Construction of a New Bivariate Mixture Lindley Power Function Distribution with Functional Shape Parameter Utilizing Non-Identical Distributions via Conditional Approach

Abstract

This paper presents a new bivariate mixture Lindley power function (BMLPF) distribution that employs a conditional approach with non-identical asymmetric distributions, distinguishing itself by the incorporation of a functional shape parameter. Various structural properties of bivariate distribution are presented, including explicit marginals, cumulative distribution function (CDF), product moments, correlation coefficients, conditional densities, moment generating functions, conditional mean, and variances. The parameters of the proposed distribution are evaluated using the maximum likelihood estimation method. To assess the effectiveness of this estimation approach, an extensive simulation study is carried out. The analysis quantifies these point estimators with their standard errors, RMSE, LCL, and UCL. This research significantly contributes to the development and application of bivariate distributions particularly in modeling and analyzing various types of data.

1. Introduction

Bivariate distribution is a foundational concept in statistics, offering powerful tools for the simultaneous comparison of two variables. The bivariate conditional approach provides insight into the relationship between the variables and their dependencies. The conditional approach is extensively employed across a wide range of distributions, including various types of generalized distributions. Initially, ref. [1] proposed a conditional approach for the development and adjustments of a bivariate gamma distribution. Subsequently, ref. [2,3,4,5,6,7] lead the way of the conditional approach for the construction of bivariate distributions. Among the various methods for construction of bivariate distribution explained by [8], a conditional approach is an effective and robust technique due to its flexibility in integrating baseline distributions with additional functions. The authors of [9,10] introduced the bivariate gamma distribution by incorporating a scale parameter function and employing two identical baseline gamma distributions with a conditional approach. The authors of [11] developed the bivariate distribution through the conditional approach to generate two parametric gamma distribution. In addition, ref. [12] introduced the bivariate Poisson distribution with univariate conditionals. The authors of [13] explained a semi-parametric model for the conditional joint distribution of bivariate estimations on the basis of the distribution regression and factorization methods, and ref. [14] applied the conditional approach to estimate weakly correlated lifetime data in a bivariate model.

The bivariate Lindley distribution is an extension of the conventionally employed Lindley distribution with two dimensions. The bivariate Lindley distribution encounters numerous ways to analyze dependent random variables. The authors of [15,16] proposed one scale parameter Lindley distribution as an extended version of traditional exponential distribution. Recently, refs. [17,18,19,20,21,22,23,24] made valuable contributions to the construction of the bivariate Lindley distributions and explored the distributions’ potential applications across various fields. The power function (PF) distribution is an incredibly compatible model and fits well for failure rates and remains versatile for evaluating lifetime models. The authors of [25] discussed the PF distribution due to its flexibility and preferred it over the other distributions. Although trustworthy work on power function distribution has been extensively produced, bivariate power function distributions have been rarely developed.

In the existing literature, bivariate distributions have been primarily constructed by incorporating scale parameter functions with identical baseline distributions, typically employing the conditional approach. However, real-world data sometimes exhibit a mixture of diverse asymmetric distributions, particularly whenever employing extreme and complex nature data. The conditional approach method is preferred over other construction methods due to its capability to mix identical or non-identical asymmetric distributions with incorporating functional parameters. In this research, we propose a bivariate mixture Lindley power function (BMLPF) model by incorporating a shape parameter function with a mixture of two non-identical asymmetric baseline distributions: power function distribution and Lindley distribution. The aim of this study is to explore the potential of this approach to effectively model dependencies while maintaining all the fundamental mathematical properties. This research provides a more generalized and flexible modeling approach for bivariate distributions, extending their applicability across various statistical and real-world domains.

2. The Proposed Mixture Lindley Power Function (MLPF) Distribution

A new bivariate mixture Lindley power function (BMLPF) distribution is developed using functional shape parameter with conditional approach, where conditional distribution is power function (PF) distribution, and marginal distribution is Lindley distribution.

Let a random variable $X$ be characterized by a Lindley distribution.

$$f_X\left(x\right)={\displaystyle \frac{k^2}{1+k}}(x+1)e^{-k x} \mathrm{Where}, x>0, k>0, 0<x<\infty$$

(1)

where $k$ is a scale parameter.

Let another random variable $Y$ have a power function distribution with a functional shape parameter $\lambda (x)$. The conditional distribution of $y$ given $\lambda (x)$ is defined as

$$f_{{\displaystyle \raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{$X$}\right.}}(y\left|\lambda (x))=\right.\lambda \left(x\right)y^{\lambda \left(x\right)-1} , 0<y<1, \lambda (x)\ge 0$$

(2)

By combining the above two Equations (1) and (2), the compound density function is expressed as

$$f_{XY}\left(x,y\right)={\displaystyle \frac{k^2}{1+k}}(x+1)e^{-k x} \lambda (x)y^{\lambda \left(x\right)-1}$$

(3)

Adjusting $\lambda \left(x\right)$ allows to derive different bivariate distributions with distinct characteristics.

By letting $\lambda \left(x\right)={\displaystyle \frac{\beta x}{\theta }}$ and incorporating it into Equation (3), the new bivariate distribution is formed with the following pdf:

$$f_{XY}\left(x,y\right)={\displaystyle \frac{{\beta k}^2}{\theta (1+k)}}x(x+1)e^{-k x}{y}^{{\displaystyle \frac{\beta x}{\theta }}-1} \mathrm{where} x,y>0 , \beta ,k,\theta >0, 0<y<1, 0<x<\infty .$$

(4)

The CDF of the newly proposed BMLPF distribution is obtained by integrating Equation (4) over the range, $0<x<x$, $0<y<y$

$$F_{X,Y}\left(x,y\right)={\displaystyle \frac{{\theta k^{2}e}^{-kx} (\theta (e^{kx}\left(k+1\right)-y^{\frac{x\beta }{\theta }}(1+k\left(x+1\right)))+\beta \mathit{log}\left(y\right)(-e^{kx}+\left(x+1\right)y^{\frac{x\beta }{\theta }})) }{\left(k+1\right){\left(\theta k-\beta \mathit{log}\left(y\right)\right)}^2}}$$

(5)

where $\beta ,k,\theta >0$.

The Figure 1 provides a visualization of 3D plots for the BMLPF distribution, utilizing different combinations of parameters. These plots depict the asymmetric nature of the proposed distribution.

Figure 2 depicts the contour plots of BMLPF distribution for different parametric values. The proposed model exhibits an asymmetric behavior.

2.1. Mathematical Properties of BMLPF Distribution

The mathematical properties for the newly generated BMLPF distribution are explained briefly in this section.

2.1.1. Product Moments

The $(r,s)th$ product moment for two random variables $X$ and $Y$ for the bivariate distribution is

$${\acute{\mu }}_{r,s}=E\left(X^r Y^s\right)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{ x^r y^{s}f}_{XY}\left(x,y\right)dx dy$$

(6)

Thus, the expression of $(r,s)th$ product moment for the joint BMLPF distribution utilizing Equation (6) is

$${\acute{\mu }}_{r,s}={\displaystyle \frac{{\left(\frac{\beta }{s \theta }\right)}^{1-r}{\left(\frac{s k \theta }{\beta }\right)}^{-r}\left(\beta -e^{\frac{s k \theta }{\beta }}(r+2)(\beta -s \theta )\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{E}[r+3,\frac{s k \theta }{\beta }]\right) \mathsf{\Gamma }[r+2]}{\beta (k+1)}}$$

The ratio moment between the random variables $X$ and $Y$are represented by two methods; one of these methods is described as

$${\acute{\mu }}_{r,-s}=E\left(X^r Y^{-s}\right)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{x}^r y^{-s} f_{XY}\left(x,y\right)dx dy$$

(7)

After simplification, the ratio moment for BMLPF is

$${\acute{\mu }}_{r,-s}={\displaystyle \frac{{s{\theta k}^2 e}^{-\frac{s k \theta }{\beta }}{(-\frac{\beta }{ s \theta })}^{-r}\mathsf{\Gamma }\left[2+r\right]\left(s \theta (r+2)\mathsf{\Gamma }[-r-2,-\frac{s k \theta }{\beta }]-\beta \mathsf{\Gamma }[-r-1,-\frac{sk\theta }{\beta }]\right)}{{\beta }^2(k+1)}}$$

Likewise, the other ratio moment for the variable $X$ and$Y$ is

$${\acute{\mu }}_{-r,s}=E\left(X^{-r} Y^s\right)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{x}^{-r} y^s f_{XY}\left(x,y\right)dx dy$$

(8)

After simplification, the expression reduces to the following form:

$${\acute{\mu }}_{-r , s}=-{\displaystyle \frac{s {{\theta k}^2 e}^{\frac{s k \theta }{\beta }}{(\frac{\beta }{s\theta })}^r\mathsf{\Gamma }\left[2-r\right]\left(s\theta \left(r-2\right)\mathsf{\Gamma }[r-2,\frac{s k \theta }{\beta }]-\beta \mathsf{\Gamma }[r-1,\frac{sk\theta }{\beta }]\right)}{{\beta }^2(k+1)}}$$

2.1.2. Conditional Moments and Variances

The conditional density of the BMLPF distribution for $X$ given $Y$is observed as

$$f_{X|Y}\left(x|y\right)={\displaystyle \frac{e^{-kx}x(x+1)y^{\frac{\beta x}{\theta }}{(k\theta -\beta log(y))}^3}{{\theta }^2((k+1)\theta -\beta log(y))}}$$

The$p\mathrm{t}\mathrm{h}$ conditional moment for $X$ given $Y$ is

$$E\left(X^p|Y\right)={\displaystyle \frac{k^{-p}\Gamma \left[p+2\right]\left((k+p+2)\theta -\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)\right){(1-\frac{\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)}{k\theta })}^{-p}}{\theta (k+2)-\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)}}$$

The conditional variance for $X$ given $Y$ is provided in the Appendix A. The conditional density for $Y$ given $X$is stated as

$$f_{Y|X}\left(y|x\right)={\displaystyle \frac{\beta x}{\theta }} y^{{\displaystyle \frac{\beta x}{\theta }}-1}$$

The$p\mathrm{t}\mathrm{h}$ conditional moment for $Y$ given $X$ is

$$E\left(Y^p|X\right)={\displaystyle \frac{\beta x}{\beta x+p\theta }}$$

The conditional variance for $Y$ given $X$ is provided in the Appendix A. The variances of $X$ and $Y$ are found as

$$var(X)={\displaystyle \frac{k^2+4k+2}{k^2{(k+1)}^2}}$$

(9)

Let $\omega =\left(\mathrm{l}\mathrm{o}\mathrm{g}\left({\displaystyle \frac{\beta }{k\theta }}\right)+\mathrm{l}\mathrm{o}\mathrm{g}\left({\displaystyle \frac{k\theta }{\beta }}\right)\right)$; after simplification

$$var\left(Y\right)={\displaystyle \frac{\left({\beta }^2\left(k+1\right)\left(\beta \left(\beta +\beta k-2\theta k\right)-2e^{\frac{2\theta k}{\beta }}\left(\beta -2\theta \right)\theta k^2\left(\mathsf{\Gamma }\left[0,\frac{2\theta k}{\beta }\right]+\mathsf{\omega }\right)\right)\right)-{\left(\beta \left(\beta +\beta k-\theta k\right)-e^{\frac{\theta k}{\beta }}\left(\beta -\theta \right)\theta k^2\left(\mathsf{\Gamma }\left[0,\frac{\theta k}{\beta }\right]+\mathsf{\omega }\right)\right)}^2}{{\beta }^4{\left(k+1\right)}^2}}$$

(10)

The covariance of two random variables $X \mathrm{a}\mathrm{n}\mathrm{d}Y$ is

$$Cov\left(x,y\right)={\displaystyle \frac{\theta \left({ e}^{\frac{\theta k}{\beta }}k\left(\beta -\theta \right)\left(\theta k\left(k+1\right)+\beta \left(k+2\right)\right)\mathsf{\Gamma }\left[0,\frac{\theta k}{\beta }\right]+\beta \left(\beta -k\left(\beta -\theta \right)-k^2\left(\beta -\theta \right)+e^{\frac{\theta k}{\beta }}k\left(\beta -\theta \right)\left(k+2\right)\mathsf{\omega }\right)\right)}{{\beta }^3{(k+1)}^2}}$$

(11)

The Pearson product moment correlation between two random variables is as follows:

$$\rho ={\displaystyle \frac{cov\left(X,Y\right)}{\sqrt{\mathrm{v}\mathrm{a}\mathrm{r}\left(\mathrm{X}\right)\mathrm{v}\mathrm{a}\mathrm{r}\left(\mathrm{Y}\right)}}}$$

(12)

Incorporating the findings from Equations (9)–(11) in Equation (12), the resultant Pearson correlation coefficient is given below as:

$$\rho ={\displaystyle \frac{\theta \left({ e}^{\frac{\theta k}{\beta }}k\left(\beta -\theta \right)\left(\theta k\left(k+1\right)+\beta \left(k+2\right)\right)\mathsf{\Gamma }\left[0,\frac{\theta k}{\beta }\right]+\beta \left(\beta -k\left(\beta -\theta \right)-k^2\left(\beta -\theta \right)+e^{\frac{\theta k}{\beta }}k\left(\beta -\theta \right)\left(k+2\right)\mathsf{\omega }\right)\right)}{{\beta }^3{\left(1+k\right)}^2\sqrt{\frac{\left(k^2+4k+2\right)\left({\beta }^2\left(k+1\right)\left(\beta \left(\beta +\beta k-2\theta k\right)-2e^{\frac{2\theta k}{\beta }}\left(\beta -2\theta \right)\theta k^2\left(\mathsf{\Gamma }\left[0,\frac{2\theta k}{\beta }\right]+\mathsf{\omega }\right)\right)\right)-{\left(\beta \left(\beta +\beta k-\theta k\right)-e^{\frac{\theta k}{\beta }}\left(\beta -\theta \right)\theta k^2\left(\mathsf{\Gamma }\left[0,\frac{\theta k}{\beta }\right]+\mathsf{\omega }\right)\right)}^2}{{\beta }^4{k}^2{\left(k+1\right)}^4}}}}$$

The corresponding marginal density function of $Y$ from above CDF is obtained as:

$$f_Y\left(y\right)={\displaystyle \frac{\beta \theta k^2(\theta (k+2)-\beta \mathrm{L}\mathrm{o}\mathrm{g}\left(y\right))}{y(k+1){(k\theta -\beta \mathrm{L}\mathrm{o}\mathrm{g}\left(y\right))}^3}}\hspace{1em}\hspace{1em}\beta ,\theta >0\hspace{1em}k>0,\hspace{1em}0<y<1$$

(13)

The univariate mixture Lindley power function (MLPF) distribution is derived by integrating BMLPF distribution. The univariate MLPF distribution has an enormous multipurpose structure which will incorporate the effect of mixed data characteristics. The univariate MLPF model will acquire the impact of new added parameters in the model analysis as well as in real life.

In Figure 3, it is observed that the marginal MLPF distribution exhibits characteristics of both exponential and bathtub families, depending on parametric values. In Figure 3a, it aligns with the exponential family. In Figure 3b,c, the marginal distribution of $y$ reflects the bathtub shape, characterized by a decreasing failure rate, followed by a steady phase and a later increase.

The CDF of MLPF distribution is expressed as

$$F_Y\left(y\right)={\displaystyle \frac{\theta k^2\left(\left(\theta +1\right)k-\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)\right)}{(k+1){(\theta k-\beta \mathrm{l}\mathrm{o}\mathrm{g}(y))}^2}}$$

(14)

2.2. Properties of MLPF Distribution

The marginal probability function of mixture Lindley power function (MLPF) distribution attained from BMLPF distribution is

$$f_Y\left(y\right)={\displaystyle \frac{{\beta \theta k}^2(\theta (k+2)-\beta log(y))}{y\left(k+1\right){(k\theta -\beta log(y))}^3}}, \beta ,\theta >0,\hspace{1em}\hspace{1em}k>0,\hspace{1em}\hspace{1em}0<y<1$$

(15)

where, $\beta ,$ $\theta$ and $k$are scale parameters.

The $mth$ moment of MLPF distribution is derived in Equation (16):

$$\begin{array}{c}E\left(Y^m\right)={\displaystyle \frac{1}{{\beta }^2(k+1)}}({\beta }^2+{\beta }^{2}k-m\beta \theta k+e^{{\displaystyle \frac{m\theta k}{\beta }}}m\theta \left(m\theta -\beta \right)k^2\Gamma \left(0,{\displaystyle \frac{m\theta k}{\beta }}\right)+e^{{\displaystyle \frac{m\theta k}{\beta }}}m\theta \left(m\theta -\beta \right)k^2\mathit{log}\left({\displaystyle \frac{\beta }{\theta k}}\right)\\ -e^{{\displaystyle \frac{m\theta k}{\beta }}}m\beta \theta k^2\mathit{log}\left({\displaystyle \frac{\theta k}{\beta }}\right)+e^{{\displaystyle \frac{m\theta k}{\beta }}}{m}^2{\theta }^2{k}^2\mathit{log}\left({\displaystyle \frac{\theta k}{\beta }}\right))\end{array}$$

(16)

$$\begin{array}{c}E\left(Y\right)={\displaystyle \frac{1}{{\beta }^2\left(k+1\right)}}({\beta }^2+{\beta }^{2}k-\beta \theta k-e^{{\displaystyle \frac{\theta k}{\beta }}}\beta \theta k^2\Gamma \left(0,{\displaystyle \frac{\theta k}{\beta }}\right)+e^{{\displaystyle \frac{\theta k}{\beta }}}{\theta }^2{k}^2\Gamma \left(0,{\displaystyle \frac{\theta k}{\beta }}\right)-e^{{\displaystyle \frac{\theta k}{\beta }}}\beta \theta k^2\mathit{log}\left({\displaystyle \frac{\beta }{\theta k}}\right)\\ +e^{{\displaystyle \frac{\theta k}{\beta }}}{\theta }^2{k}^2\mathit{log}\left({\displaystyle \frac{\beta }{\theta k}}\right)-e^{{\displaystyle \frac{\theta k}{\beta }}}\beta \theta k^2\mathit{log}\left({\displaystyle \frac{\theta k}{\beta }}\right)+e^{{\displaystyle \frac{\theta k}{\beta }}}{\theta }^2{k}^2\mathit{log}\left({\displaystyle \frac{\theta k}{\beta }}\right))\end{array}$$

$$\begin{array}{c}E\left(Y^2\right)={\displaystyle \frac{1}{{\beta }^2\left(k+1\right)}}({\beta }^2+{\beta }^{2}k-2\beta \theta k-2e^{{\displaystyle \frac{2\theta k}{\beta }}}\beta \theta k^2\Gamma \left(0,{\displaystyle \frac{2\theta k}{\beta }}\right)+4e^{{\displaystyle \frac{2\theta k}{\beta }}}{\theta }^2{k}^2\Gamma a\left(0,{\displaystyle \frac{2\theta k}{\beta }}\right)\\ -2e^{{\displaystyle \frac{2\theta k}{\beta }}}\beta \theta k^2\mathit{log}\left({\displaystyle \frac{\beta }{\theta k}}\right)+4e^{{\displaystyle \frac{2\theta k}{\beta }}}{\theta }^2{k}^2\mathit{log}\left({\displaystyle \frac{\beta }{\theta k}}\right)-2e^{{\displaystyle \frac{2\theta k}{\beta }}}\beta \theta k^2\mathit{log}\left({\displaystyle \frac{\theta k}{\beta }}\right)+4e^{{\displaystyle \frac{2\theta k}{\beta }}}{\theta }^2{k}^2\mathit{log}\left({\displaystyle \frac{\theta k}{\beta }}\right))\end{array}$$

Median

The median of MLPF as derived from Equation (14) is presented by the following expression:

$$Median=e^{{\displaystyle \frac{\theta k}{\beta +\beta k}}-\sqrt{{\displaystyle \frac{{\theta }^2{k}^2(k^2+2k+2)}{{\beta }^2{(k+1)}^2}}}}$$

Mode

The location of mode is derived from Equation (15), and when $\beta , \theta , \mathrm{and} k > 0$, the mode is positioned at

$$Mode=exp\left({\displaystyle \frac{\left(\theta \left(k+1\right)-\beta \right)\pm \sqrt{{\left(\left(\beta -\theta \left(k+1\right)\right)\right)}^2-(\theta \left(\theta k\left(k+2\right)-2\beta \left(k+3\right)\right)}}{\beta }}\right)$$

The survival and hazard rate function of the corresponding MLPF distribution is

$$S_Y\left(y\right)=1-{\displaystyle \frac{\theta k^2\left(\theta (k+1)-\beta log(y)\right)}{(k+1){(\theta k-\beta log(y))}^2}}$$

$$H_Y\left(y\right)={\displaystyle \frac{\theta k^2(\beta log(y)-\theta \left(k+2\right))}{ylog(y)(\beta log(y)-\theta k)(\beta \left(k+1\right)log(y)-\theta k(k+2))}}$$

Figure 4 and Figure 5 illustrate the survival and hazard rate function of MLPF distribution; the survival function exhibits different patterns based on parameter variations. Hazard rate function depicts the bathtub shape.

3. Reliability and Hazard Gradient Function of BMLPF Model

In this section, the evaluation of the joint behavior of the variables of bivariate distribution is effectively conducted through reliability like stress–strength model, survival, and hazard gradient functions.

3.1. Stress–Strength Model

The stress–strength model in reliability studies is utilized to evaluate the lifespan of a module described by stochastic durability $Y$ under variable stress conditions $X$. The model assumes that a component experiences failure when the exerted stress exceeds its durability. It is evident that the component operates effectively when its strength level $Y$ exceeds the stress level $X$. Consequently, the reliability of the component, denoted as R, is measured by the probability $P(X<Y)$; hence, $R=P(X<Y)$. This model is applied in various domains including engineering, biology, and medicine. The authors of [26] applied the stress–strength model to numerous bivariate distributions. For our distribution, it is calculated as

$$R=P\left(X<Y\right)$$

$$R=\underset{0}{\overset{\infty }{\int }}\underset{X}{\overset{1}{\int }}{f}_{XY}\left(x,y\right)dy dx$$

(17)

$$R={\displaystyle \frac{k^2}{k+1}}{\int }_0^{\infty }{e}^{-xk}\left(x+1\right)(1-x^{{\displaystyle \frac{x\beta }{\theta }}})dy$$

Since the integral in the reliability component cannot be evaluated analytically, numerical methods can be employed to approximate its value.

3.2. Hazard Gradient Function

The mathematical expression for the hazard rate function for two-dimensional distribution explained by [27] utilized both combined density function $f_{XY}\left(x,y\right)$ and the survival function, further applied by [28], subsequently.

$$r_{XY}\left(x,y\right)={\displaystyle \frac{f_{XY}\left(x,y\right)}{{\overline{F}}_{XY}\left(x,y\right)}}$$

(18)

$${\overline{F}}_{XY}\left(x,y\right)=1-F_X\left(x\right)-F_Y\left(y\right)+F_{XY}\left(x,y\right)$$

(19)

By combining the results from Equations (14) and (19) in Equation (18), the resultant survival function is

$${\overline{F}}_{XY}(x,y)={\displaystyle \frac{e^{-kx}(1+k+kx-\frac{y^{\frac{x\beta }{\theta }}\theta k^2(\theta (1+k+kx)-(x+1)\beta \mathrm{l}\mathrm{o}\mathrm{g}(y))}{{(k\theta -\beta \mathrm{l}\mathrm{o}\mathrm{g}(y))}^2})}{k+1}}$$

(20)

The authors of [29] explained the hazard gradient function for bivariate distribution as

$$r_{XY}\left(x,y\right)=-{({\displaystyle \frac{\partial }{\partial x}} ,{\displaystyle \frac{\partial }{\partial y}})}^t\log {\overline{F}}_{XY}(x,y)$$

(21)

$$r_1\left(x,y\right)=r(xǀY>y)=-{\displaystyle \frac{\partial }{\partial x}}\log {\overline{F}}_{XY}(x,y)$$

$$r_2\left(x,y\right)=r(yǀX>x)=-{\displaystyle \frac{\partial }{\partial y}}\log {\overline{F}}_{XY}(x,y)$$

where $r_1(x,y)$ denotes the rate of failure of $X$, satisfying the condition that $Y>y,$ given the insights into exposure of $X$ given certain information about $Y$. Correspondingly, $r_2(x,y)$ represents the failure occurrence rate in $Y$ under the condition $X>x$, proposing a perspective on the reliability of $Y$ under some specified conditions of $X$.

The vector $r_{XY}\left(x,y\right)=-{({\displaystyle \frac{\partial }{\partial x}} ,{\displaystyle \frac{\partial }{\partial y}})}^t\log {\overline{F}}_{XY}(x,y)$ indicates the hazard gradient of BMLPF distribution as

$$r_{XY}\left(x,y\right)=\left(\begin{array}{c}{\displaystyle \frac{\alpha x^{\alpha -1}(y^{\frac{\gamma x^{\alpha }}{\beta }}-1)(\beta -\gamma log(y))}{\beta (y^{\frac{\gamma x^{\alpha }}{\beta }}+e^{x^{\alpha }}-1)+\gamma log(y)}}\\ {\displaystyle \frac{e^{-x^{\alpha }}(\beta \left(y^{\frac{\gamma x^{\alpha }}{\beta }}+e^{x^{\alpha }}-1\right)+\gamma log(y))}{\beta -\gamma log(y)}}\end{array}\right)$$

4. Dependency Structure

This section provides an overview of some well-known dependency measures for BMLPF distribution.

4.1. Clayton–Oakes’s Measure of Association

The local dependence function on the basis of survival analysis defined by [30,31] is designated as

$$l\left(x,y\right)={\displaystyle \frac{f\left(x,y\right) \overline{F}(x,y)}{{\overline{F}}_1(x,y){\overline{F}}_2(x,y)}}$$

(22)

where ${\overline{F}}_1\left(x,y\right)={\displaystyle \frac{\partial }{\partial x}}{\overline{F}}_{\mathit{XY}}\left(x,y\right)$ and ${\overline{F}}_2\left(x,y\right)={\displaystyle \frac{\partial }{\partial y}}{\overline{F}}_{XY}\left(x,y\right)$

$${\overline{F}}_1\left(x,y\right)={\displaystyle \frac{e^{-kx}{k}^2(x+1)(y^{\frac{x\beta }{\theta }}01)}{k+1}}$$

(23)

$${\overline{F}}_2\left(x,y\right)=-{\displaystyle \frac{\left(\beta k^2{y}^{\frac{x\beta }{\theta }-1}{e}^{-kx}\left({\theta }^2\left(2+k+2kx+k^{2}x\left(x+1\right)\right)-\beta \theta \left(1+2{kx}^2+2x\left(k+1\right)\right)\mathrm{l}\mathrm{o}\mathrm{g}(y)+{\beta }^{2}x\left(x+1\right){\mathrm{l}\mathrm{o}\mathrm{g}(y)}^2\right)\right)}{\left(\left(k+1\right){\left(k\theta -\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)\right)}^3\right)}}$$

(24)

By incorporating Equations (4), (20), (23) and (24) into Equation (22), we derived the result as follows:

$$l\left(x,y\right)=-{\displaystyle \frac{(x{(k\theta -\beta \log (y))}^3(1+k+kx-\frac{y^{\frac{x\beta }{\theta }}\theta k^2(\theta (1+k+kx)-(x+1)\beta \mathrm{l}\mathrm{o}\mathrm{g}(y))}{{(k\theta -\beta \mathrm{l}\mathrm{o}\mathrm{g}(y))}^2}))}{(y^{\frac{x\beta }{\theta }}-1)\theta k^2({\theta }^2\left(2+k+2kx+k^{2}x(x+1)\right)-\beta \theta \left(1+2{kx}^2+2x(k+1)\right)\mathrm{l}\mathrm{o}\mathrm{g}(y)+x(x+1){\beta }^2{\mathrm{l}\mathrm{o}\mathrm{g}(y)}^2))}}$$

4.2. Conditional Probability Measure

The authors of [32] described the measure of association based on conditional probabilities to examine the relationship between two variables in a random vector $\left(X,Y\right).$ This measure is based on survival function of the joint distribution for $X$ and $Y,$ as well as their marginal survival functions.

$$\begin{array}{l}\omega \left(x,y\right)={\displaystyle \frac{P(X>x|Y>y)}{P(X>x)}}\\ \omega \left(x,y\right)={\displaystyle \frac{\overline{F}(x,y)}{\overline{F}(x,0)\overline{F}(0,y)}}\end{array}$$

(25)

For BMLPF distribution, the conditional probability measure calculation is as follows:

$$\begin{array}{l}\overline{F}\left(x,0\right)={\displaystyle \frac{e^{-kx}(1+k+kx)}{k+1}}\\ \overline{F}\left(0,y\right)={\displaystyle \frac{1+k-\frac{\theta +k\theta +\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)}{{(k\theta -\beta \mathrm{l}\mathrm{o}\mathrm{g}(y))}^2}}{k+1}}\end{array}$$

Thus, the conditional probability measure is

$$\omega \left(x,y\right)={\displaystyle \frac{(1+k)(1+k+kx-\frac{y^{\frac{x\beta }{\theta }}\theta k^2\left(\theta \left(1+k+kx\right)-\left(x+1\right)\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)\right)}{{\left(k\theta -\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)\right)}^2})}{(1+k+kx)(1+k-\frac{\theta +k\theta +\beta \mathrm{l}\mathrm{o}\mathrm{g}(y)}{{\left(k\theta -\beta \log \left(y\right)\right)}^2})}}$$

(26)

4.3. Correlation Measure

The Pearson product moment correlation coefficient between $X$ and $Y$, as determined from prior results, is

$$\rho ={\displaystyle \frac{\theta \left({ e}^{\frac{\theta k}{\beta }}k\left(\beta -\theta \right)\left(\theta k\left(k+1\right)+\beta \left(k+2\right)\right)\mathsf{\Gamma }\left[0,\frac{\theta k}{\beta }\right]+\beta \left(\beta -k\left(\beta -\theta \right)-k^2\left(\beta -\theta \right)+e^{\frac{\theta k}{\beta }}k\mathsf{\omega }\left(\beta -\theta \right)\left(k+2\right)\right)\right)}{{\beta }^3{\left(1+k\right)}^2\sqrt{\frac{\left(k^2+4k+2\right)\left({\beta }^2\left(k+1\right)\left(\beta \left(\beta +\beta k-2\theta k\right)-2e^{\frac{2\theta k}{\beta }}\left(\beta -2\theta \right)\theta k^2\left(\mathsf{\Gamma }\left[0,\frac{2\theta k}{\beta }\right]+\mathsf{\omega }\right)\right)\right)-{\left(\beta \left(\beta +\beta k-\theta k\right)-e^{\frac{\theta k}{\beta }}\left(\beta -\theta \right)\theta k^2\left(\mathsf{\Gamma }\left[0,\frac{\theta k}{\beta }\right]+\mathsf{\omega }\right)\right)}^2}{{\beta }^4{k}^2{\left(k+1\right)}^4}}}}$$

Due to the non-linearity between $X$ and $Y$, the Pearson correlation coefficient is limited in its effectiveness as observed in

Table 1. Pearson correlation for BMLPF distribution.

$\mathit{k}$	$\mathit{\beta }$	$\mathit{\theta }$	$\mathit{\rho }$	$\mathit{k}$	$\mathit{\beta }$	$\mathit{\theta }$	$\mathit{\rho }$	$\mathit{k}$	$\mathit{\beta }$	$\mathit{\theta }$	$\mathit{\rho }$
0.5	0.5	$1$	0.5531	1	0.5	$1$	0.5426	2	0.5	$1$	0.4917
		$1.5$	0.5435			$1.5$	0.5140			$1.5$	0.4478
		$2$	0.5303			$2$	0.4880			$2$	0.4136
		$2.5$	0.5164			$2.5$	0.4651			$2.5$	0.3860
		$3$	0.5029			$3$	0.4451			$3$	0.3672
	1.5	$1$	0.5303		1.5	$1$	0.5657		1.5	$1$	0.5663
		$1.5$	0.5461			$1.5$	0.5664			$1.5$	0.5477
		$2$	0.5523			$2$	0.5604			$2$	0.5279
		$2.5$	0.5539			$2.5$	0.5519			$2.5$	0.5091
		$3$	0.5531			$3$	0.5426			$3$	0.4917
	2.5	$1$	0.5010		2.5	$1$	0.5514		2.5	$1$	0.5738
		$1.5$	0.5250			$1.5$	0.5639			$1.5$	0.5693
		$2$	0.5383			$2$	0.5673			$2$	0.5594
		$2.5$	0.5461			$2.5$	0.5664			$2.5$	0.5477
		$3$	0.5505			$3$	0.5633			$3$	0.5358
	3.5	$1$	0.4775		3.5	$1$	0.5351		3.5	$1$	0.5697
		$1.5$	0.5055			$1.5$	0.5541			$1.5$	0.5738
		$2$	0.5224			$2$	0.5628			$2$	0.5704
		$2.5$	0.5335			$2.5$	0.5665			$2.5$	0.5639
		$3$	0.5409			$3$	0.5673			$3$	0.5561

. Therefore, to assess dependency in the proposed bivariate model, alternative dependency measures such as Kendall’s Tau and Spearman’s rho are more reliable.

5. Parameter Estimation

Numerous studies utilized the parameter estimation by different techniques, among which maximum likelihood estimation (MLE) is the most extensively applied technique. The authors of [33] utilized this method for the underdetermined parameters of their distribution; ref. [34] described it for estimating all the parameters jointly, as a one-step parametric method; and ref. [35] described the mathematical structure for solving non-differential linear equations and partial differential equations.

This section describes MLE for the proposed BMLPF distribution by considering the assumption that the distribution parameters are known. Let ${X_1,X}_2, {X_3,\dots X}_n$ be the random sample of size n from any family of bivariate transmuted distributions. The likelihood function based on MLE for the BMLPF distribution is

$$L=\prod _{i=1}^n{f}_{X,Y}\left(x_i,{ y}_i\right)=\prod _{i=1}^n{\displaystyle \frac{{\beta k}^2}{\theta (1+k)}}x(x+1)e^{-k x}{y}^{{\displaystyle \frac{\beta x}{\theta }}-1}$$

The log likelihood functional form of the above equation is

$$\begin{array}{c}\log \left(L\right)=n\mathrm{l}\mathrm{o}\mathrm{g}\left(\beta \right)+2n\mathrm{l}\mathrm{o}\mathrm{g}\left(k\right)-n\mathrm{l}\mathrm{o}\mathrm{g}(\theta )-n\mathrm{l}\mathrm{o}\mathrm{g}(k+1)+{\sum }_{i=1}^n\mathrm{l}\mathrm{o}\mathrm{g}(x_i)+{\sum }_{i=1}^n\mathrm{l}\mathrm{o}\mathrm{g}(x_i+1)+\\ {\sum }_{i=1}^n\mathrm{l}\mathrm{o}\mathrm{g}(e^{-k{x}_i})+{\sum }_{i=1}^n\mathrm{l}\mathrm{o}\mathrm{g}({y_i}^{{\displaystyle \frac{\beta x_i}{\theta }}-1})\end{array}$$

The partial deviations with respect to different parameters are given as

$$\begin{array}{l}{\displaystyle \frac{\partial \log \left(L\right)}{\partial \beta }}={\displaystyle \frac{n}{\beta }}+{\sum }_{i=1}^n{\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}(y_i)x_i}{\theta }}\\ {\displaystyle \frac{\partial \log \left(L\right)}{\partial {\beta }^2 }}=-{\displaystyle \frac{n}{{\beta }^2}}\\ {\displaystyle \frac{\partial \log \left(L\right)}{\partial k}}={\displaystyle \frac{2n}{k}}-{\displaystyle \frac{n}{1+k}}-{\sum }_{i=1}^n{x}_i\\ {\displaystyle \frac{\partial \log \left(L\right)}{\partial k^2}}=-{\displaystyle \frac{2n}{k^2}}+{\displaystyle \frac{n}{{(1+k)}^2}}\\ {\displaystyle \frac{\partial \log \left(L\right)}{\partial \theta }}=-{\displaystyle \frac{n}{\theta }}+{\sum }_{i=1}^n\left(-{\displaystyle \frac{\beta \mathrm{l}\mathrm{o}\mathrm{g}\left(y_i\right)x_i}{{\theta }^2}}\right)\\ {\displaystyle \frac{\partial \log \left(L\right)}{\partial {\theta }^2}}={\displaystyle \frac{n}{{\theta }^2}}+{\sum }_{i=1}^n\left({\displaystyle \frac{2\beta \mathrm{l}\mathrm{o}\mathrm{g}\left(y_i\right)x_i}{{\theta }^3}}\right)\\ {\displaystyle \frac{\partial \mathrm{l}\mathrm{o}\mathrm{g}(L)}{\partial \beta k}}=0\\ {\displaystyle \frac{\partial \log \left(L\right)}{\partial k\beta }}=0\\ {\displaystyle \frac{\partial \log \left(L\right)}{\partial \beta \theta }}={\sum }_{i=1}^n\left(-{\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}\left(y_i\right)x_i}{{\theta }^2}}\right)\\ {\displaystyle \frac{\partial \mathrm{l}\mathrm{o}\mathrm{g}(L)}{\partial \theta k}}=0\end{array}$$

The MLE for the vector of parameter$L$ are generated numerically by solving the above equations, which are derived through utilizing the likelihood function for the BMLPF distribution.

6. Generation of Random Sample

The conditional distribution approach is applied for generating random samples from BMLPF distribution. The process proceeds with the following steps:

(a)

The initial step involves generating random observation by utilizing the CDF of marginal distribution of $X:$

$$\begin{array}{l}{F}_X\left(x\right)={\displaystyle \frac{e^{-xk}(-1+e^{xk}-k+e^{xk}k-xk)}{1+k}}\\ x={\displaystyle \frac{-1-k-\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{L}\mathrm{o}\mathrm{g}\left(e^{\left(k+1\right)}(u-1)(k+1)\right)}{k}}\end{array}$$

(b)

Moving towards the next step, the CDF of $f_{Y|X}\left(y|x\right)$is derived:

$$F_{Y|X}\left(y|x\right)=y^{{\displaystyle \frac{x\beta }{\theta }}}$$

(27)

By letting $F_{Y|X}\left(y|x\right)=v$in Equation (19),

$$\begin{array}{l}y=v^{{\displaystyle \frac{\theta }{x\beta }}}\\ y=v^{{\displaystyle \frac{\theta k}{\beta \left(-1-k-\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{L}\mathrm{o}\mathrm{g}\left(e^{\left(k+1\right)}(u-1)(k+1)\right)\right)}}}\end{array}$$

7. Real Data Application

The proposed BMLPF distribution is compared with some existing models. Although models of the same characteristics as those of the proposed model are not available in the literature, certain comparable models are used. This evaluation is performed by means of comparative analysis utilizing two real-world datasets. The following bivariate densities are utilized for real datasets.

Bivariate distribution with Rayleigh and Lindley distributions [20]:

$$f_{X,Y}\left(x,y\right)={\displaystyle \frac{\alpha x}{{\phi }^2}}{e}^{-\alpha y}{e}^{{\displaystyle \frac{-x^2\left(\alpha +1\right)}{2{\phi }^2}}}\left(e^{{\displaystyle \frac{\alpha x^2}{{2\phi }^2}}}+\alpha y-1\right)$$

The authors of [36] presented bivariate Poisson generalized Lindley distributions as

$$f_{X,Y}\left(x,y\right)={\displaystyle \frac{{\mathsf{\lambda }}^{\alpha }{\omega }_1^x{\omega }_2^y\mathsf{\Gamma }\left(\alpha +x+y-1\right)\left(\left(\alpha -1\right)\left(\mathsf{\lambda }+{\omega }_1+{\omega }_2+1\right)+x+y\right)}{{\left(\mathsf{\lambda }+{\omega }_1+{\omega }_2\right)}^{\alpha +x+y}(\mathsf{\lambda }+1)x!y!\mathsf{\Gamma }(\alpha )}}$$

Bivariate Weibull distributions generated by [37]:

$$f\left(x, y\right)={\beta }_1{\beta }_2{x}^{2{\beta }_1-1}{y}^{{\beta }_2-1}exp\left[-x^{{\beta }_1} \left(1+y^{{\beta }_2} \right)\right]; x, y, {\beta }_1, {\beta }_2>0$$

The authors of [38] introduced the Bivariate exponential distribution:

$$f\left(x, y\right)=\mathsf{\lambda }x\exp \left[-x \left(\mathsf{\lambda }+y\right)\right]; x, y,\mathsf{\lambda }>0$$

The two datasets are selected for their distinct characteristics: one featuring variables with different units, indicative of mixture nature, and the other representing non-mixture distribution. Using both the datasets, the model’s flexibility and effectiveness is accessed across different types of bivariate data structures. The first real dataset includes the bivariate measurement of bone mineral density (BMD) for the dominant ulna and ulna bones recorded in gram per square centimeter (gm/cm²) from 24 children at the age of one, collected by [39] and further utilized by [40]. For comparison purposes, four other distributions are considered alongside the new proposed BMLPF distribution. The maximum likelihood estimates along with their standard errors are computed and the Akaike information criteria (AIC), and Bayesian information Criteria (BIC) are utilized to further evaluate the BMLPF model performance using the maxLik package in R (v: 4.3.2).

X	0.87	0.60	0.77	0.76	0.55	0.75	0.71	0.69	0.84	0.87	0.65	0.69
Y	0.97	0.69	0.74	0.70	0.62	0.52	0.79	0.72	0.66	0.79	0.73	0.53
X	0.67	0.82	0.75	0.66	0.69	0.88	0.58	0.80	0.54	0.80	0.57	0.59
Y	0.58	0.77	0.73	0.51	0.74	0.79	0.63	0.77	0.50	0.78	0.63	0.64

Table 2. Maximum likelihood estimates for different distributions on Data 1.

Data 1
Distribution	Parameters	Estimate	SE	Log Likelihood	AIC	BIC
New Bivariate Mixture Lindley Power Function Distribution (BMLPF)	$\beta$	2.9388	5.9841
	$\theta$	0.9296	1.8991	−7.5773	21.1546	24.6887
	$k$	1.8893	0.3076
Bivariate Distribution with Rayleigh and Lindley Distributions	$\alpha$	1.7635	0.2896
	$\phi$	0.5184	0.0564	−14.3791	32.7581	35.1142
Bivariate Poisson Generalized Lindley Distribution	$\lambda$	4.5444	0.0631
	$\alpha$	49.0902	0.7476	−41.4652	90.9303	95.6425
	${\omega }_1$	0.0671	0.7476
	${\omega }_2$	0.0631	0.0237
Bivariate Weibull Distribution	${\beta }_1$	2.0719	0.3674	−21.7882	47.5764	49.9326
	${\beta }_2$	2.4118	0.4620
Bivariate Exponential Distribution	$\lambda$	1.4035	0.2865	−35.9723	73.9446	75.1226

Table 2. Maximum likelihood estimates for different distributions on Data 1.

Data 1
Distribution	Parameters	Estimate	SE	Log Likelihood	AIC	BIC
New Bivariate Mixture Lindley Power Function Distribution (BMLPF)	$\beta$	2.9388	5.9841
	$\theta$	0.9296	1.8991	−7.5773	21.1546	24.6887
	$k$	1.8893	0.3076
Bivariate Distribution with Rayleigh and Lindley Distributions	$\alpha$	1.7635	0.2896
	$\phi$	0.5184	0.0564	−14.3791	32.7581	35.1142
Bivariate Poisson Generalized Lindley Distribution	$\lambda$	4.5444	0.0631
	$\alpha$	49.0902	0.7476	−41.4652	90.9303	95.6425
	${\omega }_1$	0.0671	0.7476
	${\omega }_2$	0.0631	0.0237
Bivariate Weibull Distribution	${\beta }_1$	2.0719	0.3674	−21.7882	47.5764	49.9326
	${\beta }_2$	2.4118	0.4620
Bivariate Exponential Distribution	$\lambda$	1.4035	0.2865	−35.9723	73.9446	75.1226

The second dataset comprises of bivariate data obtained from 51 USA cities, where X is represented as the average precipitation measured in millimeters, and Y is denoted as the average max temperature measured in degree Celsius, originally generated by USA National Clinic Data Center (NCDC) and utilized by [20].

X	99	61	86	113	96	83	57	80	79	75	70	15	62	95	81
Y	12	17	7	13	5	2	12	2	6	4	5	14	6	2	4
X	71	44	13	52	97	146	52	52	29	108	135	102	48	66	90
Y	18	4	19	5	14	20	8	11	26	1	3	10	18	6	10
X	22	72	176	107	84	83	37	67	83	36	49	39	102	66	154
Y	23	7	20	4	7	6	12	10	20	3	12	3	18	14	8
X	72	63	83	77
Y	6	22	11	8

Table 3. Maximum likelihood estimates for different distributions on Data 2.

Data 2
Distribution	Parameters	Estimates	SE	Log Likelihood	AIC	BIC
New Bivariate Mixture Lindley Power Function Distribution (BLMPF)	$\beta$	0.3739	3.8975
	$\theta$	0.7170	7.4745	−10.6927	27.3854	33.1809
	$k$	1.7725	0.1968
Bivariate Distribution with Rayleigh and Lindley Distributions	α	19.243	2.097
	φ	173.842	4.689	−473.153	950.3069	954.1705
Bivariate Poisson Generalized Lindley Distribution	$\lambda$	0.3375	2.2495
	$\alpha$	27.0658	7.1301	−63.0167	134.0335	141.7608
	${\omega }_1$	0.00966	0.0678
	${\omega }_2$	0.00131	0.0092
Bivariate Weibull Distribution	${\beta }_1$	1.5087	0.18386	−36.9440	77.8880	81.7517
	${\beta }_2$	0.4999	0.06128
Bivariate Exponential Distribution	$\lambda$	1.3027	0.1824	−60.5942	123.1885	125.1203

Table 3. Maximum likelihood estimates for different distributions on Data 2.

Data 2
Distribution	Parameters	Estimates	SE	Log Likelihood	AIC	BIC
New Bivariate Mixture Lindley Power Function Distribution (BLMPF)	$\beta$	0.3739	3.8975
	$\theta$	0.7170	7.4745	−10.6927	27.3854	33.1809
	$k$	1.7725	0.1968
Bivariate Distribution with Rayleigh and Lindley Distributions	α	19.243	2.097
	φ	173.842	4.689	−473.153	950.3069	954.1705
Bivariate Poisson Generalized Lindley Distribution	$\lambda$	0.3375	2.2495
	$\alpha$	27.0658	7.1301	−63.0167	134.0335	141.7608
	${\omega }_1$	0.00966	0.0678
	${\omega }_2$	0.00131	0.0092
Bivariate Weibull Distribution	${\beta }_1$	1.5087	0.18386	−36.9440	77.8880	81.7517
	${\beta }_2$	0.4999	0.06128
Bivariate Exponential Distribution	$\lambda$	1.3027	0.1824	−60.5942	123.1885	125.1203

To access the adequacy of the new BLMPF distribution, it is compared with the four existing distributions: two mixture distributions and two non-mixture distributions (Table 2 and Table 3). Through statistical assessment using AIC (Akaike information criteria), BIC (Bayesian information criteria), log likelihood, and SE, the proposed model shows enhanced alignment in reflecting data patterns. In Table 2, the evaluation of AIC = 21.1546 and BIC = 24.6887 values suggests that the proposed distribution Is the most appropriate model for Data 1. Likewise, Table 3 reflects the BLMPF distribution as a best fit for Data 2, as supported by its minimal AIC = 27.3854 and BIC = 33.1809 values. The significantly lower values of AIC and BIC show the optimal balance between generalizability and complexity. This exhibits efficacy of BLMPF distribution to accommodate diverse data structures with reliability and precision. Comparative analysis reveals that, for both real-life datasets, the proposed BLMPF distribution achieved the minimum AIC and BIC values, making it the most suitable model for the analyzed data.

8. Simulation Study

The simulation algorithm is also generated for the estimation of parametric values for the sample sizes n = 50, 100, 200, 500, 1000 based on 10000 replications using R package. A set of varying parameters is taken to evaluate the potential of model to remain consistent across different magnitudes. The analysis included root mean squared errors (RMSE), mean estimates, standard errors (SE), and bias and confidence intervals for each sample size. The resulting data are outlined as follows.

The

Table 4. Simulation-1 study of BMLPF distribution with combination of parameters.

$\mathit{k}=3, \mathit{\beta }=1.00, \mathit{\theta }=1.25$
	Parameters	50	100	200	500	1000
ESTIMATES	$k$	3.00019	3.00018	3.00021	3.00020	3.00019
	$\theta$	1.00029	1.00029	1.00033	1.00029	1.00031
	$\beta$	1.25036	1.25040	1.25041	1.25045	1.25038
RMSE	$k$	0.00271	0.00194	0.00388	0.00279	0.00296
	$\theta$	0.00491	0.00357	0.00497	0.00358	0.00551
	$\beta$	0.00359	0.00566	0.00597	0.00848	0.00480
SE	$k$	0.00038	0.00019	0.00027	0.00012	0.00009
	$\theta$	0.00069	0.00036	0.00035	0.00016	0.00017
	$\beta$	0.00051	0.00056	0.00042	0.00038	0.00015
BIAS	$k$	0.03324	0.01807	0.00957	0.00386	0.00165
	$\theta$	0.05222	0.02571	0.01470	0.0054	0.00305
	$\beta$	0.06719	0.04021	0.02025	0.00854	0.00377
LCL	$k$	2.99944	2.99981	2.99967	2.99996	3.00001
	$\theta$	0.99893	0.99959	0.99965	0.99998	0.99996
	$\beta$	1.24936	1.24929	1.24958	1.24971	1.25008
UCL	$k$	3.00094	3.00057	3.00075	3.00045	3.00037
	$\theta$	1.00165	1.00099	1.00102	1.00061	1.00065
	$\beta$	1.25136	1.25151	1.251243	1.25119	1.25068

provides the estimated values for the parameters corresponding to $k=3 , \beta =$1.00$,\mathrm{a}\mathrm{n}\mathrm{d} \theta =1.25$. The results indicate that the parameter estimates closely align with the initial values. It is observed that, with the increase in sample size, both the bias and SE show a decreasing trend. However, the LCL and UCL remain nearly consistent across different sample sizes.

In

Table 5. Simulation-2 study of BMLPF distribution with combination of parameters.

$\mathit{k}=3.5, \mathit{\beta }=1.25, \mathit{\theta }=2.5$
	$\mathbf{Parameters}$	50	100	200	500	1000
ESTIMATES	k	3.50020	3.50021	3.50020	3.50018	3.50019
	$\theta$	1.25028	1.25033	1.25027	1.25029	1.25033
	$\beta$	2.50038	2.50044	2.50045	2.50042	2.50042
RMSE	k	0.00300	0.00328	0.00353	0.00188	0.00297
	$\theta$	0.00368	0.00644	0.00284	0.00337	0.00643
	$\beta$	0.00614	0.00849	0.00849	0.00729	0.00727
BIAS	k	0.03800	0.01933	0.00945	0.00303	0.00193
	$\theta$	0.05406	0.02933	0.01078	0.00557	0.00319
	$\beta$	0.06861	0.04078	0.02164	0.00814	0.00415
SE	k	0.00042	0.00033	0.00025	0.00008	0.00009
	$\theta$	0.00052	0.00064	0.00020	0.00015	0.00020
	$\beta$	0.00087	0.00085	0.00060	0.00033	0.00023
LCL	k	3.49937	3.49956	3.49971	3.50002	3.50001
	$\theta$	1.24926	1.24907	1.24988	1.24999	1.24993
	$\beta$	2.49868	2.49877	2.49927	2.49978	2.49997
UCL	k	3.50103	3.50085	3.50068	3.50035	3.50038
	$\theta$	1.25131	1.25159	1.25066	1.25058	1.25073
	$\beta$	2.50208	2.50211	2.50162	2.50106	2.50087

, the second combination of parameters set at $k=3.5, \beta =1.25 \mathrm{a}\mathrm{n}\mathrm{d}, \theta =2.5$ are detailed. The outcomes reinforce that the estimation procedure reliably generates the parameter values close to their initial settings. The Bias, SE, and RMSE decline with increasing sample size, supporting the model’s strength.

The

Table 6. Simulation-3 study of BMLPF distribution with combination of parameters.

$\mathit{k}=2, \mathit{\beta }=3.55, \mathit{\theta }=1.05$
	Parameters	50	100	200	500	1000
ESTIMATES	$k$	2.00019	2.00017	2.00019	2.00017	2.00019
	$\theta$	3.55033	3.55029	3.55027	3.55029	3.55029
	$\beta$	1.05042	1.05045	1.05036	1.05035	1.05042
RMSE	$k$	0.00287	0.00140	0.00255	0.00165	0.00219
	$\theta$	0.00531	0.00345	0.00305	0.00413	0.00351
	$\beta$	0.00701	0.00691	0.00449	0.00297	0.00786
SE	$k$	0.00041	0.00014	0.00018	0.00007	0.000069
	$\theta$	0.00075	0.00035	0.00022	0.00018	0.00011
	$\beta$	0.00099	0.00069	0.00032	0.00013	0.00025
BIAS	$k$	0.03128	0.01350	0.00796	0.00343	0.00157
	$\theta$	0.06616	0.02835	0.01206	0.00545	0.00271
	$\beta$	0.07772	0.04445	0.01732	0.00658	0.00357
LCL	$k$	1.99939	1.99989	1.99984	2.00002	2.00006
	$\theta$	3.54886	3.54961	3.54985	3.5499	3.55007
	$\beta$	1.04847	1.04909	1.04974	1.05008	1.04993
UCL	$k$	2.00099	2.00045	2.00055	2.00032	2.00033
	$\theta$	3.55181	3.55096	3.55070	3.55065	3.55051
	$\beta$	1.05236	1.05180	1.05098	1.05061	1.05091

displays the estimated values for the parameters for $k=2, \beta =3.55, \theta =1.05$ for the sample sizes of n = 50, 100, 200, 500, and 1000. The results indicate that increasing sample size leads to more accurate estimates of the BMLPF distribution. The reduction in bias, SE, and RMSE across large sample sizes underscores the robustness of the maximum likelihood estimation procedure. These findings indicate BMLPF distribution achieved the minimum AIC and BIC values, making it the most suitable model for the analyzed data.

9. Conclusions

In this article, the new BMLPF distribution is generated by combining the power function distribution and Lindley distribution as non-identical asymmetric baselines by the incorporation of a functional shape parameter via a conditional approach. This study covers various properties, including product moments, conditional densities, correlations, conditional means, and variances. This study also assesses reliability, hazard functions, dependency structure, and parameter estimation through maximum likelihood estimation. Practical application of the model is presented through real-world data analysis and simulation experiments. The comparative evaluation with four alternative models depicts considerably lower values for AIC and BIC of the proposed BMLPF distribution. The results validate the adaptability of proposed distribution to model various data structures, including distributions of both mixture and non-mixture nature while maintaining reliability and consistency. The simulation study with varying sets of parameters shows parameter estimates are consistent with the baseline values. Moreover, the analysis indicates a decreasing trend of bias and SE with the increase in sample size. Further research can explore the impact of adjusting shape parameter functions to derive distinct bivariate distributions, each characterized by specific dependency structures. Future studies can expand on these findings to extend the bivariate framework into multivariate distribution models.