# A New Bimodal Distribution for Modeling Asymmetric Bimodal Heavy-Tail Real Lifetime Data

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## Abstract

**:**

## 1. Introduction

- The new density in (6) can be “unimodal and right-skewed,” “symmetric and unimodal,” and “bimodal density” with many useful shapes (see Figure 1).
- The HRF of the new model can be “monotonically increasing,” “bathtub (U-HRF),” “J-shaped (J-HRF),” “monotonically decreasing,” “increasing-constant-increasing,” “reversed J-HRF,” and “upside-down (reversed U-HRF)” (see Figure 2).
- In reliability analysis, the OBBX model could be chosen as the best model, especially in modeling asymmetric bimodal failure times data and the asymmetric bimodal right-skewed and heavy-tail survival times data as illustrated in Section 5.1 and Section 5.3, respectively.
- In medical fields, the OBBX model could be chosen as the best model, especially in modeling the bimodal right-skewed and heavy-tail cancer data, as illustrated in Section 5.2.
- In engineering, the OBBX model could be chosen as the best model, especially in modeling the asymmetric bimodal left-skewed and heavy-tail glass fibers data, as shown in Section 5.4.

## 2. Mathematical Properties

#### 2.1. Useful Representations

#### 2.2. Moments and Incomplete Moments

#### 2.3. Moment Generating Function (MGF)

#### 2.4. Residual Life and Reversed Residual Life Functions

#### 2.5. Numerical Analysis for the Mean, V$\left(W\right)$, S$\left(W\right)$, K$\left(W\right)$, and DisIx$\left(W\right).$

## 3. Maximum Likelihood Estimation

## 4. Graphical Assessment

- $N=1000$ samples of size $n{|}_{\left(n=\text{}50,\text{}100,\text{}\dots ,\text{}2000\right)}$ were generated from the OBBX distribution using (7);
- The MLEs for $N=2000$ samples, say $\left[\widehat{{\nu}_{\hslash}},\widehat{{\theta}_{\hslash}},\widehat{{\left({c}_{1}\right)}_{\hslash}},\widehat{{\left({c}_{2}\right)}_{\hslash}}\right]{|}_{\left(\hslash =1,2,\dots ,2000\right)}$ were computed.
- The SEs of the MLEs for the $2000$ samples, say $\left[{S}_{\widehat{{\nu}_{\hslash}}},{S}_{\widehat{{\theta}_{\hslash}}},{S}_{\widehat{{\left({c}_{1}\right)}_{\hslash}}},{S}_{\widehat{{\left({c}_{2}\right)}_{\hslash}}}\right]{|}_{\left(\hslash =1,2,\dots ,2000\right)}$ were computed by inverting the observed information matrix.
- The biases and mean squared errors given for $\underset{\_}{\Theta}=\nu ,\theta ,{c}_{1},{c}_{2}$. We repeated these steps for $n{|}_{\left(n=50,100,\dots ,2000\right)}$ with $\nu =1,\text{}2,\dots ,\text{}100;\text{}\theta =1,\text{}2,\dots ,\text{}100;\text{}{c}_{1}=1,\text{}2,\dots ,\text{}100;\text{}{c}_{2}=1$, 2,…, 100 to compute the biases $\left({\mathrm{Bias}}_{\underset{\_}{\Theta}}\left(n\right)\right)$ and mean squared errors ($\mathrm{MSEs}$) $\left({\mathrm{MSE}}_{h}\left(n\right)\right)$ for $\underset{\_}{\Theta}=\nu ,\theta ,{c}_{1},{c}_{2}$ and $n{|}_{\left(n=50,100,\dots ,2000\right)}$ where ${\mathrm{Bias}}_{\underset{\_}{\Theta}}\left(n\right){|}_{\left(\underset{\_}{\Theta}=\nu ,\theta ,{c}_{1},{c}_{2}\right)}=\frac{1}{1000}{\displaystyle \sum}_{\hslash =1}^{1000}\left({\widehat{\underset{\_}{\Theta}}}_{\hslash}-\underset{\_}{\Theta}\right)$ and ${\mathrm{MSE}}_{\underset{\_}{\Theta}}\left(n\right){|}_{\left(\underset{\_}{\Theta}=\nu ,\theta ,{c}_{1},{c}_{2}\right)}=\frac{1}{1000}{\displaystyle \sum}_{\hslash =1}^{1000}{\left({\widehat{\underset{\_}{\Theta}}}_{\hslash}-\underset{\_}{\Theta}\right)}^{2}.$

## 5. Applications

#### 5.1. Modeling Failure Times

#### 5.2. Modeling Cancer Data

#### 5.3. Modeling Survival Times

#### 5.4. Glass Fibers Data

## 6. Concluding Remarks

- The new density can be “unimodal and right-skewed,” “symmetric and unimodal,” and “bimodal density” with many useful shapes.
- The HRF of the new model can be “monotonically increasing,” “bathtub (U-HRF),” “J-HRF,” “monotonically decreasing,” “increasing-constant-increasing,” “reversed J-HRF,” and “upside-down (reversed U-HRF).”
- In the reliability analysis, the OBBX model could be chosen as the best model, especially for modeling the asymmetric bimodal failure times data and the asymmetric bimodal right-skewed and heavy-tail survival times data.
- In medical fields, the OBBX model could be chosen as the best model, especially for modeling the bimodal right-skewed and heavy-tail cancer data.
- In engineering, the OBBX model could be chosen as the best model, especially for modeling the asymmetric bimodal left-skewed and heavy-tail glass fibers data.
- In modeling the failure times data, the OBBX model showed its superiority against the Burr type X, odd Lindley exponentiated Weibull, Burr X exponentiated Weibull, Poisson Topp Leone Weibull, Marshall Olkin extended Weibull, Gamma Weibull Kumaraswamy Weibull, beta Weibull, transmuted modified Weibull, modified beta Weibull, Mcdonald Weibull, and the transmuted exponentiated generalized Weibull distributions.
- In modeling the cancer data, the OBBX model showed its superiority against the Burr type X, Weibull, transmuted modified Weibull, modified beta Weibull, and transmuted additive Weibull distributions.
- In modeling the survival data, the OBBX model showed its superiority against the Burr type X, transmuted Burr type X, odd Lindley exponentiated Weibull, odd Weibull–Weibull, and gamma-exponentiated exponential distributions.
- In modeling the glass fibers data, the OBBX model showed its superiority against the Burr type X, transmuted Burr type X, odd Lindley Burr type X, odd Lindley exponentiated Weibull, exponentiated Weibull, transmuted Weibull, and odd log–logistic Weibull distributions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Probability density function | |

CDF | Cumulative distribution function |

RF | Reliability function |

MGF | Moment generating function |

HRF | Hazard rate function |

MLE | Maximum likelihood estimation |

MSE | Mean square error |

P-P | Probability–probability |

TTT | Total time in test |

RV | Random variable |

BX | Burr type X |

BXII | Burr type XII |

O-G | Odd G family |

OBG | Odd Burr-G family |

OBBX | Odd Burr–Burr type X model |

PRHR | Proportional reversed hazard rate family |

O-BX | Odd Burr type X |

PRHR-BX | Proportional reversed hazard rate Burr type X model |

QF | Quantile function |

DisIx | Dispersion index |

V(W) | Variance |

S(W) | Skewness |

K(W) | Kurtosis |

EPDF | Estimated probability density function |

ECDF | Estimated cumulative distribution function |

EHRF | Estimated hazard rate function |

Q-Q | Quantile–quantile plot |

NKDE | Nonparametric kernel density estimation plot |

## Appendix A

**R Codes for Applications:**

- (c1)))^v))^theta))

- 2.
- #=======================================

**R Codes for Simulations:**

- 3.
- (c1)))^v))^theta))

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**Figure 1.**Different plots of the odd Burr BX (OBBX) probability density function (PDF) for selected parameter values.

**Figure 2.**(

**a**) Increasing HRF, (

**b**) Bathtub HRF (U-HRF), (

**c**) J-Shaped HRF (J-HRF), (

**d**) Decreasing HRF, (

**e**) Increasing–Constant–Increasing, (

**f**) Reversed J-HRF, (

**g**) Upside-down (Reversed U-HRF).

**Figure 10.**Estimated probability density function (EPDF), estimated cumulative distribution function (ECDF), probability–probability (P-P), and estimate hazard rate function (EHRF) plots for data set I.

**Table 1.**Numerical results for the mean, variance, skewness, kurtosis, and dispersion index for selected parameter values.

$\mathbf{\nu}$ | $\mathbf{\theta}$ | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | $\mathbf{E}\left(\mathit{W}\right)$ | $\mathbf{V}\left(\mathit{W}\right)$ | $\mathbf{S}\left(\mathit{W}\right)$ | $\mathbf{K}\left(\mathit{W}\right)$ | $\mathbf{DisIx}\left(\mathit{W}\right)$ |
---|---|---|---|---|---|---|---|---|

1 | 2 | 1.5 | 0.5 | 1.3958 | 1.0652 | 1.0068 | 3.6312 | 0.7631 |

1 | 1.5769 | 0.4230 | 0.4614 | 3.1031 | 0.2682 | |||

5 | 1.8855 | 0.0271 | −0.3677 | 3.7912 | 0.0144 | |||

30 | 1.9753 | 0.0008 | −0.5491 | 4.2589 | 0.0004 | |||

50 | 1.9828 | 0.0003 | −0.5608 | 4.2901 | 0.0001 | |||

2 | 0.5 | 0.5 | 1.5 | 0.5220 | 0.0704 | 0.9528 | 4.2278 | 0.1348 |

1 | 0.3768 | 0.0305 | 0.7423 | 3.9547 | 0.0809 | |||

5 | 0.1930 | 0.0070 | 0.3053 | 2.8543 | 0.0365 | |||

50 | 0.0731 | 0.0012 | 0.3940 | 2.5613 | 0.0164 | |||

100 | 0.0535 | 0.0007 | 0.4433 | 2.8959 | 0.0126 | |||

200 | 0.0389 | 0.0004 | 0.4866 | 4.0305 | 0.0095 | |||

500 | 0.0252 | 0.0002 | 0.2288 | 1.8690 | 0.0064 | |||

1000 | 0.0181 | 8.4 × 10^{−5} | 1.8057 | 1.5059 | 0.0047 | |||

5 | 5 | 0.5 | 1.8 | 0.2391 | 0.0016 | −0.4656 | 3.4042 | 0.0069 |

1 | 0.4079 | 0.0018 | −0.6485 | 3.8647 | 0.0044 | |||

5 | 0.7632 | 0.0012 | −0.7280 | 4.1308 | 0.0015 | |||

50 | 1.1437 | 0.0006 | −0.8281 | 13.3165 | 0.0005 | |||

100 | 1.2381 | 0.0005 | −0.6902 | 4.0299 | 0.0004 | |||

500 | 1.4343 | 0.0004 | −0.6746 | 3.9872 | 0.0003 | |||

1000 | 1.5111 | 0.0004 | −0.6696 | 3.9739 | 0.0002 | |||

2000 | 1.5843 | 0.0003 | −0.6657 | 3.9787 | 0.0002 | |||

15 | 10 | 10 | 0.3 | 1.2 × 10^{−5} | 41.2151 | 1.0002 | 1.0005 | 3,376,320 |

0.5 | 3.2097 | 0.0013 | −0.9760 | 4.8669 | 0.0004 | |||

1 | 1.6049 | 0.0003 | −0.9812 | 5.3298 | 0.0002 | |||

2 | 0.8024 | 8.2 × 10^{−5} | −0.9760 | 4.8665 | 0.0001 | |||

3 | 0.5350 | 3.6 × 10^{−5} | −0.9760 | 4.8591 | 6.82 × 10^{−5} | |||

5 | 1.9 × 10^{−6} | 6.5 × 10^{−7} | 414.9644 | 172,196.5 | 0.3333 |

Distribution | Estimates (SEs) | ||||
---|---|---|---|---|---|

BX(c_{1};c_{2}) | 1.181876 | 0.377525 | |||

(0.17060) | (0.02532) | ||||

OLEW(θ;c_{1};c_{2}) | 0.15935 | 0.7322 | 0.765 | ||

(0.3712) | (1.778) | (0.041) | |||

OLBX(θ;c_{1};c_{2}) | 1.45406 | 0.7543 | 0.2379 | ||

(0.9018) | (0.2530) | (0.0317) | |||

BXEW(θ;c_{1};c_{2}) | 0.63684 | 4.2622 | 0.5364 | ||

(0.356) | (1.757) | (0.0997) | |||

PTLW(θ;c_{1};c_{2}) | −5.78175 | 4.22865 | 0.65801 | ||

(1.395) | (1.167) | (0.039) | |||

MOEW(θ;c_{1};c_{2}) | 488.899 | 0.2832 | 1261.97 | ||

(189.358) | (0.013) | (351.07) | |||

GamW(θ;c_{1};c_{2}) | 2.37697 | 0.84809 | 3.5344 | ||

(0.378) | (0.00053) | (0.665) | |||

OBBX(ν;θ;c_{1};c_{2}) | 1.29102 | 3.1331 | 0.8448 | 0.1906 | |

(0.544) | (2.3251) | (0.4961) | (0.084) | ||

KumW(ν;θ;c_{1};c_{2}) | 14.4331 | 0.2041 | 34.6599 | 81.8459 | |

(27.095) | (0.042) | (17.527) | (52.014) | ||

Beta-W(ν;θ;c_{1};c_{2}) | 1.36 | 0.2981 | 34.1802 | 11.4956 | |

(1.002) | (0.06) | (14.838) | (6.73) | ||

TrMW(ν;θ;c_{1};c_{2}) | 0.2722 | 1 | 4.6 × 10^{−6} | 0.4685 | |

(0:014) | (5.2 × 10^{−5}) | (1.9 × 10^{−4}) | (0.165) | ||

MBW(ν;θ;λ;c_{1};c_{2}) | 10.1502 | 0.1632 | 57.4167 | 19.3859 | 2.0043 |

(18.697) | (0.019) | (14.063) | (10.019) | (0.662) | |

MacW(ν;θ;λ;c_{1};c_{2}) | 1.9401 | 0.306 | 17.686 | 33.6388 | 16.7211 |

(1.011) | (0.045) | (6.222) | (19.994) | (9.722) | |

TrEGW(ν;θ;λ;c_{1};c_{2}) | 4.2567 | 0.1532 | 0.0978 | 5.2313 | 1173.33 |

(33.401) | (0.017) | (0.609) | (9.792) | (6.999) |

**Table 3.**The Cramér–von Mises (CVM), Anderson–Darling (AD), and Kolmogorov–Smirnov (KS) (p-value) results for data set I.

Distribution | CVM | AD | KS (p-Value) |
---|---|---|---|

OBBX | 0.0580 | 0.5777 | 0.05602 (0.9547) |

OLEW | 0.0723 | 0.6086 | 0.87572 (<0.001) |

OLBX | 0.0792 | 0.5910 | 0.37584 (<0.001) |

BX | 0.0690 | 0.6916 | 0.07981 (0.6584) |

BXEW | 0.0744 | 0.6420 | 0.06935 (0.8139) |

PTLW | 0.1397 | 1.1939 | 0.11542 (0.8004) |

MOEW | 0.3995 | 4.4477 | 0.06170 (0.9064) |

GamW | 0.2553 | 1.9489 | 0.58482 (0.33119) |

KumW | 0.1852 | 1.5059 | 0.23917 (0.43651) |

Beta-W | 0.4652 | 3.2197 | 0.66032 (<0.001) |

TrMW | 0.8065 | 11.2047 | 0.68989 (<0.001) |

MBW | 0.4717 | 3.2656 | 0.33902 (<0.001) |

MacW | 0.1986 | 1.5906 | 0.09243 (0.81193) |

TrEGW | 1.0079 | 6.2332 | 0.22402 (<0.001) |

Distribution | Estimates (SEs) | ||||
---|---|---|---|---|---|

BX(c_{1};c_{2}) | 0.36413 | 0.04763 | |||

(0.0373) | (0.0039) | ||||

W(c_{1};c_{2}) | 9.5593 | 1.0477 | |||

(0.853) | (0.068) | ||||

OBBX(ν;θ;c_{1};c_{2}) | 5.6822 | 1.61069 | 0.0688 | 0.00067 | |

(1.0574) | (0.7171) | (0.0066) | (<0.001) | ||

TrMW(ν;θ;c_{1};c_{2}) | 0.1208 | 0.8955 | 0.0002 | 0.2513 | |

(0.024) | (0.626) | (0.011) | (0.407) | ||

MBW(ν;θ;λ;c_{1};c_{2}) | 0.1502 | 0.1632 | 57.4167 | 19.3859 | 2.0043 |

(22.437) | (0.044) | (37.317) | (13.49) | (0.789) | |

TrAW(ν;θ;λ;c_{1};c_{2}) | 0.1139 | 0.9722 | 3.09 × 10^{−5} | 1.0065 | −0.163 |

(0.032) | (0.125) | (6.12 × 10^{−3}) | (0.035) | (0.28) |

Distribution | CVM | AD | KS (p-Value) |
---|---|---|---|

OBBX | 0.0345 | 0.2038 | 0.04242 (0.9754) |

W | 0.1055 | 0.6628 | 0.2665 (0.00662) |

BX | 0.4747 | 2.7861 | 0.35516 (0.0042) |

TrMW | 0.1251 | 0.7603 | 0.15875 (0.3969) |

MBW | 0.1068 | 0.7207 | 0.25762 (0.3198) |

TrAW | 0.1129 | 0.7033 | 0.16872 (0.3376) |

Distribution | Estimates (SEs) | |||
---|---|---|---|---|

BX(c_{1};c_{2}) | 0.93658 | 0.00478 | ||

(0.1461) | (0.0004) | |||

TrBX(θ;c_{1};c_{2}) | 0.6328 | 1.03917 | 0.00417 | |

(0.2453) | (0.1445) | (0.0005) | ||

OLEW(ν;θ;c_{1};c_{2}) | 0.0018 | 0.0716 | 0.2813 | |

(0.0004) | (0.025) | (0.009) | ||

OWW(θ;c_{1};c_{2}) | 11.1576 | 0.0881 | 0.457 | |

(4.5449) | (0.036) | (0.08) | ||

GaEE(θ;c_{1};c_{2}) | 2.1138 | 2.6006 | 0.0083 | |

(1.3288) | (0.5597) | (0.005) | ||

OBBX(ν;θ;c_{1};c_{2}) | 3.24855 | 0.40541 | 0.2905 | 0.0029 |

(0.8191) | (0.1563) | (0.039) | (0.0003) |

Distribution | CVM | AD | KS (p-Value) |
---|---|---|---|

OBBX | 0.0549 | 0.34601 | 0.06916 (0.8812) |

BX | 0.1849 | 1.08578 | 0.096596 (0.5125) |

TrBX | 0.1352 | 0.79259 | 0.085663 (0.6662) |

OLEW | 0.2517 | 1.47502 | 0.999870 (<0.001) |

OWW | 0.4494 | 2.47640 | 0.658701 (<0.001) |

GaEE | 0.3150 | 1.72080 | 0.508710 (<0.001) |

Distribution | Estimates (SEs) | |||
---|---|---|---|---|

BX(c_{1};c_{2}) | 5.48597 | 0.9868 | ||

(1.1853) | (0.0540) | |||

TrBX(θ;c_{1};c_{2}) | −0.65524 | 4.78605 | 1.04504 | |

(0.19529) | (1.2831) | (0.0549) | ||

OLBX(θ;c_{1};c_{2}) | 0.65831 | 2.1019 | 0.8429 | |

(0.5112) | (1.246) | (0.058) | ||

OLEW(θ;c_{1};c_{2}) | 0.50878 | 2.534 | 1.7122 | |

(0.397) | (1.8298) | (0.0959) | ||

EW(θ;c_{1};c_{2}) | 0.67132 | 7.285 | 1.71811 | |

(0.249) | (1.707) | (0.086) | ||

TrW(θ;c_{1};c_{2}) | −0.5010 | 5.1498 | 0.6458 | |

(0.2741) | (0.6657) | (0.0235) | ||

OLLW(θ;c_{1};c_{2}) | 0.9439 | 6.0256 | 0.6159 | |

(0.2689) | (1.3478) | (0.0164) | ||

OBBX(ν;θ;c_{1};c_{2}) | 5.8221 | 6.9739 | 0.3384 | 0.1739 |

(6.1527) | (8.4085) | (0.449) | (0.2701) |

Distribution | CVM | AD | KS (p-Value) |
---|---|---|---|

OBBX | 0.2041 | 1.1216 | 0.14084 (0.1642) |

OLBX | 0.2557 | 1.4154 | 0.62469 (<0.001) |

TrBX | 0.4764 | 2.6163 | 0.19385 (0.01757) |

OLEW | 0.2711 | 1.4965 | 0.66225 (<0.001) |

BX | 0.5594 | 3.0722 | 0.21497 (0.00592 |

EW | 0.6361 | 3.4842 | 0.15895 (<0.001) |

TrW | 1.0358 | 0.1691 | 0.33359 (0.07651) |

OLLW | 1.2364 | 0.2194 | 0.59001 (0.01543) |

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## Share and Cite

**MDPI and ACS Style**

Butt, N.S.; Khalil, M.G.
A New Bimodal Distribution for Modeling Asymmetric Bimodal Heavy-Tail Real Lifetime Data. *Symmetry* **2020**, *12*, 2058.
https://doi.org/10.3390/sym12122058

**AMA Style**

Butt NS, Khalil MG.
A New Bimodal Distribution for Modeling Asymmetric Bimodal Heavy-Tail Real Lifetime Data. *Symmetry*. 2020; 12(12):2058.
https://doi.org/10.3390/sym12122058

**Chicago/Turabian Style**

Butt, Nadeem S., and Mohamed G. Khalil.
2020. "A New Bimodal Distribution for Modeling Asymmetric Bimodal Heavy-Tail Real Lifetime Data" *Symmetry* 12, no. 12: 2058.
https://doi.org/10.3390/sym12122058