An Alternative Lambert-Type Distribution for Bounded Data
Abstract
:1. Introduction
2. The New Distribution
2.1. LPHU Random Variable
- The LPHU pdf is not null at the lower end of its support, . Thus, the LPHU has a behavior similar to that of the LU and PHU pdf’s, but with the advantage that it can present unimodal and reverse-unimodal shapes;
- Equation (4) reduces to the PHU, LU, and uniform pdf’s when , and , respectively. Thus, for such parameter choices, the LPHU pdf inherits the shapes of the PHU, LU, and U pdf’s;
- For and , we observe that the equation leads to the statement that the LPHU pdf may have a critical point at
2.2. Related Distributions
- Let , where and . Then, Y follows the nonscaled Lambert-exponential distribution. See Iriarte et al. [7];
- Let , where and . Then, Y follows the Lambert–Rayleigh distribution. See Iriarte et al. [7];
- Let , where and . Then, the distribution of Y is a three-parameter distribution that reduces to the K distribution when . In this case, the cdf of Y is given by , where . Thus, we refer to this distribution as the Lambert–Kumaraswamy distribution.
2.3. Hazard Rate Function
2.4. Skewness and Kurtosis Behavior
3. Parameter Estimation
3.1. ML Estimation
3.2. Computational Guidelines
- Alternatively, the ML estimates can be obtained by solving the optimization problem , subject to , , where is given in Equation (7).
- For this, we use the optim function of the R programming language. An R code is provided in Appendix B.
- Specifically, we consider the L-BGSB-B algorithm [15], which allows us to specify the parameter space. This algorithm requires declaring a value in the parameter space to initialize the iterative process. Taking into account that the PHU distribution is a special case of the LPHU distribution, we consider and , where is the ML estimator of the shape parameter of the PHU distribution.
3.3. Simulation Study
- Generate ;
- Compute .
4. Data Analysis
4.1. Firm’s Risk Management Cost Effectiveness
4.2. Household Shared Budget for Transportation
5. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
B | beta distribution |
K | Kumaraswamy distribution |
probability density function | |
ML | maximum likelihood |
cdf | cumulative distribution function |
U | uniform distribution |
LU | Lambert uniform distribution |
PHU | proportional hazard uniform distribution |
rf | reliability function |
hrf | hazard rate function |
LPHU | Lambert proportional hazard uniform distribution |
AE | average estimate |
SD | standard deviation |
RMSE | root mean square error |
SE | standard error |
CP | coverage probability |
AD | Anderson–Darling |
CvM | Cramer–von Mises |
AIC | Akaike information criterion |
BIC | Bayesian information criterion |
Appendix A. R Codes
Appendix B. R Code
Appendix C. R Code
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Distribution | AIC | BIC | AD | CvM | ||
---|---|---|---|---|---|---|
LPHU | 0.012 | 1.889 | −174.3 | −169.7 | 0.484 | 0.616 |
(0.016) | (0.496) | |||||
K | 0.664 | 3.440 | −153.3 | −148.7 | 0.025 | 0.039 |
(0.071) | (0.620) | |||||
B | 0.612 | 3.797 | −148.2 | −143.7 | 0.009 | 0.013 |
(0.085) | (0.715) |
Non-Rejection Rate | Hit Rate | |||
---|---|---|---|---|
Distribution | AIC | BIC | ||
LPHU | 0.734 | 0.718 | 0.836 | 0.836 |
B | 0.502 | 0.479 | 0.130 | 0.130 |
K | 0.562 | 0.531 | 0.034 | 0.034 |
Distribution | AIC | BIC | ||||
---|---|---|---|---|---|---|
LPHU | 1.996 | 9.222 | −209.5 | −204.3 | 0.112 | 0.715 |
(0.259) | (0.933) | |||||
B | 1.302 | 8.217 | −203.1 | −197.8 | 0.213 | 1.308 |
(0.166) | (1.228) | |||||
K | 1.273 | 10.539 | −105.1 | −199.9 | 0.177 | 1.092 |
(0.116) | (2.385) |
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Varela, H.; Rojas, M.A.; Reyes, J.; Iriarte, Y.A. An Alternative Lambert-Type Distribution for Bounded Data. Mathematics 2023, 11, 667. https://doi.org/10.3390/math11030667
Varela H, Rojas MA, Reyes J, Iriarte YA. An Alternative Lambert-Type Distribution for Bounded Data. Mathematics. 2023; 11(3):667. https://doi.org/10.3390/math11030667
Chicago/Turabian StyleVarela, Héctor, Mario A. Rojas, Jimmy Reyes, and Yuri A. Iriarte. 2023. "An Alternative Lambert-Type Distribution for Bounded Data" Mathematics 11, no. 3: 667. https://doi.org/10.3390/math11030667
APA StyleVarela, H., Rojas, M. A., Reyes, J., & Iriarte, Y. A. (2023). An Alternative Lambert-Type Distribution for Bounded Data. Mathematics, 11(3), 667. https://doi.org/10.3390/math11030667