https://www.R-project.org (accessed on 6 January 2022).
- Dunn, P.K.; Smyth, G.K. Generalized Linear Models with Examples in R; Springer: New York, NY, USA, 2018. [Google Scholar]
- McCullagh, P.; Nelder, J.A. Generalized Linear Models; Chapman and Hall: London, UK, 1989. [Google Scholar]
- Hilbe, J. Logistic Regression Models; Chapman and Hall: New York, NY, USA, 2009. [Google Scholar]
- McCullagh, P. Tensor Methods in Statistics; Chapman and Hall: London, UK, 2018. [Google Scholar]
- Rydlewski, J.; Mielczarek, D. On the maximum likelihood estimator in the generalized beta regression model. Opusc. Math.
**2012**, 32, 761–774. [Google Scholar] [CrossRef] - Simas, A.; Barreto-Souza, W.; Rocha, A.V. Improved estimators for a general class of beta regression models. Comput. Stat. Data Anal.
**2010**, 54, 348–366. [Google Scholar] [CrossRef][Green Version] - Rustagi, J. Optimization Techniques in Statistics; Elsevier: Amsterdam, The Netherlands, 2014. [Google Scholar]
- Kosmidis, I.; Firth, D. A generic algorithm for reducing bias in parametric estimation. Electron. J. Stat.
**2010**, 4, 1097–1112. [Google Scholar] [CrossRef] - Espinheira, P.; Silva, L.; Silva, A.; Ospina, R. Model selection criteria on beta regression for machine learning. Mach. Learn. Knowl. Extr.
**2019**, 1, 427–449. [Google Scholar] [CrossRef][Green Version] - Grün, B.; Kosmidis, I.; Zeileis, A. Extended beta regression in R: Shaken, stirred, mixed, and partitioned. J. Stat. Softw.
**2012**, 48, 1–25. [Google Scholar] [CrossRef][Green Version] - Rocha, A.; Simas, A.B. Influence diagnostics in a general class of beta regression models. TEST
**2011**, 20, 95–119. [Google Scholar] [CrossRef] - Billio, M.; Casarin, R. Beta autoregressive transition Markov-switching models for business cycle analysis. Stud. Nonlinear Dyn. Econom.
**2011**, 15, 4. [Google Scholar] [CrossRef] - Pumi, G.; Valk, M.; Bisognin, C.; Bayer, F.; Prass, T. Beta autoregressive fractionally integrated moving average models. J. Stat. Plan. Inference
**2019**, 200, 196–212. [Google Scholar] [CrossRef][Green Version] - Silva, C.; Migon, H.; Correia, L. Dynamic Bayesian beta models. Comput. Stat. Data Anal.
**2011**, 55, 2074–2089. [Google Scholar] [CrossRef] - Bayer, F.; Cintra, R.; Cribari-Neto, F. Beta seasonal autoregressive moving average models. J. Stat. Comput. Simul.
**2018**, 88, 2961–2981. [Google Scholar] [CrossRef][Green Version] - Galvis, D.; Bandyopadhyay, D.; Lachos, V. Augmented mixed beta regression models for periodontal proportion data. Stat. Med.
**2014**, 33, 3759–3771. [Google Scholar] [CrossRef] [PubMed] - Ospina, R.; Ferrari, S. A general class of zero-or-one inflated beta regression models. Comput. Stat. Data Anal.
**2012**, 56, 1609–1623. [Google Scholar] [CrossRef][Green Version] - Pereira, G.; Botter, D.; Sandoval, M. The truncated inflated beta distribution. Commun. Stat. Theory Methods
**2012**, 41, 907–919. [Google Scholar] [CrossRef][Green Version] - Bonat, W.; Ribeiro, P., Jr.; Zeviani, W. Likelihood analysis for a class of beta mixed models. J. Appl. Stat.
**2015**, 42, 252–266. [Google Scholar] [CrossRef][Green Version] - de Brito Trindade, D.; Espinheira, P.; Pinto Vasconcellos, K.; Farfán Carrasco, J.; Lima, M. Beta regression model nonlinear in the parameters with additive measurement errors in variables. PLoS ONE
**2021**, 16, e0254103. [Google Scholar] [CrossRef] [PubMed] - Carrasco, J.; Ferrari, S.; Arellano-Valle, R. Errors-in-variables beta regression models. J. Appl. Stat.
**2014**, 41, 1530–1547. [Google Scholar] [CrossRef] - Figueroa-Zúñiga, J.; Arellano-Valle, R.; Ferrari, S. Mixed beta regression: A Bayesian perspective. Comput. Stat. Data Anal.
**2013**, 61, 137–147. [Google Scholar] [CrossRef][Green Version] - Mullen, K.M. Continuous global optimization in R. J. Stat. Softw.
**2014**, 60, 1–45. [Google Scholar] [CrossRef][Green Version] - Scrucca, L. GA: A package for genetic algorithms in R. J. Stat. Softw.
**2013**, 53, 1–37. [Google Scholar] [CrossRef][Green Version] - Dünder, E.; Cengíz, M. Model selection in beta regression analysis using several information criteria and heuristic optimization. J. New Theory
**2020**, 33, 76–84. [Google Scholar] - Mullen, K.; Ardia, D.; Gil, D.; Windover, D.; Cline, J. DEoptim: An R package for global optimization by differential evolution. J. Stat. Softw.
**2011**, 40, 1–26. [Google Scholar] [CrossRef][Green Version] - Hansen, N.; Ostermeier, A. Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation. In Proceedings of the IEEE International Conference on Evolutionary Computation, Nagoya, Japan, 20–22 May 1996; pp. 312–317. [Google Scholar] [CrossRef]
- Xiang, Y.; Gubian, S.; Suomela, B.; Hoeng, J. Generalized simulated annealing for efficient global optimization: The GenSA package for R. R J.
**2013**, 5, 13–21. [Google Scholar] [CrossRef][Green Version] - Franzke, B.; Kosko, B. Noise can speed Markov chain Monte Carlo estimation and quantum annealing. Phys. Rev. E
**2019**, 100, 053309. [Google Scholar] [CrossRef] [PubMed] - Geyer, C. Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics, Proceedings of 23rd Symposium on the Interface, Fairfax Station, Seattle, WA, USA, 21–24 April 1991; Interface Foundation of North America: Fairfax Station, VA, USA, 1991; pp. 156–163. [Google Scholar]
- Martino, L.; Elvira, V.; Luengo, D.; Corander, J.; Louzada, F. Orthogonal parallel MCMC methods for sampling and optimization. Digit. Signal Process.
**2016**, 58, 64–84. [Google Scholar] [CrossRef][Green Version] - El-Alem, M.; Aboutahoun, A.; Mahdi, S. Hybrid gradient simulated annealing algorithm for finding the global optimal of a nonlinear unconstrained optimization problem. Soft Comput.
**2021**, 25, 2325–2350. [Google Scholar] [CrossRef] - Xu, P.; Sui, S.; Du, Z. Application of Hybrid Genetic Algorithm Based on Simulated Annealing in Function Optimization. Int. J. Math. Comput. Sci.
**2015**, 9, 695–698. [Google Scholar] - Kaelo, P.; Ali, M. Some variants of the controlled random search algorithm for global optimization. J. Optim. Theory Appl.
**2006**, 130, 253–264. [Google Scholar] [CrossRef] - Johnson, S.G. The Nlopt Package. Version 1.2.2.2. 2020. Available online: https://nlopt.readthedocs.io/en/latest/ (accessed on 6 January 2022).
- Jones, D.; Perttunen, D.; Stuckman, E. Lipschitzian optimisation without the Lipschitz constant. J. Optim. Theory Appl.
**1993**, 79, 157–181. [Google Scholar] [CrossRef] - Gablonsky, J.; Kelley, C. A locally-biased form of the direct algorithm. J. Glob. Optim.
**2001**, 21, 27–37. [Google Scholar] [CrossRef] - Runarsson, T.; Yao, X. Search biases in constrained evolutionary optimization. IEEE Trans. Syst. Man Cybern. C
**2005**, 35, 233–243. [Google Scholar] [CrossRef][Green Version] - Bergmeir, C.; Molina, D.; Benítez, J.M. Memetic algorithms with local search chains in R: The Rmalschains package. J. Stat. Softw.
**2016**, 75, 1–33. [Google Scholar] [CrossRef] - Gilli, M.; Maringer, D.; Schumann, E. Numerical Methods and Optimization in Finance; Academic Press: Waltham, MA, USA, 2011. [Google Scholar]
- Martin-Barreiro, C.; Ramirez-Figueroa, J.A.; Cabezas, X.; Leiva, V.; Martin-Casado, A.; Galindo-Villardón, M.P. A new algorithm for computing disjoint orthogonal components in the parallel factor analysis model with simulations and applications to real-world data. Mathematics
**2021**, 9, 2058. [Google Scholar] [CrossRef] - Ramirez-Figueroa, J.A.; Martin-Barreiro, C.; Nieto, A.B.; Leiva, V.; Galindo-Villardón, M.P. A new principal component analysis by particle swarm optimization with an environmental application for data science. Stoch. Environ. Res. Risk Assess.
**2021**, 35, 1969–1984. [Google Scholar] [CrossRef] - Martin-Barreiro, C.; Ramirez-Figueroa, J.A.; Nieto, A.B.; Leiva, V.; Martin-Casado, A.; Galindo-Villardón, M.P. A new algorithm for computing disjoint orthogonal components in the three-way Tucker model. Mathematics
**2021**, 9, 203. [Google Scholar] [CrossRef] - Espinheira, L.; Ferrari, S.; Cribari-Neto, F. On beta regression residuals. J. Appl. Stat.
**2008**, 35, 407–419. [Google Scholar] [CrossRef]

**Figure 1.**Beta probability density function for $\mu \in \{0.10,0.25,0.50,0.75,0.90\}$ with $\varphi =5$ (

**left**) and $\varphi =100$ (

**right**).

**Figure 2.**Box plots of the values of the overall log-likelihood function corresponding to the beta regression mean with dots in blue indicating the sample mean and central line for the sample median using the indicated method (color).

**Figure 3.**Box plots of the values of the log-likelihood function corresponding to the beta regression mean with dots in blue indicating the sample mean and central line for the sample median using the indicated method (color) and set of parameters.

**Figure 4.**Box plots of the maximum likelihood estimates of ${\beta}_{0}$ for the indicated method, sample size, and set of parameters.

**Figure 5.**Box plots of the maximum likelihood estimates of ${\beta}_{1}$ for the indicated method, sample size, and set of parameters.

**Figure 6.**Bar plot of the average time per run in seconds for the indicated method (color) when estimating the beta regression mean by the ML method.

**Figure 7.**Bar plot of average execution time in seconds for the indicated method (color) and set of parameters when estimating the beta regression mean by the ML method.

**Figure 8.**Box plots of the estimates of ${\beta}_{0}$ for the indicated method and sample size with Sets 5 and 6.

**Figure 9.**Box plots of the estimates of ${\beta}_{1}$ for the indicated method and sample size with Sets 5 and 6.

**Figure 10.**Box plots of the estimates of $\varphi $ for the indicated method and sample size with Sets 5 and 6.

**Figure 11.**Box plots of the estimates of ${\beta}_{0}$ for the indicated method and sample size with Sets 7 and 8.

**Figure 12.**Box plots of the estimates of ${\beta}_{1}$ for the indicated method and sample size with Sets 7 and 8.

**Figure 13.**Box plots of the estimates of $\varphi $ for the indicated method and sample size with Sets 7 and 8.

**Figure 14.**Box plots of the estimates of ${\beta}_{0}$ for the indicated method and sample size with Sets 9 and 10.

**Figure 15.**Box plots of the estimates of ${\beta}_{1}$ for the indicated method and sample size with Sets 9 and 10.

**Figure 16.**Box plots of the estimates of $\varphi $ for the indicated method and sample size with Sets 9 and 10.

**Figure 17.**Bar plot of the average time per run in seconds for the indicated method (color) when estimating the beta regression mean by the ML method with Sets 5 and 6.

**Figure 18.**Bar plot of the average time per run in seconds for the indicated method (color) when estimating the beta regression mean by the ML method with Sets 7 and 8.

**Figure 19.**Bar plot of the average time per run in seconds for the indicated method (color) when estimating the beta regression mean by the ML method with Sets 9 and 10.

Parameter | |||
---|---|---|---|

Set | ${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | $\mathit{\varphi}$ |

1 | $4.0$ | $-0.8$ | 12 |

2 | $4.0$ | $-0.8$ | 148 |

3 | $-2.5$ | $-1.2$ | 12 |

4 | $-2.5$ | $-1.2$ | 148 |

Method | |||
---|---|---|---|

Indicator | cma_es | malschains | PSopt |

Number of failures | $39.03$ | $33.90$ | $7.06$ |

Percentage of failures | $48.78\%$ | $42.37\%$ | $8.82\%$ |

Parameter | ||||
---|---|---|---|---|

Set | Distribution | ${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | $\mathit{\varphi}$ |

5 | Uniform(0,1) | $-2.50$ | $-1.20$ | $0.5$ |

6 | Uniform(0,1) | $-2.50$ | $-1.20$ | $2.0$ |

7 | Normal(0,1) | $-2.05$ | $-1.20$ | $0.5$ |

8 | Normal(0,1) | $-2.50$ | $-1.20$ | $2.0$ |

9 | Student-t(3) | $1.21$ | $1.25$ | $0.5$ |

10 | Student-t(3) | $1.21$ | $1.25$ | $2.0$ |

**Table 4.**Indicators of failures for the listed method with Sets 5 and 6 of the second stage of the simulation.

Method | ||||||
---|---|---|---|---|---|---|

Sets | Indicator | crs | DEoptim | isres | optim | GenSA |

5–6 | Number of failures | $39.83$ | 0 | $39.80$ | $37.50$ | 19 |

Percentage of failures | $99.56\%$ | $0\%$ | $99.50\%$ | $93.75\%$ | $0.04\%$ | |

7–8 | Number of failures | $39.83$ | 0 | $39.80$ | $37.50$ | 19 |

Percentage of failures | $99.56\%$ | $0\%$ | $99.50\%$ | $93.75\%$ | $0.04\%$ | |

9–10 | Number of failures | $39.83$ | 0 | $39.80$ | $37.50$ | 19 |

Percentage of failures | $99.56\%$ | $0\%$ | $99.50\%$ | $93.75\%$ | $0.04\%$ |

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**MDPI and ACS Style**

Couri, L.; Ospina, R.; Silva, G.d.; Leiva, V.; Figueroa-Zúñiga, J.
A Study on Computational Algorithms in the Estimation of Parameters for a Class of Beta Regression Models. *Mathematics* **2022**, *10*, 299.
https://doi.org/10.3390/math10030299

**AMA Style**

Couri L, Ospina R, Silva Gd, Leiva V, Figueroa-Zúñiga J.
A Study on Computational Algorithms in the Estimation of Parameters for a Class of Beta Regression Models. *Mathematics*. 2022; 10(3):299.
https://doi.org/10.3390/math10030299

**Chicago/Turabian Style**

Couri, Lucas, Raydonal Ospina, Geiza da Silva, Víctor Leiva, and Jorge Figueroa-Zúñiga.
2022. "A Study on Computational Algorithms in the Estimation of Parameters for a Class of Beta Regression Models" *Mathematics* 10, no. 3: 299.
https://doi.org/10.3390/math10030299