# A Unimodal/Bimodal Skew/Symmetric Distribution Generated from Lambert’s Transformation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The LGB Distribution

#### 2.1. LGB Random Variable

**Definition**

**1.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

#### 2.2. Related Distributions

## 3. Shapes and Aliases

#### 3.1. Shapes

#### 3.2. Skewness and Kurtosis

**Proposition**

**2.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

- The LGB pdf is symmetric when α = 1, regardless of the value assumed by the parameter γ.
- The LGB distribution can be skewed (positively or negatively) depending on the value assumed by α. If $\alpha \in (0,1)$ or $\alpha \in (1,e)$, then the LGB pdf is skewed and the skewness is also controlled by the parameter γ. However, the effect of γ on skewness is important when it assumes small values.
- If the LGB pdf is unimodal, then it is asymmetric.
- In the unimodal case, the excess kurtosis ${\beta}_{2}(\gamma ,\alpha )-3$ is less than 0; that is, the LGB distribution is a platykurtic distribution.

#### 3.3. Alias Distributions

**Proposition**

**3.**

**Proof.**

## 4. Maximum Likelihood Estimator

## 5. Simulation Studies

#### 5.1. First Simulation Study

- Scenario A: $\mu =5$, $\sigma =2$, $\gamma =0.5$ and $\alpha =0.5$.
- Scenario B: $\mu =-5$, $\sigma =4$, $\gamma =0.75$ and $\alpha =1.5$.

#### 5.2. Second Simulation Study

## 6. Data Analysis

- The first dataset corresponds to 188 observations on the inflation rate (in %) registered quarterly between the years 1950 and 1996 in Canada. This dataset can be found with the name Tbrate in the R language [31].
- The second dataset refers to 128 observations on the electrical resistance (in ohms) of nectarine fruits. This data can be found with the name fruitohms in the R language [32].

## 7. Final Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Probability density function | |

cdf | Cumulative distribution function |

qf | Quantile function |

AIC | Akaike information criterion |

CAIC | Corrected Akaike information criterion |

BIC | Bayesian informetion criterion |

MN | Mixture normal |

GB | Generalized bimodal |

GSC | Gamma sinh-Cauchy |

OLLSN | Odd log-logistic skew-normal |

LGB | Lambert generalized bimodal |

## Appendix A. R Codes to Compute the qf of the LGB Distribution and Generate Pseudo-Random Numbers

`+ qgbimodal <- function(p,gamma){`

`+ n = length(p)`

`+ f = rep(0,n)`

`+ for(i in 1:n){`

`+ f[i] = uniroot(function(x,gamma)pnorm(x)-x/(gamma+1)*dnorm(x)-p[i],`

`+ c(-1e+4,1e+4),gamma=gamma)$‘root‘`

`+ }`

`+ return(f)`

`+ }`

`+`

`+ library(LambertW)`

`+`

`+ qLGB <- function(p,mu,sigma,gamma,alpha){`

`+ if(alpha==1){`

`+ mu+sigma*qgbimodal(p,gamma)`

`+ }else{`

`+ pp = 1/log(alpha)*W(log(alpha)*(p-1)/alpha)+1`

`+ mu+sigma*qgbimodal(pp,gamma)`

`+ }`

`+ }`

`+`

`+ n <- 100; p <- runif(n); mu <- 5; sigma <- 3; gamma <- 0.5; alpha <- 1.5`

`+ x <- qLGB(p,mu,sigma,gamma,alpha)`

## Appendix B. System of Equations to Minimize the Kullback–Leibler Divergence with Respect to θ_{2}

## Appendix C. Graphical Comparison of the pdf of X_{1} and X_{2} in Scenarios A to F

**Figure A1.**Pdf curves of ${X}_{1}$ (black solid line) and ${X}_{2}$ (red dashed line) in scenarios A to F.

## Appendix D. System of Equations to Obtain the ML Estimates Based on a Random Sample of Size n from a LGB(μ, σ, γ, α) Population

## Appendix E. Second Partial Derivatives of the Log-Likelihood Function for a Single Observation of the LGB Distribution

## References

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**Figure 1.**Pdf curves for the LGB distribution for $\mu =5$, $\sigma =2$, and $\alpha =1$ in the top left panel; $\mu =-5$, $\sigma =4$, and $\alpha =0.5$ in the top right panel; $\mu =5$, $\sigma =4$, and $\alpha =2$ in the bottom left panel; and $\mu =10$, $\sigma =5$, and $\gamma =1.5$ in the bottom right panel.

**Figure 2.**Critical and inflection points and pdf curve for the distributions Lambert generalized bimodal (LGB) (5,2,1.5,0.005) (black curves), LGB (5,2,1.5,0.5) (red curves), LGB (5,2,0.5,0.5) (green curves), and LGB (5,2,0.5,1) (blue curves).

**Figure 3.**Regions of unimodality (white region) and bimodality (gray region) for a LGB distribution.

**Figure 5.**Top and center panels: Skewness curves for two LGB distributions with $\alpha =1.6$ in blue color and $\alpha =0.001$ in red color. Right panel: Pdf curves for an LGB (0,1,1000,1.6) distribution (in black color) and four LGB distributions specified by $\mu =2.360$, $\sigma =1.487$, $\gamma =0.001$, and different values of $\gamma $.

**Figure 6.**Kullback–Leibler divergence curve for ${X}_{1}\sim \mathrm{LGB}(0,1,1000,1.6)$ and ${X}_{2}\sim \mathrm{LGB}(2.360,1.487,1000,0.001)$ as a function of ${\mu}_{2}$ in the left panel and as a function of ${\sigma}_{2}$ in the right panel.

**Figure 7.**Left panels: Histogram for inflation rate data and the fitted pdf curves via the ML method. Right panels: Empirical cdf for the inflation rate data and the fitted cdf curves.

**Figure 8.**Left panels: Histogram for electrical resistance data and the fitted pdf curves via the ML method. Right panels: Empirical cdf for the electrical resistance data and the fitted cdf curves.

**Table 1.**Some values for the first four raw moments of the LGB distribution considering different values of γ and α.

Parameters | Moments | ||||
---|---|---|---|---|---|

$\mathit{\gamma}$ | $\mathit{\alpha}$ | $\mathbb{E}\left(\mathit{Z}\right)$ | $\mathbb{E}\left({\mathit{Z}}^{\mathbf{2}}\right)$ | $\mathbb{E}\left({\mathit{Z}}^{\mathbf{3}}\right)$ | $\mathbb{E}\left({\mathit{Z}}^{\mathbf{4}}\right)$ |

0.2 | 0.5 | −0.5504 | 2.7657 | −2.6574 | 13.7983 |

0.4 | 1.0 | 0.0000 | 2.4285 | 0.0000 | 11.5714 |

0.6 | 2.0 | 0.7173 | 2.4001 | 3.1908 | 11.5966 |

0.8 | 0.5 | −0.4906 | 2.2032 | −2.1096 | 10.3165 |

1.0 | 1.0 | 0.0000 | 2.0000 | 0.0000 | 9.0000 |

1.2 | 2.0 | 0.6592 | 2.0483 | 2.6865 | 9.3859 |

1.4 | 0.5 | −0.455 | 1.9191 | −1.8137 | 8.5612 |

1.6 | 1.0 | 0.0000 | 1.7692 | 0.0000 | 7.6153 |

1.8 | 2.0 | 0.6229 | 1.8451 | 2.3840 | 8.1118 |

**Table 2.**Mean, variance, skewness, and amounts of critical points and inflection points of the pdf for the distributions of ${X}_{1}$ and ${X}_{2}$ in scenarios A to F.

Variable | Scenario | Mean | Variance | Skewness | Number of Critical Points | Number of Inflexion Points |
---|---|---|---|---|---|---|

${X}_{1}$ | A | 0.271 | 1.680 | 0.271 | 1 | 4 |

${X}_{2}$ | A | 0.361 | 1.834 | 0.361 | 1 | 2 |

${X}_{1}$ | B | −0.988 | 1.749 | 0.685 | 2 | 4 |

${X}_{2}$ | B | −0.954 | 2.097 | 0.430 | 1 | 2 |

${X}_{1}$ | C | 0.365 | 1.910 | −0.235 | 2 | 4 |

${X}_{2}$ | C | 0.415 | 2.107 | 0.230 | 1 | 2 |

${X}_{1}$ | D | 0.639 | 1.526 | −0.355 | 1 | 4 |

${X}_{2}$ | D | 0.648 | 1.626 | 0.262 | 1 | 2 |

${X}_{1}$ | E | 0.481 | 0.904 | −0.172 | 1 | 2 |

${X}_{2}$ | E | 0.486 | 0.912 | −0.043 | 1 | 2 |

${X}_{1}$ | F | 0.475 | 0.880 | −0.170 | 1 | 2 |

${X}_{2}$ | F | 0.476 | 0.895 | −0.052 | 1 | 2 |

**Table 3.**Averages (AE), standard deviations (SD), and root of the simulated mean square errors (RMSE) for the estimates of $\mu $, $\sigma $, $\gamma $, and $\alpha $ of the LGB distribution.

n | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\alpha}}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

AE | SD | RMSE | AE | SD | RMSE | AE | SD | RMSE | AE | SD | RMSE | |

Scenario A | ||||||||||||

100 | 5.005 | 0.329 | 0.329 | 1.992 | 0.126 | 0.126 | 0.497 | 0.260 | 0.260 | 0.517 | 0.168 | 0.169 |

200 | 5.004 | 0.234 | 0.235 | 1.993 | 0.092 | 0.091 | 0.498 | 0.177 | 0.177 | 0.508 | 0.120 | 0.119 |

300 | 5.003 | 0.192 | 0.192 | 1.995 | 0.076 | 0.076 | 0.498 | 0.139 | 0.139 | 0.504 | 0.097 | 0.097 |

500 | 5.001 | 0.148 | 0.148 | 1.997 | 0.058 | 0.058 | 0.499 | 0.106 | 0.106 | 0.503 | 0.077 | 0.077 |

1000 | 5.001 | 0.107 | 0.107 | 1.999 | 0.041 | 0.041 | 0.500 | 0.073 | 0.073 | 0.500 | 0.052 | 0.052 |

Scenario B | ||||||||||||

100 | −4.943 | 0.774 | 0.776 | 3.977 | 0.284 | 0.285 | 0.745 | 0.413 | 0.412 | 1.478 | 0.315 | 0.316 |

200 | −4.984 | 0.556 | 0.556 | 3.989 | 0.217 | 0.217 | 0.748 | 0.276 | 0.276 | 1.505 | 0.233 | 0.233 |

300 | −4.996 | 0.464 | 0.464 | 3.996 | 0.167 | 0.167 | 0.749 | 0.227 | 0.227 | 1.504 | 0.193 | 0.193 |

500 | −4.999 | 0.342 | 0.342 | 3.997 | 0.126 | 0.126 | 0.750 | 0.160 | 0.160 | 1.502 | 0.143 | 0.143 |

1000 | −5.000 | 0.236 | 0.236 | 3.999 | 0.094 | 0.094 | 0.750 | 0.115 | 0.115 | 1.500 | 0.101 | 0.101 |

**Table 4.**Averages of asymptotic standard errors (SE) and coverage probabilities (CP) for the estimates of $\mu $, $\sigma $, $\gamma $, and $\alpha $ for the LGB distribution.

n | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\alpha}}$ | ||||
---|---|---|---|---|---|---|---|---|

SE | CP | SE | CP | SE | CP | SE | CP | |

Scenario A | ||||||||

100 | 0.329 | 0.947 | 0.129 | 0.942 | 0.247 | 0.903 | 0.169 | 0.939 |

200 | 0.234 | 0.948 | 0.092 | 0.943 | 0.170 | 0.924 | 0.118 | 0.943 |

300 | 0.188 | 0.948 | 0.074 | 0.944 | 0.135 | 0.929 | 0.096 | 0.947 |

500 | 0.145 | 0.955 | 0.057 | 0.947 | 0.104 | 0.939 | 0.075 | 0.949 |

1000 | 0.103 | 0.956 | 0.041 | 0.954 | 0.073 | 0.950 | 0.053 | 0.953 |

Scenario B | ||||||||

100 | 0.798 | 0.931 | 0.293 | 0.936 | 0.388 | 0.883 | 0.336 | 0.939 |

200 | 0.559 | 0.938 | 0.205 | 0.937 | 0.258 | 0.921 | 0.237 | 0.944 |

300 | 0.455 | 0.945 | 0.168 | 0.954 | 0.210 | 0.929 | 0.193 | 0.945 |

500 | 0.347 | 0.955 | 0.128 | 0.955 | 0.158 | 0.945 | 0.148 | 0.953 |

1000 | 0.245 | 0.956 | 0.091 | 0.955 | 0.111 | 0.947 | 0.105 | 0.957 |

**Table 5.**Non-rejection rate based on modified statistics ${W}^{*}$ and ${A}^{*}$ and hit rates based on the AIC, CAIC, and BIC for the LGB distribution, when the data are simulated from the MN distribution.

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

Rate | Rate | ||||||||||

n | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ | AIC | CAIC | BIC | n | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ | AIC | CAIC | BIC |

Scenario A | Scenario C | ||||||||||

50 | 0.993 | 0.994 | 0.730 | 0.798 | 0.903 | 50 | 0.998 | 0.996 | 0.721 | 0.763 | 0.871 |

100 | 0.996 | 0.994 | 0.686 | 0.736 | 0.936 | 100 | 0.997 | 0.997 | 0.704 | 0.737 | 0.906 |

200 | 0.992 | 0.988 | 0.584 | 0.608 | 0.941 | 200 | 0.966 | 0.988 | 0.606 | 0.622 | 0.895 |

300 | 0.993 | 0.990 | 0.531 | 0.549 | 0.936 | 300 | 0.989 | 0.978 | 0.511 | 0.519 | 0.840 |

Scenario B | Scenario D | ||||||||||

50 | 0.995 | 0.993 | 0.732 | 0.793 | 0.874 | 50 | 0.980 | 0.978 | 0.507 | 0.566 | 0.728 |

100 | 0.998 | 0.996 | 0.692 | 0.719 | 0.911 | 100 | 0.951 | 0.951 | 0.427 | 0.451 | 0.696 |

200 | 0.992 | 0.990 | 0.583 | 0.608 | 0.902 | 200 | 0.892 | 0.886 | 0.297 | 0.308 | 0.600 |

300 | 0.991 | 0.984 | 0.505 | 0.525 | 0.875 | 300 | 0.781 | 0.758 | 0.198 | 0.205 | 0.464 |

**Table 6.**Non-rejection rate based on modified statistics ${W}^{*}$ and ${A}^{*}$ and hit rates based on the AIC, CAIC, and BIC for the MN distribution, when the data are simulated from the LGB distribution.

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

Rate | Rate | ||||||||||

n | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ | AIC | CAIC | BIC | n | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ | AIC | CAIC | BIC |

Scenario A | Scenario C | ||||||||||

50 | 0.995 | 0.996 | 0.302 | 0.229 | 0.122 | 50 | 0.999 | 0.999 | 0.294 | 0.225 | 0.130 |

100 | 0.995 | 0.996 | 0.244 | 0.214 | 0.065 | 100 | 0.999 | 0.999 | 0.227 | 0.199 | 0.074 |

200 | 0.995 | 0.994 | 0.229 | 0.221 | 0.044 | 200 | 0.999 | 0.999 | 0.211 | 0.198 | 0.031 |

300 | 0.996 | 0.995 | 0.207 | 0.203 | 0.031 | 300 | 0.999 | 0.999 | 0.206 | 0.197 | 0.030 |

Scenario B | Scenario D | ||||||||||

50 | 0.999 | 0.999 | 0.241 | 0.189 | 0.098 | 50 | 0.098 | 0.999 | 0.298 | 0.255 | 0.141 |

100 | 0.998 | 0.988 | 0.250 | 0.232 | 0.044 | 100 | 0.998 | 0.988 | 0.266 | 0.233 | 0.076 |

200 | 0.998 | 0.993 | 0.222 | 0.208 | 0.042 | 200 | 0.999 | 0.999 | 0.210 | 0.193 | 0.034 |

300 | 9.997 | 0.993 | 0.172 | 0.171 | 0.026 | 300 | 0.999 | 0.999 | 0.200 | 0.193 | 0.028 |

**Table 7.**The ML estimates and their standard errors (in parentheses) for each distribution fitted to the inflation rate and electrical resistance data and the values of the statistics ${W}^{*}$ and ${A}^{*}$ and of the information criteria.

Distribution | $\widehat{\mathit{\mu}}$ | ${\widehat{\mathit{\mu}}}_{2}$ | $\widehat{\mathit{\sigma}}$ | ${\widehat{\mathit{\sigma}}}_{2}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\alpha}}$ | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ | AIC | CAIC | BIC |
---|---|---|---|---|---|---|---|---|---|---|---|

Inflation rate data | |||||||||||

LGB | 6.682 | - | 2.545 | - | 0.434 | 0.183 | 0.045 | 0.248 | 962.5 | 962.7 | 975.4 |

(0.307) | (0.131) | (0.175) | (0.060) | ||||||||

MN | 2.600 | 9.086 | 2.001 | 2.026 | - | 0.777 | 0.040 | 0.246 | 965.5 | 965.9 | 981.7 |

(0.220) | (0.585) | (0.161) | (0.410) | (0.045) | |||||||

OLLSN | 0.834 | - | 4.129 | - | 3.003 | 0.772 | 0.085 | 0.558 | 973.7 | 973.9 | 986.6 |

(1.048) | (1.401) | (0.880) | (0.294) | ||||||||

GSC | 6.441 | - | 1.176 | - | 0.177 | 0.572 | 0.158 | 1.083 | 982.1 | 982.2 | 995.0 |

(0.274) | (0.104) | (0.044) | (0.065) | ||||||||

GB | 4.454 | - | 2.631 | - | 2.000 | - | 0.610 | 3.318 | 996.7 | 996.8 | 1006.4 |

(0.379) | (0.230) | (1.321) | |||||||||

LBN | 6.381 | - | 2.248 | - | - | 0.247 | 0.254 | 1.356 | 987.3 | 987.4 | 997.0 |

(0.073) | (0.065) | (0.055) | |||||||||

BN | 6.227 | - | 2.314 | - | - | - | 0.244 | 1.417 | 1046.3 | 1046.3 | 1052.8 |

(0.083) | (0.073) | ||||||||||

Electrical resistance data | |||||||||||

LGB | 5215.168 | - | 1163.235 | - | 0.075 | 0.379 | 0.089 | 0.523 | 2239.1 | 2239.4 | 2250.5 |

(96.230) | - | (51.339) | - | (0.070) | (0.102) | ||||||

MN | 3208.166 | 725.681 | 6666.668 | 1158.101 | - | 0.672 | 0.048 | 0.355 | 2241.6 | 2242.1 | 2255.8 |

(87.272) | (62.698) | (250.758) | (218.172) | (0.047) | |||||||

OLLSN | 5039.000 | - | 642.500 | - | −0.296 | 0.216 | 0.627 | 3.113 | 2277.6 | 2277.6 | 2289.0 |

(12.787) | (0.991) | (0.006) | (0.015) | ||||||||

GSC | 5024.639 | - | 5218.963 | - | 0.083 | 0.788 | 0.211 | 1.168 | 2244.2 | 2244.5 | 2255.6 |

(102.417) | (48.691) | (0.028) | (0.078) | ||||||||

GB | 4980.008 | - | 1157.944 | - | 0.085 | - | 0.297 | 1.597 | 2256.2 | 2256.4 | 2264.7 |

(89.904) | (47.155) | (0.061) | |||||||||

LBN | 5182.212 | - | 1132.943 | - | - | 0.400 | 0.085 | 0.525 | 2239.0 | 2239.1 | 2247.5 |

(60.406) | (41.002) | (0.097) | |||||||||

BN | 5085.210 | - | 1148.593 | - | - | - | 0.209 | 1.191 | 2257.0 | 2257.1 | 2262.7 |

(54.338) | (42.991) |

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## Share and Cite

**MDPI and ACS Style**

Iriarte, Y.A.; de Castro, M.; Gómez, H.W.
A Unimodal/Bimodal Skew/Symmetric Distribution Generated from Lambert’s Transformation. *Symmetry* **2021**, *13*, 269.
https://doi.org/10.3390/sym13020269

**AMA Style**

Iriarte YA, de Castro M, Gómez HW.
A Unimodal/Bimodal Skew/Symmetric Distribution Generated from Lambert’s Transformation. *Symmetry*. 2021; 13(2):269.
https://doi.org/10.3390/sym13020269

**Chicago/Turabian Style**

Iriarte, Yuri A., Mário de Castro, and Héctor W. Gómez.
2021. "A Unimodal/Bimodal Skew/Symmetric Distribution Generated from Lambert’s Transformation" *Symmetry* 13, no. 2: 269.
https://doi.org/10.3390/sym13020269