# On Construction and Estimation of Mixture of Log-Bilal Distributions

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## Abstract

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## 1. Introduction

- To construct a new two-component mixture of a LB distribution which has simple- and closed-form equations for its statistical characteristic.
- We illustrate some graphs of the unimodal and bimodal cases of the mixture model density and hazard rate functions.
- The properties of the MLBDs are obtained in explicit forms without any special mathematical functions.
- The main focus of this work is to analyze the different method of estimation and to carry out a comparative study for estimation for the mixture model. This comparison will be expressed with the help of statistical graphs.
- The feasibility and effectiveness of this model is proven through the simulation study and a real dataset.

## 2. Model Analysis and General Properties of the MLBDs

_{i}denotes the scale parameter. The CDF (Cumulative Distribution Function) of the MLBD is

#### 2.1. Mean and Variance:

#### 2.2. k^{th} Moments

^{th}Moments of theMLBDsis presented as

#### 2.3. m^{th} Order Negative Moments

^{th}Order Negative Moment can be simply obtained by substituting k with “m” in (8), as shown below

#### 2.4. Factorial Moments: The Factorial Moments Can Be Measured Using [26] Result as Given

#### 2.5. Mode and Median

#### 2.6. Incomplete Moments

^{th}Incomplete Moment of Y is

## 3. Estimation

#### 3.1. Maximum Likelihood

#### 3.2. Least Squares

#### 3.3. Weighted Least Squares

## 4. Simulation Study and Comparisons

- Utilizing various weighting factor $\delta $ and model parameters for the unimodal $\left\{\left({\xi}_{1},{\xi}_{2},\delta \right)=a\left(0.15,0.30,0.4\right),b\left(1.25,0.5,0.6\right),c\left(1.15,1.3,0.4\right)\right\}$ and bimodal $\left({\xi}_{1},{\xi}_{2},\delta \right)=e\left(1.8,0.3,0.6\right)$ scenarios, develop random samples of sizes $30,40,\dots ,800$ from the mixture model MLBDs. The random samples for the simulation are obtained in the following step.
- Start generating one variable u from the $U(0,1)$ distribution using (runif) in R.
- If $u\le \delta ,$ then we create a random variable from the first component (LBD with ${\xi}_{1}$). If $u>\delta $, we develop a random variable from the second component (LBD with ${\xi}_{2}$).
- Continue with (2) till we have the requisite sample of size n.
- Using 1000 replications, keep repeating steps 1 to 4 again. Compute the MLEs, LSEs, and WLSEs for the 1000 samples; if ${\tilde{\mathrm{\Theta}}}_{j}$ for $j=1,2,\dots ,1000,$ to acquire numerical outcomes for the simulation experiment, the statistical software R is employed. The following quantities are used to interpret the simulation results.

## 5. Empirical Studies

#### Result Reveals from the Analysis of the Dataset

- Table 2 reveals that the MLBDs distribution contains the lowest scores with the highest value of the log-likelihood function when compared to certain other distributions on all information metrics.
- Furthermore, when the distribution is the MLBDs, the value of l (.) is the highest. As a result, we can conclude that MLBDs better fits the trade share dataset.
- The PP plot in Figure 11 indicates that the proposed model is a good match and model for dataset.
- The estimated CDF and SF of the model plots are shown in Figure 11 indicate that the proposed model is a good fit for data set.
- The log-likelihood function has a global maximum root for the model parameters, as demonstrated in Figure 11.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The visualization of HRF of the MLBDs model (unimodal and bimodal cases) with given parameters.

**Figure 5.**3D profile of variance of MLBDs with rising $\delta ,{\xi}_{1}\text{}\mathrm{and}\text{}{\xi}_{2}$.

**Figure 11.**The visualization of the profile-likelihood functions for three parameters for real-life application and the P-P plot.

Data | n | Mean | Median | Standard Deviation | Skewness | Kurtosis | Min | Max |
---|---|---|---|---|---|---|---|---|

I | 61 | 0.5142 | 0.5278 | 0.1935 | 0.0059 | −0.5304 | 0.1405 | 0.9794 |

**Table 2.**Estimates (MLEs) and SEs, l (.), along with goodness-of-fit measures, associated with the model parameters, for the trade share dataset.

Distributions | MLEs | LL | AIC | BIC | CAIC | |
---|---|---|---|---|---|---|

Mixture of two one-parameter log-Bilals (MLBDs) | ${\widehat{\xi}}_{1}$ | 0.02545 | 13.26968 | −20.5394 | −14.2065 | −20.1183 |

${\widehat{\xi}}_{2}$ | 0.91683 | |||||

$\delta $ | 0.01097 | |||||

Mixture of two one-parameter unit-Lindleys | ${\widehat{\xi}}_{1}$ | 0.04300 | 12.94698 | −19.8940 | −13.5613 | −19.4729 |

${\widehat{\xi}}_{2}$ | 1.04685 | |||||

$\delta $ | 0.01713 | |||||

Mixture of two one-parameter log-X Lindleys | ${\widehat{\xi}}_{1}$ | 0.00343 | 2.7187 | 0.562600 | 6.89522 | 0.98365 |

${\widehat{\xi}}_{2}$ | 1.53000 | |||||

$\delta $ | 0.00621 |

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**MDPI and ACS Style**

Lone, S.A.; Sindhu, T.N.; Anwar, S.; Hassan, M.K.H.; Alsahli, S.A.; Abushal, T.A.
On Construction and Estimation of Mixture of Log-Bilal Distributions. *Axioms* **2023**, *12*, 309.
https://doi.org/10.3390/axioms12030309

**AMA Style**

Lone SA, Sindhu TN, Anwar S, Hassan MKH, Alsahli SA, Abushal TA.
On Construction and Estimation of Mixture of Log-Bilal Distributions. *Axioms*. 2023; 12(3):309.
https://doi.org/10.3390/axioms12030309

**Chicago/Turabian Style**

Lone, Showkat Ahmad, Tabassum Naz Sindhu, Sadia Anwar, Marwa K. H. Hassan, Sarah A. Alsahli, and Tahani A. Abushal.
2023. "On Construction and Estimation of Mixture of Log-Bilal Distributions" *Axioms* 12, no. 3: 309.
https://doi.org/10.3390/axioms12030309