EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5819 ISSN 1307-5543 – ejpam.com Published by New York Business Global A New Bivariate Transmuted Family of Distributions: Properties and Application Amani Abdullah Alsalafi1,2,∗, Saman Hanif Shahbaz1, Lutfiah Ismail Al-Turk1 1 Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Sana’a University, Sana’a, Yemen Abstract. The cubic transformation families of distributions are widely used to model complex univariate data. However, in many situations, the jointly modeling of two variables is neces- sary, making bivariate distributions essential. This paper introduces a novel family of bivariate probability distributions that extends the univariate cubic transformation family. The bivariate cubic transmuted (BCT) family of distributions is comprehensively discussed, with its statistical properties explored in detail. Within this family, the bivariate cubic transmuted Burr (BCTB) distribution is specifically analyzed. Its statistical properties are examined, and its parameters are estimated using the maximum likelihood estimation (MLE) method. To assess the performance of the estimation procedure, a Monte Carlo simulation study is conducted. Furthermore, the appli- cability of the proposed model is demonstrated by fitting it to real datasets, and its relevance is further discussed. 2020 Mathematics Subject Classifications: 60E05, 62E15, 62G30, 62F10 Key Words and Phrases: Bivariate cubic transmuted family, bivariate cubic transmuted Burr distribution, Burr distribution, maximum likelihood estimation 1. Introduction In recent decades, significant advancements have been made in the development of continuous univariate distributions and families of univariate distributions. Despite these advancements, many datasets in reliability, science, and related fields deviate from tradi- tional distributions. Consequently, there is a pressing need for modified, extended, and generalized distributions tailored to these specialized applications. One such innovation is the transformed-transformer method, formerly referred to as the exponentiated T − X family of distributions by [1]. This method has been pivotal in ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5819 Email addresses: amohammedalsalafi@stu.kau.edu.sa (A. A. Alsalafi), shmohamad2@kau.edu.sa (S. H. Shahbaz), lturk@kau.edu.sa (L. I. Al-Turk) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 2 of 28 extending traditional distributions to address complex data behaviors. Bivariate distributions also play a critical role, particularly in modeling extreme events. Return periods in bivariate distributions can be analyzed using separate single random variables or joint random variables. These return periods may involve one variable exceed- ing a certain magnitude, another variable meeting a threshold, or conditional scenarios where one variable’s behavior depends on an other’s magnitude. The mathematical explo- ration of new families of bivariate distributions is vital for enhancing their applicability to real-world problems. Among the versatile techniques in distribution theory, the transmutation method intro- duced by [2] stands out. This technique employs a functional composition of a cumulative density function (CDF) and an inverse CDF (quantile function), leading to new distribu- tions defined by the formula: F (x) = (1 + λ1)G(x)− λ1G 2(x); where λ1 ∈ [−1, 1] , x ∈ R and G(x) is the baseline distribution’s CDF. This approach has resulted in numerous novel distributions, such as the Transmuted Weibull distribution by [3], the Transmuted Pareto distribution by [4], and others. Further advancements include the cubic transmutation method. [5] studied the cubic transmuted Weibull distribution and its properties, while [6] obtained the CDF of cubic transmuted family of distrubution as: F (x) = (1 + λ1)G(x) + (λ2 − λ1)G 2(x)− λ2G 3(x); where λ1, λ2 ∈ [−1, 1] , −2 ⩽ λ1 + λ1 ⩽ 1. [7] and [8] expanded on this work by exploring parameter estimation and inference procedures. Notably, [9] introduced the cubic trans- muted Rayleigh distribution with the following CDF: F (x, σ, λ) = 1− e− 3x2 2σ2 [(1− λ)e− x2 σ2 + 3λe− x2 2σ2 − 2λ], where σ ∈ R+ is the scale parameter and λ ∈ [−1, 1] is the shape parameter. Bivariate distributions continue to evolve with innovations that integrate greater flexibility into modeling joint phenomena. [10] introduced generalized joint distributions, while [11] proposed the bivariate Gumbel-G family. These concepts were extended by incorporating transmuted families, as demonstrated by [12]. Moreover, [13] developed a bivariate trans- muted family of distributions, which provided enhanced tools for analyzing interdependent phenomena in diverse fields. The article is organized as follows: Section 2 introduced the bivariate transmuted family of distributions; Section 3 discussed the genesis of the BCT family, presenting its probability density function (PDF), cumulative distribution function (CDF), marginal dis- tributions and conditional distributions; Section 4 examined key statistical properties of the BCT family; Section 5 explored the estimation of the family parameters; Section 6 proposed the BCTB distribution; Section 7 studied the statistical properties of the BCTB A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 3 of 28 distribution; Section 8 details the parameter estimation for the proposed BCTB distribu- tion; Section 9 provided simulations and applies the BCTB distribution to a real dataset; and finally, Section 10 concluded the study with key findings and implications. 2. The Bivariate Transmuted Family of Distributions This research aims to expand the family of bivariate transformed distributions into the bivariate cubic state. Therefore, it is appropriate to discuss the family of bivariate T-X distributions. [10] introduced a new technique to derive families of continuous distributions using two different distributions. A random variable X called a transformer is used to transform another random variable, T, This family of distributions is called the T − X family of distributions. The CDF of this family is: FT−X(x) = ∫ W[G(x)] a r(u)du (1) where G(x) is CDF of baseline distribution and W (G(x)) is some function of G(x) such that W [G(x)] ∈ [a, b] W [G(x)] is differentiable and monotonically non-decreasing W [G(x)] → a as x → −∞, W [G(x)] → b as x → +∞  (2) where W [G(x)] satisfies the conditions Eq.(2). The pdf corresponding to the Eq.(1), is given by fT−X(x) = { d dx W [G(x)] } r{W [G(x)]} The joint CDF of a simple bivariate T-X family of distributions is given by [14] as: FX,Y (x, y) = ∫ W1[G1(x)] a1 ∫ W2[G2(y)] a2 r (u1, u2) du1 du2 where W1 [G1(x)] and W2 [G2(y)] have usual properties and r (u1, u2) is any bivariate distribution with suitable support for random variable U1 and U2. If (u1, u2) is a bivariate distribution such that the support of U1 and U2 is [0, 1]× [0, 1] then a simpler version of bivariate T−X family is given as FX,Y (x, y) = ∫ G1(x) 0 ∫ G2(y) 0 r (u1, u2) du1 du2. (3) [13] proposed a new family of bivariate transmuted distributions by using r(u1, u2) = 1 + λ1(1− 2u1) + λ2(1− 2u2) + 2λ3(1− u1 − u2); 0 ⩽ u1, u2 ≤ 1, A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 4 of 28 in Eq.(3) and the joint CDF of the family as: FX,Y (x, y) = G1(x)G2(y)[1 + (λ1 + λ3)(1−G1(x)) + (λ2 + λ3)(1−G2(y))], (4) Where G1(x) and G2(y) are any marginal CDF for all (x, y) ∈ R2, (λ1, λ2, λ3) are the transmutation parameters such that (λ1, λ2, λ3) ∈ [−1, 1] under these conditions: −1 ⩽ λ1 − λ2 ⩽ 1,−1 ⩽ λ1 + λ2 + 2λ3 ⩽ 1,−1 ⩽ λ1 + λ3 ⩽ 1 and −1 ⩽ λ2 + λ3 ⩽ 1. the PDF corresponding to the CDF in Eq.(4) as: fX,Y (x, y) = g1(x)g2(y)[1 + (λ1 + λ3)(1− 2G1(x)) + (λ2 + λ3)(1− 2G2(y))]. Where g1(x) and g2(y) are the PDF of any base distribution corresponding to the CDF G1(x) and G2(y), respectively. 3. The Bivariate Cubic Transmuted Families of Distributions In this section we proposed bivariate cubic transmuted (BCT) family using r(u1, u2) = 1 + λ1(1− 2u1) + λ2u1(2− 3u1) + λ3(1− 2u2) + λ4u2(2− 3u2) + λ5(1− 3u21)(1− 3u22); 0 < u1, u2 < 1. (5) This function generalizes previous models and offers more flexible dependency structures between variables while also incorporating non-linear effects to capture more complex re- lationships. It ensures the validity of probabilistic properties, making it well-suited for constructing a new bivariate distribution. In addition, it can recover traditional cases where certain parameters are set to zero, which simplifies interpretation and application. Using (5)in (3), to get the cumulative distribution function for the BCT family of distri- butions (CDF) as: FX,Y (x, y) = G1(x)G2(y)[1 + λ1 + λ3 + λ5 + (λ2 − λ1)G1(x)− (λ2 + λ5)G 2 1(x) + (λ4 − λ3)G2(y)− (λ4 + λ5)G 2 2(y) + λ5G 2 1(x)G 2 2(y)]. (6) The joint PDF corresponding to Eq. (6) is easily obtained by differentiating Eq.(6) with respect to X and Y and is fX,Y (x, y) = g1(x)g2(y)[1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + 2(λ4 − λ3)G2(y)− 3(λ4 + λ5)G 2 2(y) + 9λ5G 2 1(x)G 2 2(y)] (7) Where (λ1, λ2, λ3, λ4, λ5) ∈ [−1, 1] under these conditions:−2 ≤ λ1 + λ3 + λ5 ≤ 0,−1 ≤ λ2 − λ1 ≤ 1,−1 ≤ λ2 + λ5 ≤ 1, −1 ≤ λ3 − λ4 ≤ 1 and −1 ≤ λ4 + λ5 ≤ 1. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 5 of 28 The marginal CDF of X and Y for the BCT family of distributions, given in (6), can be easily obtained as follows: FX(x) = G1(x)[1 + λ1(1−G1(x)) + λ2G1(x)(1−G1(x))], (8) where G1(x) is the CDF of any baseline distribution and (λ1, λ2) are the transmutation parameters such that (λ1, λ2) ∈ [−1, 1] and −1 ⩽ λ1 + λ2 ⩽ 1. FY (y) = G2(y)[1 + λ3(1−G2(y)) + λ4G2(y)(1−G2(y))]. (9) where G2(y) is the CDF of any baseline distribution and (λ3, λ4) are the transmutation parameters such that (λ3, λ4) ∈ [−1, 1] and −1 ⩽ λ3 + λ4 ⩽ 1 Note that the distribution functions (8) and (9) are CDFs of the cubic transmuted family of distributions. The marginal PDF of the random variables X and Y for the bivariate cubic transmuted family of distributions corresponding to (8) and (9), respectively, are given as fX(x) = g1(x)[1 + λ1(1− 2G1(x)) + λ2G1(x)(2− 3G1(x))], (10) and fY (y) = g2(y)[1 + λ3(1− 2G2(y)) + λ4G2(y)(2− 3G2(y))], (11) where g1(x) and g2(y) are the PDF of any base distribution corresponding to the CDF G1(x) and G2(y), respectively. It is easy to see that (10) and (11) are density functions of the univariate cubic transmuted family of distributions which are CDFs of the Cubic Transmuted Family of Distributions proposed by [6]. The conditional distribution of X given Y = y is obtained as: fX|Y (x, y) = fX,Y (x, y) fY (y) , Using (7) and (11) in the above equation, the conditional distribution of X given Y = y for the bivariate cubic transmuted family of distributions is readily written as : fX|y(x, y) = g1(x)δ(x, y) [1 + λ3(1− 2G2(y)) + λ4G2(y)(2− 3G2(y))] , (12) where δ(x, y) = [1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + 2(λ4 − λ3)G2(y) − 3(λ4 + λ5)G 2 2(y) + 9λ5G 2 1(x)G 2 2(y)]. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 6 of 28 Again, the conditional PDF of Y given X = x is obtained by using fY |x(y, x) = fX,Y (x, y) fX(x) , Using the joint density function of X and Y and given in (7) the marginal density function of X given in (10), the conditional density function of Y given X = x is: fY |x(y, x) = g2(y)δ(x, y) [1 + λ1(1− 2G1(x)) + λ2G1(x)(2− 3G1(x))] . (13) Conditional distributions can be studied for any baseline distribution. In the next section, some properties of the proposed bivariate cubic transmuted family of distributions will be discussed. 4. Statistical Properties Some useful properties will be included in the following subsection. 4.1. The Product and Ratio Moments The (r, s)th product moment of the random variables X and Y, following bivariate cubic transmuted families of distributions are obtained in the next theorem. Theorem 1. Let X and Y have joint bivariate cubic transmuted families of distributions then the (r, s)th product moment of X and Y is µr,s x,y = E(XrY s) = ηµr xµ s y + δ2µ r x(2:2)µ s y − δ6µ r x(3:3)µ s y + δ4µ r xµ s y(2:2) − δ7µ r xµ s y(3:3) + λ5µ r x(3:3)µ s y(3:3). Where η = (1+λ1+λ3+λ5) ,δ2 = (λ2−λ1), δ4 = (λ4−λ3), δ6 = (λ2+λ5), δ7 = (λ4+λ5), µr,s x,y represents the joint raw moment of the random variables X and Y, µs y(2:2), µ s y(3:3) are the sth moment of larger observation in sample of size 2 and 3 from G2(y), µ r x(2:2), µ r x(3:3) are the rth moment of larger observation in sample of size 2 and 3 from G2(x). Proof. The product moments are defined as: µr,s x,y = E(XrY s) = ∫ ∞ −∞ ∫ ∞ −∞ xrysfX,Y (x, y)dxdy. Now using the bivariate cubic transmuted family PDF from (7), The product moments for the bivariate cubic transmuted family of distributions are written as E(XrY s) = (1 + λ1 + λ3 + λ5)µ r xµ s y + (λ2 − λ1)µ r x(2:2)µ s y − (λ2 + λ5)µ r x(3:3)µ s y + (λ4 − λ3)µ r xµ s y(2:2) − (λ4 + λ5)µ r xµ s y(3:3) + λ5µ r x(3:3)µ s y(3:3). which completes the proof. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 7 of 28 The moment of the (r, s)th ratio of the random variables X and Y is obtained in the following. Result. Let X and Y follow the bivariate cubic transmuted family of distribution The (r, s)th ratio moments of the random variables X and Y is µr,−s x,y = E( Xr Y s ) = ηµr xµ −s y + (λ2 − λ1)µ r x(2:2)µ −s y − (λ2 + λ5)µ r x(3:3)µ −s y + (λ4 − λ3)µ r xµ −s y(2:2) − (λ4 + λ5)µ r xµ −s y(3:3) + λ5µ r x(3:3)µ −s y(3:3), where µ−s y is sth negative moment of Y and µ−s y(2:2), µ −s y(3:3) are sth negative moment of larger observation in a sample of size 2 and 3 from G2(y). 4.2. The Conditional Moments In the following, the conditional moments are derived for the bivariate cubic trans- muted family of distributions. The conditional moment rth of X given Y = y is obtained in the following theorem. Theorem 2. Let X and Y follow the bivariate cubic transmuted family of distributions. E(Xr/y) = 1 Φ(y) [(η + 2(λ4 − λ3)G2(y)− 3(λ4 + λ5)G 2 2(y))µ r x + (λ2 − λ1)µ r x(2:2) − (λ2 + λ5 − 3λ5G 2 2(y))µ r x(3:3)], (14) where Φ(y) = (1 + λ3) + 2(λ4 − λ3)G2(y)− 3λ4G 2 2(y), η = 1+λ1+λ3+λ5 , µr x is rth raw moment of X and µr x(2:2) , µ r x(3:3) is rth raw moment of larger observation in a sample of size 2 and 3 respectively from G1(x). Proof. The rth conditional moment of X given Y = y is obtained by E(Xr|y) = ∫ ∞ −∞ xrf(x|y)dx. Using f(x—y) from (12), the above integral becomes E(Xr/y) = ∫ ∞ −∞ xr 1 + λ3 − 2λ3G2(y) + 2λ4G2(y)− 3λ4G2 2(y) [1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + 2(λ4 − λ3)G2(y) − 3(λ4 + λ5)G 2 2(y) + 9λ5G 2 1(x)G 2 2(y)]g1(x)dx. After simplifying, we can easily get (14). Again, the conditional moment sth of Y given X = x is given in the following theorem. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 8 of 28 Theorem 3. Let X and Y be two random variables having joint bivariate cubic transmuted family of distributions then the sth conditional moment of Y given X = x is E(Y s|x) = 1 Φ(x) [ (η + 2(λ2 − λ4)G1(x)− 3(λ2 + λ5)G 2 1(x))µ s y + (λ4 − λ3)µ s y(2:2) − (λ4 + λ5 + 3λ5G 2 2(x))µ s y(3:3) ] , (15) where Φ(x) = (1 + λ1) + 2(λ2 − λ1)G1(x)− 3λ2G 2 1(x), η = 1 + λ1 + λ3 + λ5 , µs y is rth raw moment of y and µs y(2:2) , µ s y(3:3) is rth raw moment of larger observation in a sample of size 2 and 3 respectively from G2(y). Proof. The sth conditional moment of Y given X = x is defined as E(Y s|x) = ∫ ∞ −∞ ysf(y|x)dy, where f(y—x) is the conditional distribution of Y given X = x, given in (13). Using the conditional distribution, the above integral becomes E(Y s|x) = ∫ ∞ −∞ ys [1 + λ1 − 2λ1G1(x) + 2λ2G1(x)− 3λ2G2 1(x)] [ 1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + 2(λ4 − λ3)G2(y) − 3(λ4 + λ5)G 2 2(y) + 9λ5G 2 1(x)G 2 2(y) ] g2(y)dy. This, in simplification, becomes (15) and hence the proof is complete. 4.3. The Bivariate Reliability and Hazard Rate Functions The reliability function of the BCT family is given by this theorem. Theorem 4. Let X and Y be two random variables having a bivariate cubic transmuted family of distributions then the bivariate reliability function of X and Y is R(x, y) = 1 +G1(x)(ηG2(y)− δ1) + δ2G 2 1(x)(1 +G2(y)) +G3 1(x)(λ2 − δ6G2(y)) − δ3G2(y) + δ4G 2 2(y)(1−G1(x)) +G3 2(y)(λ4 − δ7G1(x) + λ5G 3 1(x)), (16) where δ1 = (1+λ1),δ2 = (λ2−λ1), δ3 = (1+λ3), δ4 = (λ3−λ4), η = (1+λ1+λ3+λ5), δ6 = (λ2 + λ5), δ7 = (λ4 + λ5). Proof. The bivariate reliability function of X and Y is defined as R(x, y) = 1− [FX(x) + FY (y)− FX,Y (x, y)] . Using (8), (9) and (6) in the above equation, the bivariate reliability function for the bivariate cubic transmuted family of distribution completes the proof. The bivariate reliability function can be obtained for different choices of the baseline CDF G1(x) and G2(y) A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 9 of 28 The hazard rate is an important function in reliability analysis since it shows changes in the probability of failure over the lifetime of a component and is also known as the failure or failure rate function. The bivariate hazard rate function of X and Y is defined as: h(x, y) = fX,Y (x, y) R(x, y) , (17) by using the bivariate density function (7) and the bivariate reliability function (16) in (17), we can derive the bivariate hazard rate function for the bivariate cubic transmuted family of distributions. 4.4. Random Number Generation The random sample from the bivariate cubic transmuted family of distributions can be generated by using the conditional distribution method. In this method, the random sample from a bivariate distribution is generated by using two steps which are illustrated below: Step 1: Obtain a random sample using the marginal distribution of X. This is done by equating the distribution function of x, given in (8), to p1, where p1 is a uniform random variable, and solving it for x, that is obtain x by solving (1 + λ1)G1(x) + (λ2 − λ1)G 2 1(x)− λ2G 3 1(x) =p1 δ1G1(x) + δ2G 2 1(x)− λ2G 3 1(x) =p1, or λ2G 3 1(x)− δ2G 2 1(x)− δ1G1(x) + p1 = 0. Writing G (x) = w above equation can be written as λ2w 3 − δ2w 2 − δ1w + p1 = 0, or t1w 3 + t2w 2 + t3w + p1 = 0, where t1 = λ2, t2 = −δ2, t3 = −δ1. In simplification, this can be written as w = G(x) = − t2 3t1 − 2 1 3 ζ1 3t1ζ 1 3 3 + ζ 1 3 3 3× 2 1 , 3t1 where ζ1 = ∣∣−t22 + 3t1t3 ∣∣, ζ2 = −t32 + 9t1t2t3 − 27t21p1, and A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 10 of 28 ζ3 = ζ2 + √ 4ζ31 + ζ22 . The quantile function of the marginal distribution of X in the bivariate cubic trans- muted family of distributions is: x = G−1[− t2 3t1 − 2 1 3 ζ1 3t1ζ 1 3 3 + ζ 1 3 3 3× 2 1 3 t1 ]. (18) and can be computed for any baseline distribution. The random sample from the bivari- ate cubic transmuted family of distributions can be obtained by using uniform random numbers in ζ2 above and then using the quantile function. Step 2: For the application of the second step the conditional CDF of Y given X = x is needed and is obtained as: FY |X(Y |X) = ∫ y −∞ fY |x(u|x)du = ∫ y −∞ g2(u) σ [η + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + 2(λ4 − λ3)G2(u)− 3(λ4 + λ5)G 2 2(u) + 9λ5G 2 1(x)G 2 2(u)]du, Where η = (1 + λ1 + λ3 + λ5) and σ = [1 + λ1 − 2λ1G1(x) + 2λ2G1(x)− 3λ2G 2 1(x)] Making the transformation w = G2(u), w 2 = G2 2(u) then dw = g2(u)du FY |X(Y |X) = 1 σ ∫ G2(y) 0 [η + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + 2(λ4 − λ3)w − 3(λ4 + λ5)w 2 + 9λ5G 2 1(x)w 2]dw = G2(y) σ [η + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + (λ4 − λ3)G2(y)− (λ4 + λ5)G 2 2(y) + 3λ5G 2 1(x)G 2 2(y)]. (19) Now using random observation obtained from (18) in (19), and then setting FY |X(Y |X) is equal to u, where u is a uniform random number, and the random observation y is obtained by solving u = G2(y) σ [η + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x) + (λ4 − λ3)G2(y)− (λ4 + λ5)G 2 2(y) + 3λ5G 2 1(x)G 2 2(y)], for y.The solution of the above equation in terms of G2(y) is uσ = G2(y)[η + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x)] +G2 2(y)(λ4 − λ3)−G3 2(y)[(λ4 + λ5) + 3λ5G 2 1(x)], A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 11 of 28 Writing G2(y) = w1 above equation can be written as r1w 3 1 + r2w 2 1 + r3w1 + p2 = 0, where r1 = [(λ4 + λ5) + 3λ5G 2 1(x)], r2 = −(λ4 − λ3),r3 = −[η + 2(λ2 − λ1)G1(x)− 3(λ2 + λ5)G 2 1(x)], p2 = uσ. In simplification, this can be written as w1 = G2(y) = − r2 3r1 − 2 1 3ϖ1 3r1ϖ 1 3 3 + ϖ 1 3 3 3× 2 1 3 r1 , where ϖ1 = ∣∣−r22 + 3r1r3 ∣∣, ϖ2 = −r32 + 9r1r2r3 − 27r21p2, and ϖ3 = ϖ2 + √ 4ϖ3 1 +ϖ2 2, So y = G−1 2 [− r2 3r1 − 2 1 3ϖ1 3r1ϖ 1 3 3 + ϖ 1 3 3 3× 2 1 3 r1 ]. (20) Thus, the random observation y can be obtained from (20) for any baseline distribution G2(y) and different parameter choices. 4.5. The Dependence Measures In this section, three important dependence measures are obtained for the bivariate cubic transmuted family of distributions are obtained. These dependence measures include Kendall’s τ , Spearman’s ρ, and local dependence measures. The results for the measures of dependence are derived in the following subsections. 4.5.1. Kendall’s Tau Coefficient Kendall’s τ for the BCT family proposal in the following theorem: Theorem 5. Let X and Y be two random variables that have a bivariate cubic transmuted family of distributions. The Kendall’s τ for the variables X and Y are then given as: τ = 5 90 [ (2λ1 + λ2)(2λ3 + λ4)− 3λ5(15− λ1 + λ2 − λ3 + λ4) ] . (21) Proof. The Kendall’s The Kendall’s τ coefficient for two continuous random variables is computed by using: τ = 4 ∫ ∞ −∞ ∫ ∞ −∞ FX,Y (x, y)fX,Y (x, y)dxdy − 1, A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 12 of 28 Using (6) and (7), the above integral becomes τ = 4 ∫ ∞ −∞ ∫ ∞ −∞ [G1(x)G2(y)[1 + λ1 − λ1G1(x) + λ2G1(x)− λ2G 2 1(x) + λ3 − λ3G2(y) + λ4G2(y)− λ4G 2 2(y) + λ5 − λ5G 2 1(x)− λ5G 2 2(y) + λ5G 2 1(x)G 2 2(y)] × g1(x)g2(y)[1 + λ1 − 2λ1G1(x) + 2λ2G1(x)− 3λ2G 2 1(x) + λ3 − 2λ3G2(y) + 2λ4G2(y)− 3λ4G 2 2(y) + λ5 − 3λ5G 2 1(x)− 3λ5G 2 2(y) + 9λ5G 2 1(x)G 2 2(y)]]dxdy − 1, Making the transformation u = G1(x) and v = G2(y), we then interpret as: τ = 4 ∫ 1 0 ∫ 1 0 [[ uv(1 + λ1 + λ3 + λ5) + (λ2 − λ1)u 2v − (λ2 + λ5)u 3v − (λ3 − λ4)uv 2 − (λ4 + λ5)uv 3 + λ5u 3v3 ] × [ (1 + λ1 + λ3 + λ5) + 2(λ2 − λ1)u− 3(λ2 + λ5)u 2 − 2(λ3 − λ4)v − 3(λ4 + λ5)v 2 + 9λ5u 2v2 ]] dudv − 1. In simplification, the above integral reduces to (21), and hence the proof is complete. 4.5.2. Spearman’s ρ Spearman’s ρ is another measure of dependence between two variables. The Spearman coefficient ρ for the bivariate cubic transmuted family of distributions in the following theorem Theorem 6. Let X and Y be two random variables having a bivariate cubic transmuted family of distributions then the Spearman’s ρ for X and Y ρ = − 1 300 [ 25(2λ1 + λ2)(2λ3 + λ4)− λ5(15− λ1 + λ2)(15− λ3 + λ4) ] , (22) Proof. The Spearman’s ρ for two continuous random variables is obtained as: ρ = 12 ∫ ∞ −∞ ∫ ∞ −∞ [FX,Y (x, y)− FX(x)FY (y)]fX(x)fY (y)dxdy. Now, using (6), (8), (9), (10), (11) and making the transformation u = G1(x) and v = G2(y) the above equation becomes ρ = 12 ∫ 1 0 ∫ 1 0 [ [(uv + λ1uv − λ1u 2v + λ2u 2v − λ2u 3v + λ3uv − λ3uv 2 + λ4uv 2 − λ4uv 3 + λ5uv − λ5u 3v − λ5uv 3 + λ5u 3v3) − (uv + λ1uv − λ1u 2v + λ2u 2v − λ2u 3v)(1 + λ3 − λ3v + λ4v − λ4v 2)] × (1 + λ1 − 2λ1u+ 2λ2u− 3λ2u 2)(1 + λ3 − 2λ3v + 2λ4v − 3λ4u 2)]dudv. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 13 of 28 which, on simplification, becomes (22) and hence the proof is complete. It can be easily seen that Kendall’s τ is always larger than Spearman’s ρ for the bivariate cubic transmuted family of distributions. 4.6. The Local Dependence Measure A local dependence measure defines the strength of the local association between two random variables X and Y . The bivariate local dependence function for the bivariate cubic transmuted family of distribution is the following theorem. Theorem 7. The local dependence function for the bivariate cubic transmuted family of distribution is γ(x, y) = (5184)δ2δ6δ4δ7λ5g1(x)g2(y)G1(x)G2(y) [η −+2δ2G1(x)− 3δ6G2 1(x)− 2δ4G2(y)− 3δ7G2 2(y) + 9λ5G2 1(x)G 2 2(y)] 2 , (23) where η = 1 + λ1 + λ3 + λ5, δ2 = λ2 − λ1, δ6 = λ2 + λ5, δ4 = λ3 − λ4, δ7 = λ4 + λ5 Proof. The local bivariate dependence function for two continuous random variables has been defined by [15] as γ(x, y) = ∂2 ∂x∂y logf(x, y), Using the bivariate density function, given in (7), the above equation becomes γ(x, y) = ∂2 ∂x∂y log [ g1(x)g2(y)[1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)G1(x) − 3(λ2 + λ5)G 2 1(x)− 2(λ3 − λ4)G2(y)− 3(λ4 + λ5)G 2 2(y) + 9λ5G 2 1(x)G 2 2(y)] ] . which, in simplification, becomes (23) and hence the proof is complete. The local dependence function can be computed for given values of the parameters and different choices of baseline distributions. 5. Estimation of Parameters In this section, the MLE of the parameters is discussed for the bivariate cubic trans- muted family of distributions under the assumption that all the parameters of the baseline distributions G1(x) and G2(y) are known. Let X1, X2, , Xn be a random sample of size n taken from the bivariate density in (7), then the corresponding log-likelihood function can be written as: ℓ = n∑ i=1 log(g1(xi)) + n∑ i=1 log(g2(yi)) + n∑ i=1 log [ 1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)G1(xi) − 3(λ2 + λ5)G 2 1(xi)− 2(λ3 − λ4)G2(yi)− 3(λ4 + λ5)G 2 2(yi) + 9λ5G 2 1(xi)G 2 2(yi) ] . (24) A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 14 of 28 The MLEs of λ1, λ2, λ3, λ4 and λ5 which maximize the log-likelihood function in (24) are obtained by simultaneous solution of the likelihood equations. Now, the derivatives of (24) writ the unknown parameters are ∂ℓ ∂λ1 = n∑ i=1 1− 2G1(xi) Φ(xi, yi) , (25) ∂ℓ ∂λ2 = n∑ i=1 2G1(xi)− 3G2 1(xi) Φ(xi, yi)) , (26) ∂ℓ ∂λ3 = n∑ i=1 1− 2G2(yi) Φ(xi, yi)) , (27) ∂ℓ ∂λ4 = n∑ i=1 2G2(yi)− 3G2 2(yi) Φ(xi, yi) , (28) and ∂ℓ ∂λ5 = n∑ i=1 1− 3G2 2(yi)− 3G2 1(xi) + 9G2 1(xi)G 2 2(yi) Φ(xi, yi) . (29) where Φ(xi, yi) = [ 1+λ1+λ3+λ5+2(λ2−λ1)G1(xi)−3(λ2+λ5)G 2 1(xi)−2(λ3−λ4)G2(yi)− 3(λ4 + λ5)G 2 2(yi) + 9λ5G 2 1(xi)G 2 2(yi) ] . The MLE of λ1,λ2, λ3, λ4 and λ5 is obtained by equating (25), (26),(27), (28) and (29) to zero by solving the resulting equations numerically. 6. The Bivariate Cubic Transmuted Burr Distribution In this section, the bivariate cubic transmuted Burr distribution (BCTB) is proposed by using the distribution function of Burr-XII distribution as the baseline distribution in the bivariate cubic transmuted family of distributions, given in (6) Suppose that the two random variables X and Y have the Burr distributions with the following CDF: G1(x) = 1− (1 + xa1)−b1 ;x, a1, b1 ≥ 0, (30) and G2(y) = 1− (1 + ya2)−b2 ; y, a2, b2 ≥ 0. (31) Using the above CDF in (6), the distribution function of the BCTB distribution proposed is: FBCTB(x, y) = TxTy [ 1 + λ1 + λ3 + λ5 + (λ2 − λ1)Tx− (λ2 + λ5)Tx 2 − (λ3 − λ4)Ty − (λ4 + λ5)Ty 2 + λ5Tx 2Ty2 ] , (32) A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 15 of 28 where Tx = 1− (1 + xa1)−b1 and Ty = 1− (1 + ya2)−b2 . In Figure (1), the CDF of the BCTB distribution increases monotonically, with higher values appearing in the upper right corners of each graph. The surface curvature varies across the plots, reflecting the influence of the parameters on the dependence structure and shape of the distribution. Figure 1: The CDF of BCTB distribution for x, y, a1, a2, b1, b2 > 0. The density function of the BCTB distribution correspond- A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 16 of 28 ing to (32) is obtained by using the following PDF of Burr distribution g1(x) = a1b1x a1−1(1 + xa1)−(b1+1);x, a1, b1 ≥ 0, (33) and g2(y) = a2b2y a2−1(1 + ya2)−(b2+1); y, a2, b2 ≥ 0. (34) in (7), the joint density function of the proposed BCTB distribution is given as fBCTB(x, y) = a1a2b1b2x a1−1ya2−1(1 + xa1)−(b1+1)(1 + ya2)−(b2+1) [ 1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)Tx− 3(λ2 + λ5)Tx 2 − 2(λ3 − λ4)Ty − 3(λ4 + λ5)Ty 2 + 9λ5Tx 2Ty2 ] , (35) where (b1, b2) > 0, are the scale parameters, (a1, a2) > 0 are the shape parameters, (λ1, λ2, λ3, λ4, λ5) are the transmutation parameters such that (λ1, λ2, λ3, λ4, λ5) ∈ [−1, 1] under these conditions:−1 ≤ 1 + λ1 + λ3 + λ5 ≤ 1,−1 ≤ λ2 − λ1 ≤ 1,−1 ≤ λ2 + λ5 ≤ 1, −1 ≤ λ3 − λ4 ≤ 1 and −1 ≤ λ4 + λ5 ≤ 1. In Figure (2), PDF plot of the BCTB distribution is shown. Among the six plots in Figure (2), some show centrally located peaks, indicating symmetric or balanced distribu- tions. Other plots exhibit off-centered peaks or more pronounced slopes, reflecting skewed distributions or stronger interactions between the two variables. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 17 of 28 Figure 2: The PDF of BCTB distribution The following special cases can be immediately obtained from the distribution and density functions given in (32) and (35): 1. we can obtain the bivariate cubic transmuted Lomax distribution (BCTL for short) by setting a1 = a2 = 1 in (32) and (35). 2. we can obtain the bivariate cubic transmuted log-logistic distribution (BCTLog-L for short) by setting b1 = b2 = 1 in (32) and (35). A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 18 of 28 The marginal CDF of X for BCTB distribution is obtained by using (30) in (8), as: FMCT (x) = Tx [ 1− λ2(1 + xa1)−2b1 + (λ1 + λ2)(1 + xa1)−b1 ] . Now using (31) in (9). The marginal CDFs of the random variable Y for the BCTB distribution is FMCT (y) = Ty [ 1− λ4(1 + ya2)−2b2 + (λ3 + λ4)(1 + ya2)−b2 ] . Using the PDF of Burr distribution (33) and (30) in (10), the marginal density function of X for the BCTB distribution is fX(x) = a1b1x a1−1 (1 + xa1)(b1+1) [1 + λ1 + 2Tx(λ2 − λ1)− 3Tx2λ2, (36) Again, the marginal PDF of Y for the BCTB distribution are obtained by using (34) and (31) in (11), and is given as fY (y) = a2b2y a2−1 (1 + ya2)(b2+1) [1 + λ3 − 2Ty(λ3 − λ4)− 3Ty2λ4]. (37) The conditional PDF of the random variable X given Y = y is obtained using (35) and (37)in (12). The conditional distribution of X given Y = y for the BCTB distribution is fBCTB(x|y) = a1b1x a1−1 (1 + xa1)(b1+1)∆BCTB(y) [1 + λ1 + λ3 + λ5 + 2Tx(λ2 − λ1)− 3Tx2(λ2 + λ5)− 2Ty(λ3 − λ4)− 3Ty2(λ4 + λ5) + 9λ5Tx 2Ty2], where ∆BCTB(y) = [1 + λ3 − 2Ty(λ3 − λ4)− 3Ty2λ4], and The conditional distribution of Y given X = x for BCTB distribution is obtained by using (35) and (36) in (13) and is fBCTB(y|x) = a2b2y a2−1 (1 + ya2)(b2+1)∆BCTB(x) [ 1 + λ1 + λ3 + λ5 + 2Tx(λ2 − λ1)− 3Tx2(λ2 + λ5) − 2Ty(λ3 − λ4)− 3Ty2(λ4 + λ5) + 9λ5Ty 2Tx2 ] , where ∆BCTB(x) = [1 + λ1 − 2(λ1 − λ2)Tx− 3λ2Tx 2]. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 19 of 28 7. Statistical Properties This section discusses some important distributional properties of the BCTB distribu- tion and the special distributions obtained from the proposed BCTB distribution. 7.1. The Conditional Moments In the following, conditional moment rth for the BCTB distribution is obtained using the raw momen rth and the rth moment of the second and third order statistics in a sample of sizes 2 and 3 for the BurrXII distribution is given in the following result. Result: For a fixed positive integer r rth raw moment of a Burr random variable, Z, with two positive shape parameters, namely a , b and rth raw moment of its second-order statistics in a sample of size 2 and 3 are, respectively, given as µr Z = E(Zr) = bB(1 + r a , b− r a ), (38) µr Z(2:2) = E(Zr (2:2)) = 2bB(1 + r a , b− r a )− 2bB(1 + r a , 2b− r a ) (39) , and µr Z(3:3) = E(Zr (3:3)) = 6bB(1+ r a , b− r a )− 12bB(1+ r a , 2b− r a )+ 6bB(1+ r a , 3b− r a ), (40) where B(a, b) = Γ(a)Γ(b) Γ(a+b) is the complete beta function, µr Z represents the rth raw moment (also known as the non-central moment) of the random variable Z, µr Z(2:2) and µr Z(3:3) the rth raw moment of the largest order statistic in a sample of size 2 and 3 from the distribution of Z. The conditional moment rth for the BCTB distribution is derived in the following theorem. Theorem 8. If random variables X and Y have joint BCTB distribution then the rth conditional moment of X given Y = y is µr X|y = b1 ΦBCTB(y) [ A1 ·B ( 1 + r a1 , b1 − r a1 ) +A2 ·B ( 1 + r a1 , 2b1 − r a1 ) +A3 ·B ( 1 + r a1 , 3b1 − r a1 )] , (41) where: A1 = η + 2(λ4 − λ3)G2(y)− 3(λ4 + λ5)G 2 2(y) + 2(λ2 − λ1)− 6(λ2 + λ5 − 3λ5G 2 2(y)), A2 = −2(λ2 − λ1) + 12(λ2 + λ5 − 3λ5G 2 2(y)), A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 20 of 28 A3 = −6(λ2 + λ5 − 3λ5G 2 2(y)). Here, B(·, ·) is the Beta function and ΦBCTB(y) = (1 − λ3 − λ4 + 2(2λ4 + λ3)(1 + ya2)−b2 − 3λ4(1 + ya2)−2b2 is the normalization constant. Proof. The rth conditional moment of X given Y = y when X and Y have the bivariate cubic transmuted family of distributions. are given in (14) as µr X|y = E(Xr/y) = 1 Φ(y) [( η + 2(λ4 − λ3)G2(y)− 3(λ4 + λ5)G 2 2(y) ) µr x + (λ2 − λ1)µ r x(2:2) − ( λ2 + λ5 − 3λ5G 2 2(y) ) µr x(3:3) ] , where Φ(y) = (1 + λ3) + 2(λ4 − λ3)G2(y)− 3λ4G 2 2(y), η = 1+ λ1 + λ3 + λ5. Now, for the BCTB distribution, the random variable X is Burr(a1, b1) and the random variable Y is Burr(a2, b2) hence, using (38) and (39), µr x(2:2), µ r x(3:3)the rth raw moment of X and rth raw moment of larger observation in a sample of size 2 and 3 for X are, respectively, given as µr x = E(Xr) = b1B(1 + r a1 , b1 − r a1 ) (42) , µr x(2:2) = E(Xr (2:2)) = 2b1B(1 + r a1 , b1 − r a1 )− 2b1B(1 + r a1 , 2b1 − r a1 ) (43) and µr x(3:3) = E(Xr (3:3)) = 6b1B(1 + r a1 , b1 − r a1 )− 12b1B(1 + r a1 , 2b1 − r a1 ) + 6b1B(1 + r a1 , 3b1 − r a1 ). (44) Furthermore, G2(y) is given in (31) and hence ΦBCTB(y) = (1−λ3−λ4+2(2λ4+λ3)(1+ ya2)−b2 − 3λ4(1 + ya2)−2b2 . Using (31), (42),(43) and (44) in (14), the conditional moment rth of the random variable X given Y = y for the BCTB distribution and for simplification becomes (41) and hence the proof is complete. 7.2. Random Number Generation The random sample from the BCTB distribution is generated using the following two steps. Step 1: Using the inverse of distribution function of Burr distribution (30) in (18), the random observation for x is obtained as x = [(1− u1) − 1 b1 − 1] 1 a1 , (45) A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 21 of 28 where u1 = − t2 3t1 − 2 1 3 ζ1 3t1ζ 1 3 3 + ζ 1 3 3 3× 2 1 3 t1 , where t1 = λ2, t2 = −δ2, t3 = −δ1 and ζ1, ζ2, ζ3, δ1, δ2 are defined earlier. Random sample X of the BCTB distribution can be generated using (45) for various parameters choices a1, b1, λ1, and λ2. Step 2: Using the inverse distribution function of Burr distributions (31) in (20), the random observation of Y from the bivariate transmuted Burr distribution is generated as y = [(1− u2) − 1 b2 − 1] 1 a2 , (46) where u2 = − r2 3r1 − 2 1 3ϖ1 3r1ϖ 1 3 3 + ϖ 1 3 3 3× 2 1 3 r1 , where r1, and r2, ϖ1, ϖ3 are defined earlier. The random sample for Y from the BCTB distribution can be obtained using (46). 8. Estimation of Parameters let X1, X2, , Xn be a random sample of size n be taken from the joint density function of the BCTB distribution in (35), then the corresponding log-likelihood function for BCTB distribution can be written as: ℓ = n · log(a1) + n · log(a2) + n · log(b1) + n · log(b2) + (a1 − 1) n∑ i=1 log(xi) + (a2 − 1) n∑ i=1 log(yi)− (b1 + 1) n∑ i=1 log(1 + xa1i )− (b2 + 1) n∑ i=1 log(1 + ya2i ) + n∑ i=1 log [ 1 + λ1 + λ3 + λ5 + 2(λ2 − λ1)Txi − 3(λ2 + λ5)Tx 2 i − 2(λ3 − λ4)Tyi − 3(λ4 + λ5)Ty 2 i + 9λ5Tx 2 iTy 2 i ] . (47) The MLE of a1, a2, b1, b2, λ1, λ2, λ3, λ4and λ5 are obtained by maximizing the log likelihood function (47). The derivatives of (47) with respect to unknown parameters are ∂ℓ ∂a1 = n a1 + n∑ i=1 log(xi)− (b1 + 1) n∑ i=1 xa1i log(xi) 1 + xa1i + n∑ i=1 2b1x a1 i (1 + xa1i )−1−b1 log(xi) [ δ2 − 3δ6Txi + 9λ5TxiTy 2 i ] δ(xi, yi) , (48) A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 22 of 28 ∂ℓ ∂a2 = n a2 + n∑ i=1 log(yi)− (b2 + 1) n∑ i=1 ya2i log(yi) 1 + ya2i + n∑ i=1 −2b2y a2 i (1 + ya2i )−1−b2 log(yi) [ δ4 + 3δ7Tyi − 9λ5Tx 2 iTyi ] δ(xi, yi) , (49) ∂ℓ ∂b1 = n b1 − n∑ i=1 log(1 + xa1i ) + n∑ i=1 2(1 + xa1i )−b1 log(1 + xa1i ) [ δ2 − 3δ6Txi + 9λ5TxiTy 2 i ] δ(xi, yi) , (50) ∂ℓ ∂b2 = n b2 − n∑ i=1 log(1 + ya2i ) + n∑ i=1 2(1 + ya2i )−b2 log(1 + ya2i ) [ δ4 + 3δ7Tyi − 9λ5Tx 2 iTyi ] δ(xi, yi) , (51) ∂ℓ ∂λ1 = n∑ i=1 1− 2Txi δ(xi, yi) , (52) ∂ℓ ∂λ2 = n∑ i=1 2Txi − 3Tx2i δ(xi, yi) , (53) ∂ℓ ∂λ3 = n∑ i=1 1− 2Tyi δ(xi, yi) , (54) ∂ℓ ∂λ4 = n∑ i=1 2Tyi − 3Ty2i δ(xi, yi) , (55) ∂ℓ ∂λ5 = n∑ i=1 1− 3Tx2i − 3Ty2i + 9Tx2iTy 2 i δ(xi, yi) , (56) where δ(xi, yi) = [ 1 + λ1 + λ3 + λ5 + 2Txi(λ2 − λ1) − 3(λ2 + λ5)Tx 2 i − 2(λ3 − λ4)Tyi − 3(λ4 + λ5)Ty 2 i + 9λ5Tx 2 iTy 2 i ] . The corresponding MLE of the parameter vector Θ = (a1, a2, b1, b2, λ1, λ2, λ3, λ4, λ5) is obtained by equating the above derivatives to zero and solving the resulting equations numerically. 9. Simulation and Numerical Studies In this section, a simulation study is carried out to observe the performance of the MLE procedure. In addition, the BCTB distribution has been applied to real-life data sets to investigate its applicability. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 23 of 28 9.1. Simulation Study This subsection, the method of the MLE for the parameters of the BCTB distribution parameters is obtained through a Monte Carlo simulation study using the R-package ”max- Lik” and ”stats4”. The sample size n=20, 50, 100 and 200 is considered, and m the number of samples is set to 10000. The ML estimates of the parameters a1, a2, b1, b2, λ1, λ2, λ3, λ4 and λ5 are obtained numerically as the following ste: 1. The ML estimates of parametersa1, a2, b1, b2, λ1, λ2, λ3, λ4 and λ5 are computed by solving the system of non-linear equations from (48) to (56), simultaneously. 2. The bias and MSE of the estimates are calculated using the following: Bias(Θ̂) = ¯̂ Θ−Θ, MSE(Θ̂) = V ar(Θ̂) + (Bias(Θ̂))2, where Θ = (a1, a2, b1, b2, λ1, λ2, λ3, λ4, λ5) , Θ̂ = (â1, â2, b̂1, b̂2, λ̂1, λ̂2, λ̂3, λ̂4, λ̂5) and ¯̂ Θ is the mean of Θ. It is clear from the summarized results of the simulation study for different parame- ters of Table (1) that when the sample size increases, the estimates of all parameters improve. That is, the bias and the MSE width decrease as the sample sizes increase. A . A . A lsa la fi , S . H . S h a h b a z, L . I. A l-T u rk / E u r. J . P u re A p p l. M a th , 1 8 (2 ) (2 0 2 5 ), 5 8 1 9 2 4 o f 2 8 Table 1: Simulation Results for the BCTB Distribution n=20 n=50 n=100 n=200 Parameters Estimates Bias MSE Estimates Bias MSE Estimates Bias MSE Estimates Bias MSE a1 = 2.5 2.5001 0.0183 0.0130 2.5002 0.0366 0.0136 2.4998 -0.0366 0.0134 2.5001 0.0183 0.0119 a2 = 1.25 1.2501 0.0183 0.0131 1.2498 -0.0366 0.0159 1.2500 0.0000 0.0063 1.2500 0.0000 0.0137 b1 = 3.0 2.9997 -0.0549 0.0141 3.0001 0.0183 0.0119 3.0000 0.0000 0.0066 3.0000 0.0000 0.0099 b2 = 2.0 2.0000 0.0000 0.0083 1.9999 -0.0183 0.0091 2.0002 0.0366 0.0131 1.9999 -0.0183 0.0090 λ1 = 0.1 0.1000 0.0000 0.0057 0.1000 0.0000 0.0058 0.1000 0.0000 0.0058 0.1000 0.0000 0.0058 λ2 = 0.2 0.2000 0.0000 0.0116 0.2001 0.0183 0.0115 0.2002 0.0366 0.0115 0.2000 0.0000 0.0116 λ3 = 0.25 0.2498 -0.0366 0.0145 0.2499 -0.0183 0.0144 0.2501 0.0183 0.0145 0.2499 -0.0183 0.0143 λ4 = −0.15 -0.1499 0.0183 0.0086 -0.1500 0.0000 0.0086 -0.1501 -0.0183 0.0087 -0.1500 0.0000 0.0086 λ5 = 0.15 0.1500 0.0000 0.0087 0.1501 0.0183 0.0087 0.1501 0.0366 0.008 0.1501 0.0183 0.0086 n=20 n=50 n=100 n=200 Parameters Estimates Bias MSE Estimates Bias MSE Estimates Bias MSE Estimates Bias MSE a1 = 3.0 3.0001 0.0183 0.0119 3.0000 0.0000 0.0144 3.0001 0.0183 0.0072 3.0001 0.0183 0.0072 a2 = 2.0 2.0000 0.000 0.0099 2.0000 0.0000 0.0088 1.9999 -0.0183 0.0128 2.0002 0.0366 0.0107 b1 = 1.0 0.9999 -0.0183 0.0074 1.0001 0.0183 0.0115 1.0000 0.0000 0.0124 1.0001 0.0183 0.0087 b2 = 1.5 1.5000 0.0000 0.0076 1.5000 0.0000 0.0106 1.4999 -0.0183 0.0137 1.4999 -0.0183 0.0113 λ1 = 0.01 0.0100 0.0000 0.0006 0.0100 0.0000 0.0006 0.0100 0.0000 0.0006 0.0100 0.0000 0.0006 λ2 = 0.04 0.0400 0.0000 0.0023 0.0400 0.0000 0.0023 0.0400 0.0000 0.0023 0.0400 0.0000 0.0023 λ3 = 0.05 0.0500 0.0000 0.0029 0.0500 0.0000 0.0029 0.0500 0.0000 0.0029 0.0500 0.0000 0.0029 λ4 = −0.25 -0.2499 0.018 0.0144 -0.2501 -0.0183 0.0144 -0.2499 0.0183 0.0144 -0.2503 -0.0549 0.0145 λ5 = 0.03 0.0300 0.0000 0.0017 0.0300 0.0000 0.0017 0.0300 0.0000 0.0017 0.0300 0.0000 0.0017 A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 25 of 28 9.2. Real Data Applications In the following, the BCTB distribution is applied to model a real data set. The data set is analyzed using the proposed BCTB distribution along with several other distribu- tions. The “bbmle” R package is used for most numerical computations. To perform a comparative analysis among the suggested models and to evaluate the goodness of fit, we compute AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion). Recall that smaller values of these selection criteria indicate better goodness-of-fit for the distribution. We have applied the BCTB distribution to model a real data set. The data were modeled using the BCTB distribution proposed for the other distributions. The other distributions that we have used in the study are the bivariate Burr (BB) distribution by [16], the Gumbel bivariate Burr (GBB) and the bivariate transmuted Burr (BTB) distri- bution by [13]. To check the applicability, real-life application have been conducted for the proposed BCTB distribution, which are described by the following data set. A data set with GNI for all countries from 2016 (X) and 2017 (Y) is useful for analyzing economic growth trends, understanding disparities, and conducting cross-country comparisons. Table (2) gives some descriptive statistics of the data. Table 2: Summary Statistics for Dataset Min. Q1 Median Mean Q3 Max. X 0.0644 0.3839 1.1118 1.773906 2.43105 11.8088 Y 0.0663 0.39695 1.11 1.798834 2.51975 11.6818 The results of MLE and SE are listed in Table (3) for the BCTB distribution and the BTB distribution. From Table (4), we can see that the proposed BCTB distribution has the smallest AIC and BIC values, and therefore is considered the best fit for the data based on goodness-of-fit criteria. A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 26 of 28 Table 3: MLEs and SEs for selected Distribution Distribution Parameter Estimate SE BCTB a1 1.862 0.04113 a2 1.837 0.04218 b1 1.811 0.03470 b2 2.066 0.05780 λ1 −0.2454 0.06176 λ2 −0.2483 0.06157 λ3 0.002906 0.1127 λ4 0.009309 0.09424 λ5 −0.3706 0.00001056 BTB k 9.8129 8.3886 c 0.8827 8.3886 λ1 -0.1280 5.6453 λ2 -0.1280 14.5339 λ3 -0.8720 8.3886 BB P 0.04161 0.01231 a1 0.009436 0.01974 a2 0.0002489 0.0005095 B1 9.860 2.029 B2 12.88 2.649 GBB K1 0.00000016335 0.0007430 K2 0.0000001892 0.0008853 c1 0.07567 0.05968 c2 0.06122 0.06796 γ 1.000 0.2391 Table 4: Selection Criteria for Selected Distributions Distribution LogLik AIC BIC BCTB 5272.72 -10527.44 -10498.17 BTB 78.8980 -143.7970 -175.0395 BB -473.7103 957.4205 968.6631 GBB 48.5202 -87.0404 -118.2829 A. A. Alsalafi, S. H. Shahbaz, L. I. Al-Turk / Eur. J. Pure Appl. Math, 18 (2) (2025), 5819 27 of 28 10. Conclusions In this paper, we construct a novel family of distributions, named the Bivariate Cu- bic Transmuted (BCT) family of distributions. This family introduced a new approach to modeling and analyzing multivariate data with greater flexibility and precision. Ex- plicit mathematical expressions for the probability density function (PDF), the cumulative distribution function (CDF), and key statistical properties of the proposed family are rigor- ously derived. General parameter estimation techniques for the BCT family are provided using the maximum likelihood estimation (MLE) method, ensuring accurate parameter inference. As a specific case, we introduced and thoroughly investigated the Bivariate Cubic Transmuted Burr (BCTB) distribution, a member of this family. We established and explored its theoretical properties, including marginal distributions, dependence structure, and moments. 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