# An Alternative One-Parameter Distribution for Bounded Data Modeling Generated from the Lambert Transformation

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## Abstract

**:**

## 1. Introduction

## 2. Lambert-Uniform Distribution

#### 2.1. LU Random Variable

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

#### 2.2. Related Distributions

#### 2.3. Skewness and Kurtosis

**Proposition**

**2.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

#### 2.4. ML Estimation

## 3. Quantile Regression Model

#### 3.1. The LU Model

#### 3.2. ML Estimation

## 4. Simulation Studies

#### 4.1. First Simulation Study

#### 4.2. Second Simulation Study

- Definition of covariates: Generate ${w}_{1}={({w}_{11},\dots ,{w}_{1n})}^{t}$, ${w}_{2}={({w}_{21},\dots ,{w}_{2n})}^{t}$ and ${w}_{3}={({w}_{31},\dots ,{w}_{3n})}^{t}$, where $({w}_{1j},{w}_{2j})$ follows a bivariate normal distribution with parameters ${\mu}_{1}={\mu}_{2}=0$, ${\sigma}_{1}={\sigma}_{2}=1$ and $\rho =0.7$, with $j=1,\dots ,n$ and ${w}_{3}$ is a binary variable with probability of success depending on the variable ${w}_{1}$ through the logistic function, that is, ${w}_{3j}\sim \mathrm{Bernoulli}\left({p}_{j}\right)$, where ${p}_{j}=1/[1+exp(-{w}_{1j})]$ $j=1,\dots ,n$.
- Definition of scenarios: We considered two scenarios, A and B, where in both we picked ${\beta}_{0}=-2$, ${\beta}_{1}=0.1$, ${\beta}_{2}=0.5$ and ${\beta}_{3}=-2.5$. Regarding the choices for q, we chose the values 0.25 for Scenario A and 0.75 for Scenario B.
- Simulate the response variable: Generate ${({u}_{1},\dots ,{u}_{n})}^{t}$, ${u}_{j}\sim \mathrm{uniform}(0,1)$, $j=1,\dots ,n$, and calculate$$\begin{array}{c}\hfill {x}_{j}=\frac{{\eta}_{j}}{log\left(\right)open="("\; close=")">\frac{1-q}{1-{\eta}_{j}}}{W}_{0}\left(\right)open="["\; close="]">log\left(\right)open="("\; close=")">\frac{1-q}{1-{\eta}_{j}}& \frac{{u}_{j}-1}{{\eta}_{j}}{\left(\right)}^{\frac{1-{\eta}_{j}}{1-q}}1/{\eta}_{j}\end{array}$$

## 5. Data Analysis

#### 5.1. Data from an LU Population

#### 5.2. Peak Horizontal Acceleration Data

#### 5.3. Risk Managements Practice Data

## 6. Final Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

U | Uniform |

P | Power |

B | Beta |

K | Kumaraswamy |

ASHN | Arcsecant-hyperbolic-normal |

LU | Lambert-uniform |

ML | Maximum likelihood |

cdf | Cumulative distribution function |

Probability density function | |

qf | Quantile function |

## Appendix A. Lambert W Function

- If $z<-1/e$, then no solution exists in the reals.
- If $z\in (-1/e,0)$, then there are two solutions given by the principal branch, ${W}_{0}\left(z\right)$, and the non-principal branch, ${W}_{-1}\left(z\right)$.
- If $z\ge 0$, then the solution is unique, ${W}_{0}\left(z\right)={W}_{-1}\left(z\right)$.

## Appendix B. The AEs Obtained in the Simulation Studies Presented in Section 4.1 and Section 4.2 with Sample Sizes Less Than 100

**Table A1.**The AEs obtained in the simulation study of Section 4.1.

Sample Size | |||||||||
---|---|---|---|---|---|---|---|---|---|

Scenario | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |

A ($\alpha =0.5$) | 0.566 | 0.534 | 0.523 | 0.508 | 0.520 | 0.509 | 0.498 | 0.510 | 0.512 |

B ($\alpha =1.5$) | 1.619 | 1.535 | 1.530 | 1.512 | 1.507 | 1.521 | 1.516 | 1.504 | 1.509 |

**Table A2.**The AEs obtained in Scenarios A and B of the simulation study in Section 4.2.

Sample Size | |||||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |

Scenario A ($q=0.25$) | |||||||||

${\beta}_{0}$ | −2.049 | −2.019 | −2.005 | −2.011 | −2.005 | −1.984 | −2.006 | −2.015 | −1.994 |

${\beta}_{1}$ | 0.090 | 0.097 | 0.125 | 0.089 | 0.107 | 0.113 | 0.108 | 0.098 | 0.104 |

${\beta}_{2}$ | 0.604 | 0.529 | 0.515 | 0.520 | 0.511 | 0.501 | 0.495 | 0.496 | 0.502 |

${\beta}_{3}$ | 2.558 | −2.555 | −2.563 | −2.529 | −2.535 | −2.557 | −2.515 | −2.496 | −2.528 |

Scenario B ($q=0.75$) | |||||||||

${\beta}_{0}$ | −2.275 | −2.110 | −2.063 | −2.046 | −2.034 | −2.008 | −2.026 | −2.028 | −2.009 |

${\beta}_{1}$ | −0.035 | 0.089 | 0.114 | 0.096 | 0.108 | 0.110 | 0.107 | 0.099 | 0.105 |

${\beta}_{2}$ | 0.621 | 0.507 | 0.503 | 0.503 | 0.499 | 0.495 | 0.489 | 0.491 | 0.497 |

${\beta}_{3}$ | −2.275 | −2.454 | −2.494 | −2.491 | −2.503 | −2.529 | −2.493 | −2.482 | −2.511 |

## Appendix C. Second Partial Derivatives of the Log-Likelihood Function Given in Equation (7)

## Appendix D. Histograms of the Data Considered in Section 5.2 and Section 5.3

**Figure A1.**(

**Left**) Histograms of the peak horizontal acceleration data. (

**Right**) Histogram of the response FI (the measure of the firm’s risk management cost effectiveness).

## Appendix E. Estimates and other Fit Measures for the Sample Associated with Figure 7 of Section 5.1

**Table A3.**The parameter estimates (with standard errors in parentheses); the ℓ, AIC, CAIC and BIC values; and the p-values of the AD and CvM goodness-of-fit tests for the SU, P, MOEU, B, K and LU distributions fitted to simple generated data.

Parameter | LU | K | B | MOEU | P | SU |
---|---|---|---|---|---|---|

$\alpha $ | 0.009 | 0.914 | 0.914 | 0.131 | 0.438 | 1.000 |

(0.003) | (0.050) | (0.065) | (0.012) | (0.025) | (0.113) | |

$\beta $ | - | 4.078 | 4.291 | - | - | - |

(0.399) | (0.375) | |||||

ℓ | 232.3 | 227.0 | 226.4 | 216.3 | 136.5 | 143.4 |

AIC | −462.7 | −450.0 | −448.8 | −430.7 | −271.0 | −284.8 |

BIC | −459.0 | −442.6 | −441.4 | −427.0 | −267.3 | −281.1 |

AD | 0.915 | 0.485 | 0.399 | 0.005 | <0.001 | <0.001 |

CvM | 0.962 | 0.624 | 0.473 | 0.027 | <0.001 | <0.001 |

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**Figure 2.**Plots of the skewness and kurtosis coefficients of the LU distribution (red color) and the U distribution (circle).

**Figure 3.**The AE, SD, SE, RMSE and CP for each of the 1000 estimates of $\alpha $ obtained in the scenarios $\alpha =0.5$ (

**top**) and $\alpha =1.5$ (

**bottom**), under the different sample sizes.

**Figure 4.**The AE, SD, SE and RMSE for each of the 1000 estimates of the coefficients $\beta $s obtained in Scenario A, under the different sample size.

**Figure 5.**The AE, SD, SE and RMSE for each of the 1000 estimates of the coefficients $\beta $s obtained in Scenario B, under the different sample size.

**Figure 6.**The CPs for the estimates of the coefficients $\beta $s in: Scenario A (

**left**); and Scenario B (

**right**).

**Figure 7.**Histogram for a single sample generated from the LU($0.01$) population fitted with the LU, K, MOEU, SU, B and K distributions.

**Figure 8.**QQ-plots: (

**a**) LU distribution; (

**b**) K distribution; (

**c**) B distribution; (

**d**) MOEU distribution; (

**e**) P distribution; and (

**f**) SU distribution.

**Figure 9.**Coefficient estimates and its 95% confidence intervals for variables AS, CA, SI, IN, CE and SO in different LU quantile regression models considering $q=0.1,0.2,\dots ,0.9$ and response variable FI.

**Table 1.**Non-rejection and hit rates for the LU, K, MOEU, SU, B and K distributions obtained from the 1000 samples generated.

Non-Rejection Rate | Hit Rate | |||
---|---|---|---|---|

Distribution | AD | CvM | AIC | BIC |

LU | 0.997 | 0.999 | 0.865 | 0.969 |

P | 0.000 | 0.000 | 0.000 | 0.000 |

MOEU | 0.117 | 0.418 | 0.002 | 0.002 |

SU | 0.000 | 0.000 | 0.000 | 0.000 |

B | 0.982 | 0.986 | 0.041 | 0.020 |

K | 0.993 | 0.995 | 0.092 | 0.009 |

**Table 2.**The parameter estimates (with standard errors in parentheses), the ℓ, AIC, CAIC and BIC values and the p-values of the AD and CvM goodness-of-fit tests for the SU, P, MOEU, B, K and LU distributions fitted to the peak horizontal acceleration data.

Parameter | LU | K | B | MOEU | P | SU |
---|---|---|---|---|---|---|

$\alpha $ | 0.005 | 0.890 | 0.877 | 0.112 | 0.412 | 1.000 |

(0.002) | (0.062) | (0.080) | (0.013) | (0.030) | (0.170) | |

$\beta $ | - | 4.423 | 4.699 | - | - | - |

(0.571) | (0.533) | |||||

ℓ | 158.5 | 157.0 | 156.6 | 149.3 | 98.1 | 91.7 |

AIC | −315.1 | −310.1 | −309.3 | −296.7 | −194.3 | −181.5 |

BIC | −311.9 | −303.7 | −302.9 | −293.5 | −191.1 | −178.2 |

AD | 0.978 | 0.882 | 0.486 | 0.069 | <0.001 | <0.001 |

CvM | 0.965 | 0.884 | 0.576 | 0.140 | <0.001 | <0.001 |

**Table 3.**The ℓ, AIC, CAIC and BIC values for the ASHN, K and LU quantile regression models fitted to the risk managements practice data and the p-values of the AD and CvM tests for the randomize residuals.

Criterion | p-Value for the | |||||
---|---|---|---|---|---|---|

q | Model | ℓ | AIC | BIC | AD Test | CvM Test |

0.25 | ASHN | 80.3 | −144.7 | −126.4 | <0.001 | <0.001 |

K | 97.9 | −179.8 | −161.5 | 0.166 | 0.198 | |

LU | 107.8 | −201.6 | −185.6 | 0.134 | 0.231 | |

0.5 | ASHN | 80.1 | −144.2 | −125.9 | <0.001 | <0.001 |

K | 98.8 | −181.6 | −163.3 | 0.150 | 0.169 | |

LU | 108.1 | −202.2 | −186.2 | 0.154 | 0.252 | |

0.75 | ASHN | 81.8 | −147.7 | −129.4 | <0.001 | <0.001 |

K | 99.9 | −183.9 | −165.6 | 0.151 | 0.147 | |

LU | 108.6 | −203.2 | −187.2 | 0.158 | 0.230 |

**Table 4.**Coefficient estimates for the LU quantile regression model fitted to the risk managements practice data and significance tests of individual regression coefficients.

q | Parameter | Estimate | SE | z | p-Value |
---|---|---|---|---|---|

0.25 | Intercept | 1.619 | 1.543 | 1.049 | 0.293 |

AS | −0.022 | 0.017 | −1.311 | 0.189 | |

CA | 0.318 | 0.314 | 1.013 | 0.310 | |

SI | −0.774 | 0.164 | −4.717 | <0.001 | |

IN | 3.494 | 0.888 | 3.932 | <0.001 | |

CE | −0.044 | 0.112 | −0.399 | 0.689 | |

SO | −0.009 | 0.028 | −0.322 | 0.747 | |

0.50 | Intercept | 2.730 | 1.601 | 1.705 | 0.088 |

AS | −0.022 | 0.018 | −1.263 | 0.206 | |

CA | 0.311 | 0.322 | 0.965 | 0.334 | |

SI | −0.802 | 0.167 | −4.788 | <0.001 | |

IN | 3.644 | 0.842 | 4.323 | <0.001 | |

CE | −0.044 | 0.115 | −0.388 | 0.697 | |

SO | −0.009 | 0.029 | −0.333 | 0.738 | |

0.75 | Intercept | 3.850 | 1.715 | 2.245 | 0.024 |

AS | −0.023 | 0.019 | −1.173 | 0.240 | |

CA | 0.293 | 0.340 | 0.863 | 0.387 | |

SI | −0.855 | 0.175 | −4.869 | <0.001 | |

IN | 3.946 | 0.826 | 4.777 | <0.001 | |

CE | −0.045 | 0.122 | −0.374 | 0.707 | |

SO | −0.010 | 0.031 | −0.347 | 0.728 |

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## Share and Cite

**MDPI and ACS Style**

Iriarte, Y.A.; de Castro, M.; Gómez, H.W.
An Alternative One-Parameter Distribution for Bounded Data Modeling Generated from the Lambert Transformation. *Symmetry* **2021**, *13*, 1190.
https://doi.org/10.3390/sym13071190

**AMA Style**

Iriarte YA, de Castro M, Gómez HW.
An Alternative One-Parameter Distribution for Bounded Data Modeling Generated from the Lambert Transformation. *Symmetry*. 2021; 13(7):1190.
https://doi.org/10.3390/sym13071190

**Chicago/Turabian Style**

Iriarte, Yuri A., Mário de Castro, and Héctor W. Gómez.
2021. "An Alternative One-Parameter Distribution for Bounded Data Modeling Generated from the Lambert Transformation" *Symmetry* 13, no. 7: 1190.
https://doi.org/10.3390/sym13071190