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 , then , that is, the BS distribution is a homogeneous family;
- P2:
- If , then , 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:
- L2: Loss Function 2. Second, we consider the well-known quadratic loss function stated as:
- 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 well-know asymmetric loss function that we may consider. This is the linear exponential (known as LINEX) loss function given by:
- L4: Loss Function 4. The fourth function is defined as:
- L5: Loss Function 5. The fifth and last loss function considered here is expressed as:
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
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Function | Output |
---|---|
rbs() | A sample of size n from the BS distribution |
logp.beta() | The logarithm of the marginal posterior distribution of |
rbeta.post() | A random sample from the marginal posterior distribution of using the random walk Metropolis–Hastings algorithm |
bss.dt.bs() | An integer representing the optimal sample size for estimating of the BS distribution and the acceptance rate for the random walk Metropolis–Hastings algorithm |
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 | ||||||||||||
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 | |
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 | |
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 | ||||||||||||
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 | |
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 | |
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 | ||||||||||||
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 | |
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 | |
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.; Santos-Neto, 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, Santos-Neto 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 Santos-Neto, 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