Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions
Abstract
:1. Introduction
2. The Birnbaum–Saunders Model
2.1. Properties
 P1:
 If $X\sim \mathrm{BS}(\alpha ,\beta )$, then $cX\sim \mathrm{BS}(\alpha ,c\beta )$, that is, the BS distribution is a homogeneous family;
 P2:
 If $X\sim \mathrm{BS}(\alpha ,\beta )$, then $1/X\sim \mathrm{BS}(\alpha ,1/\beta )$, that is, the BS distribution is invariant under the reciprocal transformation. This property can be important in financial applications.
2.2. Inference
3. Optimal Sample Size
3.1. Determining the Optimal Sample Size
Algorithm 1: Estimation of the minimized Bayes risk. 

3.2. Loss Functions
 L1: Loss Function 1. The first loss function is defined as:$$L(\mu ,{d}_{n})=\mu {d}_{n},$$
 L2: Loss Function 2. Second, we consider the wellknown quadratic loss function stated as:$$L(\mu ,{d}_{n})={(\mu {d}_{n})}^{2}.$$
 L3: Loss Function 3. The loss functions L1 and L2 suffer from two disadvantages in practical applications: both are symmetric and unbounded. In the list of bounded loss functions that might be considered, we may include those suggested in [29,30]. However, in our case, these loss functions are not simple to deal with. Nevertheless, there is a simple wellknow asymmetric loss function that we may consider. This is the linear exponential (known as LINEX) loss function given by:$$L(\mu ,{d}_{n})=exp(\ell ({d}_{n}\mu ))\ell ({d}_{n}\mu )1,$$$${d}_{n}^{*}=\frac{1}{\ell}log\left(\right)open="("\; close=")">\mathrm{E}\left(\right)open="["\; close="]">\mathrm{exp}(\ell \mu ){\mathit{x}}_{n}.$$
 L4: Loss Function 4. The fourth function is defined as:$$L(\mu ,{d}_{n})=\rho \tau +{(a\mu )}^{+}+{(\mu b)}^{+},$$$$\mathrm{E}\left[L(\mu ,{d}_{n}^{*})\right{\mathit{x}}_{n}]=\mathrm{E}\left[\mu {\delta}_{\mu}\left({A}_{{b}^{*}}\right)\right{\mathit{x}}_{n}]\mathrm{E}\left[\mu {\delta}_{\mu}\left({A}_{{a}^{*}}\right)\right{\mathit{x}}_{n}],$$$$\widehat{\mathrm{E}}\left[L(\mu ,{d}_{n}^{*})\right{\mathit{x}}_{n}]=\frac{1}{N}\sum _{j=1}^{N}({\mu}_{j}{\delta}_{{\mu}_{j}}\left({A}_{{b}^{*}}\right){\mu}_{j}{\delta}_{{\mu}_{j}}\left({A}_{{a}^{*}}\right)).$$
 L5: Loss Function 5. The fifth and last loss function considered here is expressed as:$$L(\mu ,{d}_{n})=\gamma \tau +\frac{{(\mu m)}^{2}}{\tau},$$$$\mathrm{E}\left[L(\mu ,{d}_{n}^{*})\right{\mathit{x}}_{n}]=2{\gamma}^{1/2}\sqrt{\mathrm{Var}\left(\mu \right{\mathit{x}}_{n})}.$$
4. Computational Aspects and Empirical Applications
4.1. Computer Characteristics
4.2. The samplesizeBS Functions
4.3. Optimal Sample Sizes
4.4. Illustrative Example
5. Discussion, Conclusions and Future Research
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
 Spiegelhalter, D.J.; Abrams, K.R.; Myles, J.P. Bayesian Approaches to Clinical Trials and HealthCare Evaluation; Wiley: Chichester, UK, 2004. [Google Scholar]
 Etzioni, R.; Kadane, J.B. Optimal experimental design for another’s analysis. J. Am. Stat. Assoc. 1993, 88, 1404–1411. [Google Scholar] [CrossRef]
 Sahu, S.K.; Smith, T.M.F. A Bayesian method of sample size determination with practical applications. J. R. Stat. Soc. A 2006, 169, 235–253. [Google Scholar] [CrossRef]
 Parmigiani, G.; Inoue, L. Decision Theory: Principles and Approaches; Wiley: New York, NY, USA, 2009. [Google Scholar]
 Islam, A.F.M.S.; Pettit, L.I. Bayesian sample size determination using LINEX loss and linear cost. Commun. Stat. Theory Methods 2012, 41, 223–240. [Google Scholar] [CrossRef]
 Islam, A.F.M.S.; Pettit, L.I. Bayesian sample size determination for the bounded LINEX loss function. J. Stat. Comput. Simul. 2014, 84, 1644–1653. [Google Scholar] [CrossRef]
 Santis, F.D.; Gubbiotti, S. A decisiontheoretic approach to sample size determination under several priors. Appl. Stoch. Model. Bus. Ind. 2017, 33, 282–295. [Google Scholar] [CrossRef]
 Costa, E.G. Sample Size for Estimating the Organism Concentration in Ballast Water: A Bayesian Approach. Ph.D. Thesis, Department of Statistics, Universidade de São Paulo, São Paulo, Brazil, 2017. (In Portuguese). [Google Scholar]
 Birnbaum, Z.W.; Saunders, S.C. A new family of life distributions. J. Appl. Probab. 1969, 6, 319–327. [Google Scholar] [CrossRef]
 Bourguignon, M.; Leao, J.; Leiva, V.; SantosNeto, M. The transmuted Birnbaum–Saunders distribution. REVSTAT Stat. J. 2017, 15, 601–628. [Google Scholar]
 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]
 SantosNeto, M.; Cysneiros, F.J.A.; Leiva, V.; Barros, M. Reparameterized Birnbaum–Saunders regression models with varying precision. Electron. J. Stat. 2016, 10, 2825–2855. [Google Scholar] [CrossRef]
 Reyes, J.; BarrancoChamorro, I.; Gallardo, D.I.; Gomez, H.W. Generalized modified slash Birnbaum—Saunders distribution. Symmetry 2018, 10, 724. [Google Scholar] [CrossRef][Green Version]
 GomezDeniz, E.; Gomez, L. The Rayleigh Birnbaum Saunders distribution: A general fading model. Symmetry 2020, 12, 389. [Google Scholar] [CrossRef][Green Version]
 Desousa, M.; Saulo, H.; Leiva, V.; SantosNeto, M. On a new mixturebased regression model: Simulation and application to data with high censoring. J. Stat. Comput. Simul. 2020, 90, 2861–2877. [Google Scholar] [CrossRef]
 Sanchez, L.; Leiva, V.; Galea, M.; Saulo, H. Birnbaum–Saunders quantile regression and its diagnostics with application to economic data. Appl. Stoch. Models Bus. Ind. 2021, 37, 53–73. [Google Scholar] [CrossRef]
 Villegas, C.; Paula, G.A.; Leiva, V. Birnbaum–Saunders mixed models for censored reliability data analysis. IEEE Trans. Reliab. 2011, 60, 748–758. [Google Scholar] [CrossRef]
 Marchant, C.; Leiva, V.; Cysneiros, F.J.A. A multivariate loglinear model for Birnbaum–Saunders distributions. IEEE Trans. Reliab. 2016, 65, 816–827. [Google Scholar] [CrossRef]
 Arrue, J.; Arellano, R.; Gomez, H.W.; Leiva, V. On a new type of Birnbaum–Saunders models and its inference and application to fatigue data. J. Appl. Stat. 2020, 47, 2690–2710. [Google Scholar] [CrossRef]
 Balakrishnan, N.; Kundu, D. Birnbaum–Saunders distribution: A review of models, analysis, and applications. Appl. Stoch. Model. Bus. Ind. 2019, 35, 4–49. [Google Scholar] [CrossRef][Green Version]
 Bourguignon, M.; Ho, L.L.; Fernandes, F.H. Control charts for monitoring the median parameter of Birnbaum–Saunders distribution. Qual. Reliab. Eng. Int. 2020, 36, 1333–1363. [Google Scholar] [CrossRef]
 Costa, E.G.; SantosNeto, M.; Leiva, V. samplesizeBS: Bayesian Sample Size in a DecisionTheoretic Approach for the Birnbaum–Saunders, R Package Version 1.11; 2020. Available online: www.github.com/santosneto/samplesizeBS (accessed on 21 May 2021).
 Aykroyd, R.G.; Leiva, V.; Marchant, C. Multivariate Birnbaum–Saunders distributions: Modelling and applications. Risks 2018, 6, 21. [Google Scholar] [CrossRef][Green Version]
 Leiva, V. The Birnbaum–Saunders Distribution; Academic Press: New York, NY, USA, 2016. [Google Scholar]
 Ng, H.; Kundu, D.; Balakrishnan, N. Modified moment estimation for the twoparameter Birnbaum–Saunders distribution. Comput. Stat. Data Anal. 2003, 43, 283–298. [Google Scholar] [CrossRef]
 Wang, M.; Sun, X.; Park, C. Bayesian analysis of Birnbaum–Saunders distribution via the generalized ratioofuniforms method. Comput. Stat. 2016, 31, 207–225. [Google Scholar] [CrossRef]
 Leiva, V.; Ruggeri, F.; Saulo, H.; Vivanco, J.F. A methodology based on the Birnbaum–Saunders distribution for reliability analysis applied to nanomaterials. Reliab. Eng. Syst. Saf. 2017, 157, 192–201. [Google Scholar] [CrossRef]
 Raiffa, H.; Schlaifer, R. Applied Statistical Decision Theory; Harvard University Press: Boston, MA, USA, 1961. [Google Scholar]
 Spiring, F.A. The reflected normal loss function. Can. J. Stat. 1993, 21, 321–330. [Google Scholar] [CrossRef]
 Leung, B.P.K.; Spiring, F.A. The inverted beta loss function: Properties and applications. IEE Trans. 2002, 34, 1101–1109. [Google Scholar] [CrossRef]
 Zellner, A. Bayesian estimation and prediction using asymmetric loss functions. J. Am. Stat. Assoc. 1986, 81, 446–451. [Google Scholar] [CrossRef]
 Rice, K.M.; Lumley, T.; Szpiro, A.A. Trading Bias for Precision: Decision Theory for Intervals and Sets. Working Paper 336, UW Biostatistics. 2008. Available online: https://biostats.bepress.com/uwbiostat/paper336/ (accessed on 21 May 2021).
 Hsieh, H.K. Estimating the critical time of the inverse Gaussian hazard rate. IEEE Trans. Reliab. 1990, 39, 342–345. [Google Scholar] [CrossRef]
 Puentes, R.; Marchant, C.; Leiva, V.; Figueroa, J.I.; Ruggeri, F. Predicting PM2.5 and PM10 levels during critical episodes management in Santiago, Chile, with a bivariate Birnbaum–Saunders loglinear model. Mathematics 2021, 9, 645. [Google Scholar] [CrossRef]
 Leiva, V.; Saulo, H.; Souza, R.; Aykroyd, R.G.; Vila, R. A new BISARMA time series model for forecasting mortality using weather and particulate matter data. J. Forecast. 2021, 40, 346–364. [Google Scholar] [CrossRef]
 Martinez, S.; Giraldo, R.; Leiva, V. Birnbaum–Saunders functional regression models for spatial data. Stoch. Environ. Res. Risk Assess. 2019, 33, 1765–1780. [Google Scholar] [CrossRef]
 Huerta, M.; Leiva, V.; Liu, S.; Rodriguez, M.; Villegas, D. On a partial least squares regression model for asymmetric data with a chemical application in mining. Chemom. Intell. Lab. Syst. 2019, 190, 55–68. [Google Scholar] [CrossRef]
 Carrasco, J.M.F.; Finiga, J.I.; Leiva, V.; Riquelme, M.; Aykroyd, R.G. An errorsinvariables model based on the Birnbaum–Saunders and its diagnostics with an application to earthquake data. Stoch. Environ. Res. Risk Assess. 2020, 34, 369–380. [Google Scholar] [CrossRef]
Function  Output 

rbs()  A sample of size n from the BS distribution 
logp.beta()  The logarithm of the marginal posterior distribution of $\beta $ 
rbeta.post()  A random sample from the marginal posterior distribution of $\beta $ using the random walk Metropolis–Hastings algorithm 
bss.dt.bs()  An integer representing the optimal sample size for estimating $\mu $ of the BS distribution and the acceptance rate for the random walk Metropolis–Hastings algorithm 
$\mathit{\rho}/\mathit{\gamma}/\mathit{\ell}$  ${\mathit{a}}_{1}=8$  ${\mathit{a}}_{1}=10$  ${\mathit{a}}_{1}=13$  ${\mathit{a}}_{1}=15$  

$\mathit{c}=\mathbf{0.001}$  $\mathit{c}=\mathbf{0.01}$  $\mathit{c}=\mathbf{0.1}$  $\mathit{c}=\mathbf{0.001}$  $\mathit{c}=\mathbf{0.01}$  $\mathit{c}=\mathbf{0.1}$  $\mathit{c}=\mathbf{0.001}$  $\mathit{c}=\mathbf{0.01}$  $\mathit{c}=\mathbf{0.1}$  $\mathit{c}=\mathbf{0.001}$  $\mathit{c}=\mathbf{0.01}$  $\mathit{c}=\mathbf{0.1}$  
Loss function L1  
651  117  23  436  80  13  267  47  9  210  36  6  
641  121  22  429  77  14  267  47  8  209  36  6  
627  140  21  429  77  13  268  47  8  209  37  6  
Loss function L2  
2096  641  176  1130  317  108  542  144  33  381  88  21  
2129  697  200  1198  326  97  558  138  42  380  89  23  
2075  622  218  1182  292  81  530  139  32  360  89  21  
Loss function L3  
$\ell =2.00$  1787  354  61  1111  229  41  631  138  24  467  101  18 
1826  363  62  1115  235  40  640  135  25  468  101  18  
1820  352  61  1112  229  41  616  134  25  454  97  19  
$\ell =1.00$  929  190  34  553  122  22  310  69  13  226  51  9 
924  197  34  556  122  22  311  69  13  227  51  9  
925  190  34  552  116  22  308  67  13  222  49  9  
$\ell =0.50$  465  101  19  281  63  12  153  35  7  111  25  5 
463  101  19  275  63  12  155  35  7  111  25  4  
471  99  18  275  59  12  146  33  6  105  23  4  
Loss function L4  
$\rho =0.10$  279  53  9  175  31  7  103  18  3  79  14  2 
271  54  10  171  31  6  106  18  3  79  14  2  
284  53  10  168  33  5  103  18  3  80  14  2  
$\rho =0.05$  187  37  8  118  22  4  70  13  2  54  9  0 
197  37  8  121  22  4  71  12  2  55  9  0  
184  40  7  121  22  4  71  13  2  55  9  0  
$\rho =0.01$  82  18  3  54  9  0  30  5  0  22  4  0 
85  19  3  52  9  0  29  5  0  22  4  0  
83  18  3  49  9  0  30  5  0  22  4  0  
Loss function L5  
$\gamma =1.00$  1461  271  51  899  171  30  556  103  18  441  78  13 
1472  292  55  942  162  31  561  99  18  438  78  13  
1460  282  56  883  162  30  554  101  18  433  78  13  
$\gamma =0.50$  1208  203  39  684  132  23  427  78  14  337  59  10 
1179  201  38  690  130  23  434  78  14  335  59  10  
1183  213  42  693  134  24  436  80  14  338  60  10  
$\gamma =0.25$  796  166  32  538  106  18  333  59  10  259  46  8 
859  171  30  540  99  19  331  62  11  260  47  8  
894  167  32  531  101  18  333  60  10  260  46  8 
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Costa, E.; SantosNeto, M.; Leiva, V. Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions. Symmetry 2021, 13, 926. https://doi.org/10.3390/sym13060926
Costa E, SantosNeto M, Leiva V. Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions. Symmetry. 2021; 13(6):926. https://doi.org/10.3390/sym13060926
Chicago/Turabian StyleCosta, Eliardo, Manoel SantosNeto, and Víctor Leiva. 2021. "Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions" Symmetry 13, no. 6: 926. https://doi.org/10.3390/sym13060926