EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3945-3972 ISSN 1307-5543 – ejpam.com Published by New York Business Global A New Method of Generating Truncated Bivariate Families of Distributions Eftekhar Alsulami1,2,∗, Lutfiah Al-Turk1, Muhammad Qaiser Shahbaz1 1 Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia Abstract. The modeling of complex data has attracted several researchers for the quest of gener- ating new probability distributions. The joint modeling of two variables asks for some additional complexities as a bivariate distribution is needed. The field of research in developing bivariate families of distributions is somewhat new. In certain situations, the domain of data is restricted and some truncated distribution is required. Several univariate truncated families of distributions are available for modeling of a single variable but the bivariate truncated families of distributions has not been studied and in this paper, we have proposed a new bivariate truncated families of distributions. A specific sub-family has been proposed by using the bivariate Burr as a base- line distribution, resulting in a bivariate truncated Burr family of distributions. Some important statistical properties of the proposed family has been studied, which include the marginal and conditional distributions, bivariate reliability, and bivariate hazard rate functions. The maximum likelihood estimation for the parameters of the family is also carried out. The proposed bivariate truncated Burr family of distributions is studied for the Burr baseline distributions, giving rise to the bivariate truncated Burr-Burr distribution. The new bivariate truncated Burr-Burr distri- bution is explored in detail and several statistical properties of the new distribution are studied, which include the marginal and conditional distributions, product, ratio, and conditional moments. The maximum likelihood estimation for the parameters of the proposed distribution is done. The proposed bivariate truncated Burr-Burr distribution is used to model some real data sets. It is found that the proposed distribution performs better than the other distributions considered in this study. 2020 Mathematics Subject Classifications: 60E05, 62E15, 62G30, 62F10 Key Words and Phrases: Truncated Distributions; Bivariate Distributions; Burr Distribution; Lomax Distribution; Bivariate Burr Distribution; Maximum Likelihood Estimation ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5507 Email addresses: ekalsulami@uj.edu.sa (E. Alsulami), lturk@kau.edu.sa (L. Al-Turk), mkmohamad@kau.edu.sa (M. Q. Shahbaz) https://www.ejpam.com 3945 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3946 1. Introduction Probability distributions are an important tool in statistical sciences. There are sit- uations when complex data behavior is beyond the scope of the standard probability models and hence some extensions are needed. Several practical situations arise where one is interested in the simultaneous modeling of two or more phenomena. In these sit- uations, the univariate distributions are not applicable, rather, some suitable bivariate or multivariate distributions are required. Investigation of the bivariate and multivariate distributions is not widespread as compared with the univariate probability distributions, but in the last few decades, various researchers have proposed a significant number of bivariate distributions from their marginals. Over the last few decades, various techniques for generating new probability distributions have been proposed. One of these techniques is the transformed-transformer approach, which was first introduced by [7], as an extensive family of univariate distributions. This family of distributions provides several families as special cases. The beta-generated family of distributions, proposed by [15], is another popular family of distributions. This family has attracted various authors to propose new probability distributions. Some of these are the beta-normal distribution, introduced by [15], the beta-exponential distribution by [23], the beta-Weibull distribution by [16], and the beta-Pareto distribution by [3], among others. Recently, [6] have introduced a new method of generating truncated T − X families of distributions including the right-truncated and left-truncated families of distributions. These families are used to construct new generalized families of continuous distributions. In comparison of the univariate distributions, the bivariate distributions have attracted a less number of researchers. The bivariate distribution is defined as the joint distribution of two random variables. Various approaches are available to generate a bivariate distribution from the univariate marginals. A simple method has been proposed by [17] to generate a bivariate distribu- tion from given univariate marginals. The joint cumulative distribution function of this bivariate distribution is F (x1, x2) = G(x1)G(x2) [1 + α {1−G(x1)} {1−G(x2)}] , (1) where G(x1) and G(x2) are any marginal cdf ′s and α is some parameters. A bivariate beta family of distributions has been proposed by [28] by using the bivariate beta distribution of [26]. The bivariate Gamma distribution is a popular bivariate distribution. The joint density function is proposed by [22] as fX1 ,X2 (x1, x2) = a b+c Γ(b)Γ(c) xb−1 1 (x2 − x1) c−1e−ax2 ; 0 < x1 < x2 (2) where a, b, c > 0. The marginal distributions of X1 and X2 are gamma distributions, with shape parameters b and b+ c, respectively, and the scale a. Over the past years, the truncated family of distributions has received attention from E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3947 researchers. A truncated Fréchet family of distributions was proposed by [1] by using the truncated Fréchet distribution over (0, 1). A truncated inverted Kumaraswamy generated family was obtained by [9]. A truncated Burr-G family of distribution was obtained by [19] by using the truncated Burr distribution on (0,1). A Truncated Weibull-G (TW-G) family of distributions was obtained by [24] as an alternative to beta-G (B-G) family of distributions with more flexible hazard rate and greater reliability. The family of life-time models using a truncated negative binomial distribution was introduced by [23]. A trun- cated Lomax distribution was proposed by [18] by using a truncated Lomax distribution on [0, 1]. A truncated Cauchy power-G family of distributions was proposed by [4] and some important properties of the proposed family were studied. The moment estimation for the parameters of the truncated Weibull distribution was studied by [20]. Estimation of the parameters of truncated Gamma distribution was studied by [12] . The truncated Birnbaum-Saunders distribution was studied by [2]. An Erlang-Truncated Exponential distribution was proposed by [14] as an extension of the standard one parameter exponen- tial distribution, this distribution results from the mixture of Erlang distribution and the left-truncated one-parameter exponential distribution. The transmuted Erlang truncated exponential distribution was proposed by [25]. This new distribution extends the two- parameter Erlang truncated exponential distribution. A half-logistic inverted Topp–Leone family of distributions was proposed by [8]. In this paper, we have proposed a new method of generating bivariate truncated fami- lies of distributions. The proposed method will be used to generate a bivariate truncated Burr family of distributions. The outline for the paper follows. The bivariate truncated family of distributions is introduced in the following section. The bivariate truncated Burr family of distribution is proposed in section 3. The various statistical properties of the pro- posed bivariate truncated Burr family of distributions are explored in Section 4 alongside the maximum likelihood estimation for the parameters. A bivariate truncated Burr-Burr distribution is proposed in Section 5 and some useful properties of the proposed distribu- tion are given in Section 6. In Section 7, three real datasets have been used to study the suitability of the the proposed bivariate Burr-Burr distribution. Finally, conclusions are given in Section 8. 2. Bivariate Truncated Family of Distribution A truncated distribution represents a conditional distribution that is confined to the do- main of a random variable under specific circumstances. Recently, [6] have proposed some univariate truncated families of distributions. They have proposed two right-truncated families of distributions with cumulative distribution functions FR1T−X (x) = 1 RT (a) ∫ W1[G(x)] 0 r (t) dt (3) and FR2T−X (x) = 1 RT (a) ∫ a W2[G(x)] r (t) dt, (4) E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3948 where r (t) is density function of some random variable defined on R+, RT (a) is cumulative distribution function of T at a, W1 [G (x)] and W2 [G (x)] are any real valued functions of G (x) such that W1 (0) → 0,W1 (1) → a,W2 (0) → a and W2 (1) → 0. It has been shown by [6] that W1 (x) = ax and W2 (x) = a (1− x) can be some suitable choices to propose the new families of distributions. Recently, [5] have proposed a bivariate extension of the truncated family of distributions. The joint cdf of the proposed bivariate family of distributions is FR1T1T2−X1X2 (x1, x2) = 1 R (a1, a2) ∫ W11[G1(x1)] 0 ∫ W21[G2(x2)] 0 r (t1, t2) dt1dt2, (5) where r (t1, t2) is some bivariate distribution with domain on R2 +, R (a1, a2) is joint distri- bution function of (T1, T2) at (a1, a2), W11 [G1 (x1)] and W21 [G2 (x2)] are some functions of G (x1) and G (x2) such that W11 (0) = W21 (0) → 0,W11 (1) → a1 and W21 (1) → a2. A simpler version is also proposed by [5] and the joint distribution function of this version is given as FR1T1T2−X1X2 (x1, x2) = 1 R (a1, a2) ∫ a1G1(x1) 0 ∫ a2G2(x2) 0 r (t1, t2) dt1dt2. (6) The density function corresponding to above joint distribution function is fR1 (x1, x2) = a1a2g1 (x1) g2 (x2) r [a1G1 (x1) , a2G2 (x2)] R (a1, a2) , (7) where g1 (x1) and g2 (x2) are density functions corresponding to G1 (x1) and G2 (x2) . A bivariate truncated Burr family of distributions has been proposed by [5] by using r(t1, t2) = α(α+ 1)β1β2x (β1−1) 1 x (β2−1) 2 (1 + xβ1 1 + xβ2 2 )α+2 (8) in 6. In the next section, we will propose the bivariate truncated Burr family(BTBF ) of distributions by using the Burr distribution as a generating distribution in the bivariate truncated family of distributions. Various characteristics of the new family of distribu- tions will be studied. The maximum likelihood estimation (MLE) of the new family of distributions will also be discussed. E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3949 3. Bivariate Truncated Burr Family of Distribution In this section, we have proposed the BTBF of distributions by using the Burr distri- bution as a generator. Suppose X1 and X2 are two random variables having a bivariate Burr distribution with the joint probability density function (pdfs) f(x1, x2) = α(α+ 1)β1β2x (β1−1) 1 x (β2−1) 2 (1 + xβ1 1 + xβ2 2 )α+2 (9) where α ≥ 0 is the scale parameter,(β1, β2) ≥ 0 are the shape parameters. The distribution function corresponding to 9 is F (x1, x2) = 1− (1 + xβ1 1 )−α − (1 + xβ2 2 )−α + (1 + xβ1 1 + xβ2 2 )−α (10) The joint cumulative distribution function (cdf) of the proposed right BTBF of distribu- tion is obtained by using 9 in 6 which on simplifying becomes FR(x1, x2) = 1− (1 + (a1G1(x1)) β1)−α − (1 + (a2G2(x2)) β2)−α + (∆1(x1, x2)) −α γ , (11) where γ = 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α + (1 + aβ1 1 + aβ2 2 )−α and ∆1(x1, x2) = 1 + [a1G1(x1i)] β1 + [a2G2(x2i)] β2 . The pdf of the right BTBF of distribution corresponding to 11 is fR(x1, x2) = 1 γ α(α+ 1)β1β2a β1 1 aβ2 2 g1 (x1) g2 (x2)G1 (x1) β1−1G2 (x2) β2−1 (∆1(x1, x2)) −(α+2) (12) In the following section, some characteristics of the BTBF distribution will be presented. 4. Statistical Properties of Bivariate Truncated Burr Family of Distributions This section discusses several statistical properties of the BTBF . These include, the marginal distributions, the conditional distributions, the bivariate reliability ad hazard rate functions, etc.. 4.1. The Marginal Distributions The marginal pdf of X1 can be readily obtain from 12 as fTB(x1) = ∫ ∞ 0 fR(T1T2−X1X2)(x1, x2)dx2 E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3950 fTB(x1) = αβ1a β1 1 g1 (x1)G1 (x1) β1−1 γ [ [1 + (a1G1(x1)) β1 ]−(α+1) − [1 + (a1G1(x1)) β1 + aβ2 2 ]−(α+1) ] . (13) Similarly, the marginal pdf of the random variable X2 is fTB(x2) = αβ2a β2 2 g2 (x2)G2 (x2) β2−1 γ [ [1 + (a2G2(x2)) β2 ]−(α+1) − [1 + (a2G2(x2)) β2 + aβ1 1 ]−(α+1) ] , (14) where g1(x1) and g2(x2) are the density function of any baseline distribution correspond- ing to G1(x1) and G2(x2). The marginal cdf of X1 of the BTBF of distributions is FTB(x1) = 1− [ 1 + (a1G1(x1)) β1 ]−α − [ 1 + aβ2 2 ]−α + [ 1 + (a1G1(x1)) β1 + aβ2 2 ]−α γ . (15) Similarly, the marginal cdf of the second random variable X2 is FTB(x2) = 1− [ 1 + (a2G2(x2)) β2 ]−α − [ 1 + aβ1 1 ]−α + [ 1 + (a2G2(x2)) β2 + aβ1 1 ]−α γ , (16) where G1(x1) and G2(x2) are the cdf ′s of any baseline distributions. 4.2. The Conditional Distributions The conditional distribution of X1 given X2 = x2 is f(x1|x2) = (α+ 1)β1a β1 1 g1(x1)G1(x1) β1−1[∆1(x1, x2)] −(α+2) [1 + (a2G2(x2))β2 ] −(α+1) − [1 + (a2G2(x2))β2 + aβ1 1 ]−(α+1) . (17) Similarly,the conditional distribution of X2 given X1 = x1 is obtained as f(x2|x1) = (α+ 1)β2a β2 2 g2(x2)G2(x2) β2−1[∆1(x1, x2)] −(α+2) [1 + (a1G1(x1))β1 ] −(α+1) − [1 + (a1G1(x1))β1 + aβ2 2 ]−(α+1) . (18) The conditional distributions can be studied for any baseline distribution. 4.3. The Bivariate Reliability and Hazard Rate Function The reliability function provides the chance that an item or system will function at any time t (for more details see [27]). The bivariate reliability function (BRF ) of X1 and X2 is defined as R(x1, x2) = 1− [FX1(x1) + FX2(x2)− FX1,X2(x1, x2)] E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3951 Using Eqs. 15, 16, and 11 in the above equation, the BRF for the BTBF of distributions is R(x1, x2) =1− 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α + ( 1 + (a1G1(x1)) β1 + aβ2 2 )−α γ + [∆1(x1, x2)] −α − [1 + (a2G2(x2)) β2 + aβ1 1 ]−α γ (19) where G1(x1) and G2(x2) any baseline cdf ′s. The hazard rate function provides the instantaneous rate at which an item or system will fail, given that it has already survived a specific length of time. The bivariate hazard rate function(BHRF ) (see [10])is H(x1, x2) = f(x1, x2) R(x1, x2) By using the pdf and BRF of the BTBF of distribution from Eqs.12 and 19, the BHRF for the BTBF of distribution is H(x1, x2) = 1 γ α(α+ 1)β1β2a β1 1 aβ2 2 g1 (x1) g2 (x2)G1 (x1) β1−1G2 (x2) β2−1 [∆1(x1, x2)] −(α+2) ÷ [ 1− 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α + ( 1 + (a1G1(x1)) β1 + aβ2 2 )−α γ + [∆1(x1, x2)] −α − [1 + [a2G2(x2)] β2 + aβ1 1 ]−α γ ] (20) The BHRF can be calculated for any baseline distribution. 4.4. Maximum Likelihood Estimation This section discusses the maximum likelihood estimation for the parameters of the BTBF of distributions. The likelihood function for the proposed BTBF of distributions is LF =αn(α+ 1)nanβ1 1 anβ2 2 βn 1 β n 2 [ 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α + (1 + aβ1 1 + aβ2 2 )−α ]−n × n∏ i=1 g1(x1) n∏ i=1 g2(x2) n∏ i=1 G1(x1) β1−1 n∏ i=1 G2(x2) β2−1 n∏ i=1 [ 1 + [a1G1(x1)] β1 + [a2G2(x2)] β2 ]−(α+2) . E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3952 The log of likelihood function is ℓ =n ln(α) + n ln(α+ 1) + nβ1 ln(a1) + nβ2 ln(a2) + n ln(β1) + n ln(β2) + n∑ i=1 ln[g1(x1i)] + n∑ i=1 ln[g2(x2i)] + (β1 − 1) n∑ i=1 ln[G1(x1i)] + (β2 − 1) n∑ i=1 ln[G2(x2i)] − (α+ 2) n∑ i=1 ln(∆1(x1, x2))− n ln(γ), (21) where γ and ∆1(x1, x2) are earlier defined. The derivatives of log of likelihood function w.r.t α,a1,a2,β1 and β2 are ∂ℓ ∂α = n α + n α+ 1 − n∑ i=1 ln [∆1(x1, x2)]− n γ [ (1 + aβ1 1 )−α ln(1 + aβ1 1 ) + (1 + aβ2 2 )−α ln(1 + aβ2 2 ) −(1 + aβ1 1 + aβ2 2 )−α ln(1 + aβ1 1 + aβ2 2 ) ] (22) ∂ℓ ∂a1 = nβ1 a1 − (α+ 2) n∑ i=1 β1a β1−1 1 G1(x1i) β1 ∆1(x1, x2) − nαβ1a β1−1 1 γ − [ (1 + aβ1 1 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (23) ∂ℓ ∂a2 = nβ2 a2 − (α+ 2) n∑ i=1 β2a β2−1 2 G2(x2i) β2 ∆1(x1, x2) − nαβ2a β2−1 2 γ − [ (1 + aβ2 2 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (24) ∂ℓ ∂β1 =n ln(a1) + n β1 + n∑ i=1 ln(G1(x1i))− (α+ 2) n∑ i=1 [a1G1(x1i)] β1 ln[a1G1(x1i)] ∆1(x1, x2) − nαaβ1 1 ln a1 γ[ (1 + aβ1 1 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (25) and ∂ℓ ∂β2 =n ln(a2) + n β2 + n∑ i=1 ln(G2(x2i))− (α+ 2) n∑ i=1 [a2G2(x2i)] β2 ln[a2G2(x2i)] ∆1(x1, x2) − nαaβ2 2 ln a2 γ[ (1 + aβ2 2 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (26) The maximum likelihood estimators of α, a1, a2, β1 and β2 are obtained by equating the derivatives in (22), (23), (24), (25), and (26) to zero and solving the resulting equations. E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3953 The solution is done by using some numerical method. In the next section we will study the bivariate truncated Burr family of distributions for baseline Burr distribution. The resulting distribution is is called bivariate truncated Burr-Burr distribution. 5. The Bivariate Truncated Burr-Burr Distribution A family of twelve cumulative distribution functions have been proposed by [11] and the Burr distribution is one of these. Several distributions are related with the Burr dis- tribution; including the Lomax, the logistic, the log-logistic, the exponential, and Weibull distributions. In this section, we have introduced the Bivariate Truncated Burr-Burr (BTBB) distribution, by using the Burr distribution as a baseline distribution in the pro- posed BTBF of distributions. We have given some desirable statistical properties of this new distribution. The maximum likelihood estimation for the parameters of the proposed BTBB distribution is also discussed. The BTBB distribution is proposed by using the following cdf of the Burr distribution in Eq. 11. F (x1) = 1− 1 (1 + xβ1 2 )α and F (x2) = 1− 1 (1 + xβ2 2 )α . (27) The cdf of the BTBB is FBTBB(x1, x2) = 1 γ [ 1− [1 + (a1(1− (1 + xβ1 1 )−α))β1 ]−α − [1 + (a2(1− (1 + xβ2 2 )−α))β2 ]−α + [∆2(x1, x2)] −α ] , (28) where ∆2(x1, x2) = [1 + (a1(1− (1 + xβ1 1 )−α))β1 + (a2(1− (1 + xβ2 2 )−α))β2 ] The pdf for the bivariate truncated Burr-Burr distribution corresponding to Eq. 28 is fBTBB(x1, x2) = α3(α+ 1)aβ1 1 aβ2 2 β2 1β 2 2x β1−1 1 xβ2−1 2 (1 + xβ1 1 )−(α+1)(1 + xβ2 2 )−(α+1) γ × [ [1− (1 + xβ1 1 )−α]β1−1[1− (1 + xβ2 2 )−α]β2−1[∆2(x1, x2)] −(α+2) ] , (29) where γ and ∆2(x1, x2) are defined earlier. In the following section, we have discussed some properties of the BTBB distribution. 6. Properties of the Bivariate Truncated Burr-Burr Distribution In this section, some statistical properties of the BTBB distribution are discussed. The include marginal and conditional distributions, joint and ratio moments, marginal and inverse moments, conditional moments, bivariate reliability and hazard rate function and maximum likleihood estimation of the parameters. E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3954 6.1. The Marginal and Conditional Distributions The marginal distributions of X1 and X2 are readily written from Eq. (28) as FTBB(x1) = 1− [1 + a1(1− (1 + xβ1 1 )−α))β1 ]−α − [1 + aβ2 2 ]−α + [∆1(x1)] −α γ , (30) and FTBB(x2) = 1− [1 + a2(1− (1 + xβ2 2 )−α))β2 ]−α − [1 + aβ1 1 ]−α + [∆1(x2)] −α γ , (31) where ∆1(x1) = 1+(a1(1−(1+xβ1 1 )−α))β1+aβ2 2 and ∆1(x2) = 1+(a2(1−(1+xβ2 2 )−α))β2+ aβ1 1 The marginal density function for X1 and X2 are determined from Eq. (29) as fTBB(x1) = α2β2 1a β1 1 xβ1−1 1 (1 + xβ1 1 )−(α+1)[1− (1 + xβ1 1 )−α]β1−1 γ × [ [1 + (a1(1− (1 + xβ1 1 )−α))β1 ]−(α+1) − [∆1(x1)] −(α+1) ] (32) and fTBB(x2) = α2β2 2a β2 2 xβ2−1 2 (1 + xβ2 2 )−(α+1)[1− (1 + xβ2 2 )−α]β2−1 γ × [ [1 + (a2(1− (1 + xβ2 2 )−α))β2 ]−(α+1) − [∆1(x2)] −(α+1) ] (33) Using the density and distribution functions of the Burr distribution in Eq.17, the condi- tional distribution of X1 given X2 = x2 for the BTBB distribution is fBTBB(x1|x2) = α(α+ 1)aβ1 1 β2 1x β1−1 1 (1 + xβ1 1 )−(α+1)[1− (1 + xβ1 1 )−α]β1−1∆2(x1, x2) −(α+2) [1 + (a2(1− (1 + xβ2 2 )−α))β2 ]−(α+1) − [∆1(x2)]−(α+1) . (34) Again using the density and distribution functions of Burr distribution in Eq.18, the conditional distribution of X2 given X1 = x1 is fBTBB(x2|x1) = α(α+ 1)aβ2 2 β2 2x β2−1 2 (1 + xβ2 2 )−(α+1)[1− (1 + xβ2 2 )−α]β2−1∆2(x1, x2) −(α+2) [1 + (a1(1− (1 + xβ1 1 )−α))β1 ]−(α+1) − [∆1(x1)]−(α+1) (35) where ∆2(x1, x2) is earlier defined. E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3955 6.2. The Joint and Ratio Moments The (r, s)th joint moment of X1 and X2 for the BTBB distribution is obtained as E(Xr 1 Xs 2) = ∫ ∞ 0 ∫ ∞ 0 xr1 x s 2 f(x1, x2)dx1dx2 Using the joint density function ofBTBB distribution, the joint moment is E(Xr 1 Xs 2) = ∫ ∞ 0 ∫ ∞ 0 α3(α+ 1)aβ1 1 aβ2 2 β2 1β 2 2x r+β1−1 1 xs+β2−1 2 (1 + xβ1 1 )−(α+1)(1 + xβ2 2 )−(α+1) γ × [ [1− (1 + xβ1 1 )−α]β1−1[1− (1 + xβ2 2 )−α]β2−1[∆2(x1, x2)] −(α+2) ] dx1dx2 Let u = 1− (1 + xβ1 1 )−α and v = 1− (1 + xβ2 2 )−α, then E(Xr 1 Xs 2) = α(α+ 1)aβ1 1 aβ2 2 β1β2 γ ∫ 1 0 ∫ 1 0 uβ1−1 vβ2−1 [ (1− u) −( 1 αβ1 ) − 1 ]r [ (1− v) −( 1 αβ2 ) − 1 ]s[ 1 + (a1 u) β1 + (a2 v) β2 ]−(α+2) dudv Simplifying above, the joint moments for the BTBB distribution is E(Xr 1X s 2) = α(α+ 1)aβ1 1 aβ2 2 β1β2 γ ∞∑ n=0 ( −(α+ 2) n ) a β2[−(α+2)−n] 2 [ ∞∑ n,k=0 (−1)naβ1k 1 ( r n )( n k ) B ( β1(k + 1), n− r αβ1 + 1 ) ∞∑ n=0 (−1)n ( s n ) B ( β2[−(α+ 2)− n+ 1], n− s αβ2 + 1 )] (36) where B(a, b) is the complete Beta function. The correlation coefficient between X1 and X2 can be computed for the BTBB distribution. The correlation coefficients are given in Table 1 below for α = 1. It is easy to see that the correlation coefficient increases with an increase in the values of the parameters. The (r, s)th ratio moment of X1 and X2 for the BTBB distribution is E ( Xr 1 Xs 2 ) = ∫ ∞ 0 ∫ ∞ 0 α3(α+ 1)aβ1 1 aβ2 2 β2 1β 2 2x r+β1−1 1 x−s+β2−1 2 (1 + xβ1 1 )−(α+1)(1 + xβ2 2 )−(α+1) γ × [ [1− (1 + xβ1 1 )−α]β1−1[1− (1 + xβ2 2 )−α]β2−1[∆2(x1, x2)] −(α+2) ] dx1dx2 E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3956 = α(α+ 1)aβ1 1 aβ2 2 β1β2 γ ∫ 1 0 ∫ 1 0 uβ1−1 vβ2−1 [ (1− u) −( 1 αβ1 ) − 1 ]r [ (1− v) −( 1 αβ2 ) − 1 ]−s[ 1 + (a1 u) β1 + (a2 v) β2 ]−(α+2) dudv After simplify, the ratio moment for the BTBB distribution is E ( Xr 1 Xs 2 ) = α(α+ 1)aβ1 1 aβ2 2 β1β2 γ ∞∑ n=0 ( −(α+ 2) n ) a β2[−(α+2)−n] 2 [ ∞∑ n,k=0 (−1)naβ1k 1 ( r n )( n k ) ×B ( β1(k + 1), n− r αβ1 + 1 ) ∞∑ n=0 (−1)n ( −s n ) B ( β2[−(α+ 2)− n+ 1], n+ s αβ2 + 1 )] . (37) The marginal and inverse moments for the BTBB distribution are discussed in the next subsection. Table 1: Correlation Coefficient for the BTBB Distribution. β1 = 3 β2 = 3 a2 a1 2 3 4 5 2 0.189 0.238 0.259 0.272 3 0.238 0.314 0.350 0.369 4 0.259 0.350 0.396 0.421 5 0.272 0.369 0.421 0.451 β1 = 4 β2 = 5 a2 a1 2 3 4 5 2 0.352 0.379 0.377 0.374 3 0.429 0.493 0.489 0.479 4 0.446 0.535 0.535 0.533 5 0.448 0.546 0.563 0.557 β1 = 6 β2 = 6 a2 a1 2 3 4 5 2 0.472 0.495 0.486 0.479 3 0.495 0.566 0.561 0.552 4 0.486 0.561 0.566 0.563 5 0.479 0.552 0.563 0.563 E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3957 6.3. The Marginal and Inverse Moments The rth marginal moment of X1 for the BTBB distribution is obtained by using Eq.32 as E(Xr 1) = ∫ ∞ 0 xr1 α2β2 1a β1 1 xβ1−1 1 (1 + xβ1 1 )−(α+1)[1− (1 + xβ1 1 )−α]β1−1 γ × [ [1 + (a1(1− (1 + xβ1 1 )−α))β1 ]−(α+1) − [1 + (a1(1− (1 + xβ1 1 )−α))β1 + aβ2 2 ]−(α+1) ] dx1 = α β1 a β1 1 γ ∫ 1 0 uβ1−1 [ (1− u) −1 αβ1 − 1 ]r − [ [1 + (a1u) β1 ]−(α+1)[1 + (a1u) β1 + aβ2 2 ]−(α+1) ] du Simplifying above, the rth marginal moment is E(Xr 1) = α β1 a β1 1 γ ∞∑ n=0 (−1)n a β1[−(α+1)−n] 1 ( −(α+ 1) n )( r n ) B ( β1[−(α+ 2)− n+ 1], n− r αβ1 + 1 ) × [ 1− [1 + aβ2 2 ]n ] . (38) Similarly, the marginal distribution in Eq.33 can be used to obtained the sth marginal moment of the random variable X2 for the BTBB distribution as E(Xs 2) = ∫ ∞ 0 xs2 α2β2 2a β2 2 xβ2−1 2 (1 + xβ2 2 )−(α+1)[1− (1 + xβ2 2 )−α]β2−1 γ × [ [1 + (a2(1− (1 + xβ2 2 )−α))β2 ]−(α+1) − [1 + (a2(1− (1 + xβ2 2 )−α))β2 + aβ1 1 ]−(α+1) ] dx2. Solving the integral, the sth marginal moment is E(Xs 2) = α β2 a β2 2 γ ∞∑ n=0 (−1)n a β2[−(α+1)−n] 2 ( −(α+ 1) n )( s n ) B ( β2[−(α+ 2)− n+ 1], n− s αβ2 + 1 ) × [ 1− [1 + aβ1 1 ]n ] . (39) The rth and sth inverse moments can be readily written from the marginal moments by replacing r with −r and s with −s. The conditional moments for the BTBB distribution are obtained in the following sub- section. E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3958 6.4. The Conditional Moments The rth conditional moment of X1 given X2 = x2 for the BTBB distribution is ob- tained by using Eq.34 as E(Xr 1 |X2) = ∫ ∞ 0 α(α+ 1)aβ1 1 β2 1x r+β1−1 1 (1 + xβ1 1 )−(α+1)[1− (1 + xβ1 1 )−α]β1−1 [1 + (a2(1− (1 + xβ2 2 )−α))β2 ]−(α+1) − [∆1(x2)]−(α+1) [ ∆2(x1, x2) ]−(α+2) dx1 = (α+ 1)aβ1 1 β1 [1 + (a2(1− (1 + xβ2 2 )−α))β2 ]−(α+1) − [∆1(x2)]−(α+1) × ∫ 1 0 uβ1−1[(1− u) −1 αβ1 − 1]r[ 1 + (a1u) β1 + [a2(1− (1 + xβ2 2 )−α)]β2 ]−(α+2) du After simplification, we have E(Xr 1 |X2) = (α+ 1) aβ1 1 β1 [1 + (a2(1− (1 + xβ2 2 )−α))β2 ]−(α+1) − [∆1(x2)]−(α+1) ∞∑ n=0 (−1)na β1[−(α+2)−n] 1[ 1 + [a2(1− (1 + xβ2 2 )−α)]β2 ]n(−(α+ 2) n )( r n ) B ( β1[−(α+ 2)− n+ 1], n− r αβ1 + 1 ) . (40) Again, the sth conditional moment X2 given X1 = x1 for the BTBB distribution is obtained by using Eq.35 as E(Xs 2 |X1) = ∫ ∞ 0 α(α+ 1)aβ2 2 β2 2x s+β2−1 2 (1 + xβ2 2 )−(α+1)[1− (1 + xβ2 2 )−α]β2−1 [1 + (a1(1− (1 + xβ1 1 )−α))β1 ]−(α+1) − [∆1(x1)]−(α+1) [ ∆2(x1, x2) ]−(α+2) dx2 = (α+ 1)aβ2 2 β2 [1 + (a1(1− (1 + xβ1 1 )−α))β1 ]−(α+1) − [∆1(x1)]−(α+1) ∫ 1 0 vβ2−1[(1− v) −1 αβ2 − 1]s[ 1 + (a2v) β2 + [a1(1− (1 + xβ1 1 )−α)]β1 ]−(α+2) dv Simplifying, the sth conditional moment of X2 given X1 = x1 is E(Xs 2 |X1) = (α+ 1) aβ2 2 β2 [1 + (a1(1− (1 + xβ1 1 )−α))β1 ]−(α+1) − [∆1(x1)]−(α+1) ∞∑ n=0 (−1)na β2[−(α+2)−n] 2[ 1 + [a1(1− (1 + xβ1 1 )−α)]β1 ]n(−(α+ 2) n )( s n ) B ( β2[−(α+ 2)− n+ 1], n− s αβ2 + 1 ) . (41) The conditional moments are functions of random variables.. E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3959 6.5. The Bivariate Reliability Characteristics and Hazard Rate Function Using Eq.19, the BRF of BTBB distribution is RBTBB(x1, x2) =1− [ 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α γ ] − [ [∆1(x1)] −α + [∆1(x2)] −α − [∆2(x1, x2)] −α γ ] (42) Now using Eqs.(29) and 42 in Eq.20, the BHRF for the BTBB distribution is obtained as HBTBB(x1, x2) = [ α3(α+ 1)aβ1 1 aβ2 2 β2 1β 2 2x β1−1 1 xβ2−1 2 (1 + xβ1 1 )−(α+1)(1 + xβ2 2 )−(α+1) γ ] × [ 1− (1 + xβ1 1 )−α]β1−1[1− (1 + xβ2 2 )−α]β2−1∆2(x1, x2) −(α+2) ] × [ 1− [ 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α γ ] − [ [∆1(x1)] −α + [∆1(x2)] −α − [∆2(x1, x2)] −α γ ]]−1 . (43) where γ , ∆1(x1) ,∆1(x2) and ∆2(x1, x2) are defined earlier. Table 2 contains graphs of the hazard rate function for different values of the parameters. The graph indicate that the HRF diminishes with decreases in α, a1, and a2. Conversely, increasing these parameters results in an elevation of the HRF . E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3960 Table 2: Bivariate Hazard Rate Function for BTBB Distribution α=0.25 ,β1=2, β2=2.5, a1=2, a2=3 α=0.25 , β1=2, β2=2.5 ,a1=5, a2=6 α=0.5 ,β1=3,β2=3,a1=2, a2=3 α=0.5 ,β1=3,β2=3,a1=5, a2=6 α=0.75 ,β1=1.5,β2=2,a1=2, a2=3 α=0.75 ,β1=1.5,β2=2,a1=5, a2=6 E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3961 6.6. Maximum Likelihood Estimation This subsection contains the maximum likelihood estimation for the parameters of the BTBB distribution. Let X1, X2, ..., Xn represent a size-n random sample from the BTBB distribution. The likelihood function is LH =αn(α+ 1)nanβ1 1 anβ2 2 βn 1 β n 2 [ 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α + (1 + aβ1 1 + aβ2 2 )−α ]−n × n∏ i=1 [ αβ1x (β1−1) 1 (1 + xβ1 1 )−(α+1) ] n∏ i=1 [ αβ2x (β2−1) 2 (1 + xβ2 2 )−(α+1) ] × n∏ i=1 [ 1− (1 + xβ1 1 )−α ]β1−1 n∏ i=1 [ 1− (1 + xβ2 2 )−α ]β2−1 × n∏ i=1 [ 1 + (a1(1− (1 + xβ1 1 )−α)β1 + (a2(1− (1 + xβ2 2 )−α)β2 ]−(α+2) . (44) The log of likelihood function is l =3n ln(α) + n ln(α+ 1) + nβ1 ln(a1) + nβ2 ln(a2) + 2n ln(β1) + 2n ln(β2) −n ln [ 1− (1 + aβ1 1 )−α − (1 + aβ2 2 )−α + (1 + aβ1 1 + aβ2 2 )−α ] +(β1 − 1) n∑ i=1 ln(x1i) + (β2 − 1) n∑ i=1 ln(x2i)− (α+ 1) n∑ i=1 ln(1 + xβ1 1i ) −(α+ 1) n∑ i=1 ln(1 + xβ2 2i ) + (β1 − 1) n∑ i=1 ln(1− (1 + xβ1 1i ) −α) +(β2 − 1) n∑ i=1 ln(1− (1 + xβ2 2i ) −α)− (α+ 2) n∑ i=1 ln [ ∆(x1, x2) ] (45) By maximizing the log-likelihood function in Eq. (45), the MLEs of α, a1, a2, β1, and β2 are found. The derivatives of the log-likelihood function with respect to the unknown parameters are ∂l ∂α = 3n α + n α+ 1 − n∑ i=1 ln(1 + xβ1 1i )− n∑ i=1 ln(1 + xβ2 2i )− n∑ i=1 ln(∆2(x1, x2))− n n∑ i=1 λ γ +(β1 − 1) n∑ i=1 ln(1 + xβ1 1i )(1 + xβ1 1i ) −α 1− (1 + xβ1 1i ) −α + (β2 − 1) n∑ i=1 ln(1 + xβ2 2i )(1 + xβ2 2i ) −α 1− (1 + xβ2 2i ) −α +(α+ 2)β1 n∑ i=1 ln(1 + xβ1 1i )(1 + xβ1 1i ) −α(a1(1− (1 + xβ1 1i ) −α))β1−1 ∆2(x1, x2) +(α+ 2)β2 n∑ i=1 ln(1 + xβ2 2i )(1 + xβ2 2i ) −α(a2(1− (1 + xβ2 2i ) −α))β2−1 ∆2(x1, x2) (46) E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3962 where ∆2(x1, x2) and γ earlier defined and λ = (1 + aβ1 1 )−α ln(1 + aβ1 1 ) + (1 + aβ2 2 )−α ln(1 + aβ2 2 )− (1 + aβ1 1 + aβ2 2 )−α ln(1 + aβ1 1 + aβ2 2 ) ∂l ∂a1 = nβ1 a1 − (α+ 2) n∑ i=1 β1(a1(1− (1 + xβ1 1i ) −α))β1−1(1− (1 + xβ1 1i ) −α) ∆2(x1, x2) −n α aβ1−1 1 β1 γ [ (1 + aβ1 1 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (47) ∂l ∂a2 = nβ2 a2 − (α+ 2) n∑ i=1 β2(a2(1− (1 + xβ2 2i ) −α))β2−1(1− (1 + xβ2 2i ) −α) ∆2(x1, x2) −n α aβ2−1 2 β2 γ [ (1 + aβ2 2 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (48) ∂l ∂β1 =n ln(a1) + 2n β1 + n∑ i=1 ln(x1i)− (α+ 1) n∑ i=1 xβ1 1i ln(x1i)) 1 + xβ1 1i + n∑ i=1 ln(1− (1 + xβ1 1i ) −α) + α(β1 − 1) n∑ i=1 (1 + xβ1 1i ) ln(x1i) 1− (1 + xβ1 1i ) −α −(α+ 2) aβ1 1 n∑ i=1 [ 1− (1 + xβ1 1i ) −α)β1−1 ] [∆3(x1) + ∆2(x1)] ∆2(x1, x2) −n α aβ1 1 ln(a1) γ [ (1 + aβ1 1 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (49) where ∆2(x1) = aβ1 1 (1− (1 + xβ1 1i ) −α) ln(a1(1− (1 + xβ1 1i ) −α)) and ∆3(x1) = αβ1x β1 1i ln(x1i)(1 + xβ1 1i ) −(α+1). ∂l ∂β2 =n ln(a2) + 2n β2 + n∑ i=1 ln(x2i)− (α+ 1) n∑ i=1 xβ2 2i ln(x2i)) 1 + xβ2 2i + n∑ i=1 ln(1− (1 + xβ2 2i ) −α) + α(β2 − 1) n∑ i=1 (1 + xβ2 2i ) ln(x2i) 1− (1 + xβ2 2i ) −α −(α+ 2) aβ2 2 n∑ i=1 [ 1− (1 + xβ2 2i ) −α)β2−1 ] [∆3(x2) + ∆2(x2)] ∆2(x1, x2) −n α aβ2 2 ln(a2) γ [ (1 + aβ2 2 )−(α+1) − (1 + aβ1 1 + aβ2 2 )−(α+1) ] (50) where ∆2(x2) = aβ2 2 (1− (1 + xβ2 2i ) −α) ln(a2(1− (1 + xβ2 2i ) −α)) and E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3963 ∆3(x2) = αβ2x β2 2i ln(x2i)(1 + xβ2 2i ) −(α+1). The MLEs for α, a1, a2, β1, and β2 are obtained by equating the above derivatives to zero and numerically solving the resulting equations. 7. Real Data Application In this section, we have checked the suitability of the proposed BTBB distribution by using three data sets. These data sets have been modeled by using the BTBB distribution alongside five other distributions. The competing distributions that we have used in the study are bivariate Burr (BB) distribution by Durling [13],bivariate Lomax (BL), bivariate log logistic (BL-L), bivariate transmuted Burr (BTB), and bivariate transmuted Weibull (BTW) distribution. The distributions are fitted by computing the MLEs of the parameters. To compare the performance of the proposed BTBB distribution with the competing distributions, we calculated Akaike’s information criteria (AIC) and Bayesian information criteria (BIC) for each model. The best model has the smallest AIC and BIC. The application on the real data sets are given in the following. 7.1. Patients Data This data set refers to 30 patients set from [21]. The first recurrence time is represented by X1, and the asecond recurrence time is represented by X2. Table 3 contains the MLE of the parameters while Table 4 provides the computed values of AIC and BIC. It is easy to see, from Table 4, that The BTBB distribution has the smallest values of AIC and BIC, and hence is the best fit for this data. 7.2. Gross National Income Data This data represents the Gross National Income of all countries of the World having 191 observations. In the data set, X1 represents the Gross National Income for year 2016 andX2 represents the Gross National Income for 2017. TheMLEs of various distributions are given in Table 5 and the computed values of AIC and BIC are given in Table 6. From Table 6, it is clear that the BTBB distribution is the best fitted distribution as it has the smallest values of AIC and BIC. 7.3. Breaking Strength of Fluid Data The third data set is Breaking Strength of Fluid, and its size is 70. In this data, X1 denotes the force while X2 denotes the fluid’s breaking strength. The MLE′s of different distributions are given in Table 7 while Table 8 contains the computed values of AIC and BIC. From Table8, it is obvious that the BTBB distribution is the best fit for the third data also since it has the smallest values of AIC and BIC values. In order to assess whether the BTBB is an appropriate model, we have given plots of E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3964 original data and the fitted BTBB distribution in Table 9. From these plots, We con- clude that the BTBB distribution provides a good fit to the three data sets. Table 3: Maximum Likelihood Estimation and Standard Error for Given Distributions. Distribution Parameter Estimate Standard Error BTBB a1 a2 α β1 β2 6.099633 1.196423 0.035319 2.942372 17.900089 3.757314 0.079680 0.004062 1.055427 1.115601 BB α β1 β1 0.24009 1.01058 1.00785 0.05489 0.14462 0.14352 BL α 0.2437 0.04343 BL-L β1 β1 0.48789 0.48784 0.05733 0.05713 BTB c k λ1 λ2 λ3 0.61384 0.57343 -0.10532 -0.66142 -0.06115 0.13757 0.16657 4.61707 4.19430 4.32286 BTW α1 α2 θ1 θ2 λ1 λ2 λ3 0.14116 0.08136 0.09075 0.02337 -0.07125 -0.02464 -0.09762 0.01599 0.01173 0.04044 0.01651 2.46306 2.47251 3.21034 Table 4: Akaike’s and Bayesian Information Criteria for Given Distributions Distribution Log-Lik AIC BIC BTBB -352.2676 714.5352 721.5412 BB -367.7602 741.5204 745.724 BL -367.7632 737.5263 738.9276 BL-L -391.7709 787.5418 790.3442 BTB -381.0651 772.1303 779.1362 BTW -455.7447 925.4893 935.2978 E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3965 Table 5: Maximum Likelihood Estimation and Standard Error for Given Distributions Distribution Parameter Estimate Standard Error BTBB a1 a2 α β1 β1 4.451e+03 7.187e+03 3.767e-02 2.855 2.718 1.024 1.274 3.447e-03 5.850e-03 5.607e-02 BB α β1 β1 1.02615 1.73827 1.72734 0.07057 0.09080 0.09005 BL α 1.26400 0.07984 BL-L β1 β1 1.74835 1.73769 0.08739 0.08659 BTB c k λ1 λ2 λ3 1.39380 1.06489 -0.61765 -0.80989 0.65903 0.06815 0.07529 2.97112 2.97112 2.09854 BTW α1 α2 θ1 θ2 λ1 λ2 λ3 0.63299 0.41041 0.52677 0.32809 -0.92304 -0.28514 0.39789 0.03652 0.02767 0.04261 0.03480 2.89542 2.96639 2.95426 Table 6: Akaike’s and Bayesian Information Criteria for Given Distributions Distribution Log-Lik AIC BIC BTBB -427.0594 864.1188 870.2132 BB -505.2134 1016.427 1020.083 BL -567.7754 1137.551 1138.77 BL-L -505.2836 1014.567 1017.005 BTB -610.8848 1231.77 1248.031 BTW -805.9473 1625.895 1634.427 E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3966 Table 7: Maximum Likelihood Estimation and Standard Error for Given Distributions Distribution Parameter Estimate Standard Error BTBB a1 a2 α β1 β1 0.88335 1.14359 0.13561 5.64257 3.11552 0.20265 0.33553 0.02215 0.73521 0.48151 BB α β1 β1 0.22861 1.71784 1.41722 0.03475 0.15556 0.16461 BL α 0.36525 0.04217 BL-L β1 β1 0.78856 0.91949 0.06185 0.07211 BTB c k λ1 λ2 λ3 0.75154 0.70208 -0.12896 -0.80981 -0.07487 0.08629 0.10160 33.69200 27.06451 30.5993 BTW α1 α2 θ1 θ2 λ1 λ2 λ3 0.116303 0.129937 0.025941 0.021107 -0.115249 0.017064 -0.060330 0.012094 0.012523 0.009192 0.006740 1.896561 1.896561 2.389490 Table 8: Akaike’s and Bayesian Information Criteria for Given Distributions Distribution Log-Lik AIC BIC BTBB -616.712 1243.424 1254.666 BB -626.6695 1259.339 1266.084 BL -623.3973 1248.795 1251.043 BL-L -672.6314 1349.263 1353.76 BTB –672.4948 1354.99 1366.232 BTW -873.6064 1761.213 1776.952 E. Alsulami, L. Al-Turk, M. Q. Shahbaz / Eur. J. Pure Appl. Math, 17 (4) (2024), 3945-3972 3967 Table 9: Bivariate Histograms and Fitted Distribution for Three Data Sets. Data Observed Histogram Fitted Distribution First Data Second Data Third Data REFERENCES 3968 8. Conclusion In this paper, we have presented a new truncated bivariate families of distributions. The new families of distributions are explored by using the Burr distribution as a gen- erator, giving rise to the bivariate truncated Burr family of distributions. The bivariate truncated Burr family of distributions is further investigated by using the Burr distribu- tion as a baseline distribution, resulting in a bivariate truncated Burr-Burr distributions. We have investigated various statistical characteristics of the proposed bivariate Burr-Burr distribution. Additionally, we have used three real data sets with the bivariate truncated Burr-Burr distributions. We have found that the proposed bivariate truncated Burr-Burr distribution performs well for modeling the given data. 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Journal of Statistical Distributions and Applications, 1(1):1–26, 2014. Appendix Real Data Sets 1- Patients Data X1 8, 23, 22, 447, 30, 24, 7, 511, 53, 15, 7, 141, 96, 149, 536, 17, 185, 292, 22, 15, 152, 402, 13, 39, 12, 113,132, 34, 2, 130. X2 16, 13, 28, 318, 12, 245, 9, 30, 196, 154, 333, 8, 38, 70,25, 4, 117, 114, 159, 108, 362, 24, 66, 46, 40, 201, 156, 30, 25, 26. 2- Gross National Income Data X1 0.1822, 1.1512, 1.3809, 4.6252, 0.5956, 2.0302, 1.7857, 0.8350, 4.3637, 4.4443, 1.5751, 2.6632, 4.1918, 0.3509, 1.5622, 1.5765, 4.1588, 0.7419, 0.2010, 0.7574, 0.6621, 1.1353, 1.5455, 1.3730, 7.6870, 1.7759, 0.1600, 0.0721, 0.5829, 0.3246, 0.3280, 4.2664, 0.0644, 0.1850, 2.1768, 1.4354, 1.3050, 0.1396, 0.6630, 0.0792, 1.4490, 2.1088, 0.7487, 3.0955, 2.9400, 0.3323, 4.7209, 0.3268, 0.8756, 1.3282, 1.0234, 1.0185, 0.7663, 2.1316, 0.1700, 2.7645, 0.7702, 0.1603, 0.8080, 4.0066, 3.8702, 1.6623, 0.1510, 0.8785, 4.5203, 0.3889, 2.4284, 1.2460, 0.7191, 0.1779, APPENDIX 3971 0.1540, 0.7278, 0.1681, 0.4096, 5.5809, 2.4337, 4.4971, 0.6026, 1.0437, 1.8544, 1.8446, 5.0475, 3.2273, 3.4733, 0.7832, 3.8267, 0.8320, 2.2054, 0.2898, 0.3002, 3.5122, 7.4109, 0.3113, 0.5822, 2.3685, 1.3011, 0.3124, 0.0667, 0.8876, 9.7355, 2.6884, 6.5460, 0.1339, 0.1053, 2.4968, 1.2717, 0.1901, 3.3025, 0.5006, 0.3520, 1.9468, 1.6623, 0.3789, 0.5311, 1.0618, 1.5961, 0.7149, 0.1098, 0.5282, 0.9582, 1.8652, 0.2334, 4.6711, 3.3679, 0.5145, 0.0898, 0.5326, 6.7340, 3.8129, 0.5155, 1.3466, 0.5410, 1.8494, 0.3398, 0.8424, 1.1635, 0.8729, 2.4983, 2.6521, 11.8088, 2.1060, 2.3843, 0.1744, 2.3792, 1.1441, 1.0358, 0.5804, 0.2894, 5.1329, 0.2297, 1.2877, 2.5334, 0.1216, 7.8427, 2.8546, 2.9161, 0.1850, 1.1948, 0.1115, 3.3307, 1.1118, 0.4015, 1.3413, 4.7378, 5.7636, 0.2432, 0.3164, 0.2557, 1.4971, 1.2557, 0.8045, 0.1407, 0.5447, 2.9396, 1.0192, 2.3500, 1.4890, 0.5752, 0.1654, 0.7593, 6.8121, 3.8680, 5.4104, 1.9502, 0.6135, 0.2928, 1.2570, 0.5589, 0.1480, 0.3522, 0.1677. X2 0.1824, 1.1886, 1.3802, 4.7574, 0.5790, 2.0764, 1.8461, 0.9144, 4.3560, 4.5415, 1.5600, 2.6681, 4.1580, 0.3677, 1.5843, 1.6323, 4.2156, 0.7166, 0.2061, 0.8065, 0.6714, 1.1716, 1.5534, 1.3755, 7.6427, 1.8740, 0.1650, 0.0702, 0.5983, 0.3413, 0.3315, 4.3433, 0.0663, 0.1750, 2.1910, 1.5270, 1.2938, 0.1399, 0.5694, 0.0796, 1.4636, 2.2162, 0.7524, 3.1568, 3.0588, 0.3481, 4.7918, 0.3392, 0.8344, 1.3921, 1.0347, 1.0355, 0.6868, 1.9513, 0.1750, 2.8993, 0.7620, 0.1719, 0.8324, 4.1002, 3.9254, 1.6431, 0.1516, 0.9186, 4.6136, 0.4096, 2.4648, 1.2864, 0.7278, 0.2067, 0.1552, 0.7447, 0.1665, 0.4215, 5.8420, 2.5393, 4.5810, 0.6353, 1.0846, 1.9130, 1.7789, 5.3754, 3.2711, 3.5299, 0.7846, 3.8986, 0.8288, 2.2626, 0.2961, 0.3042, 3.5945, 7.0524, 0.3255, 0.6070, 2.5002, 1.3378, 0.3255, 0.0667, 1.1100, 9.7336, 2.8314, 6.5016, 0.1358, 0.1064, 2.6107, 1.3567, 0.1953, 3.4396, 0.5125, 0.3592, 2.0189, 1.6944, 0.3843, 0.5554, 1.0103, 1.6779, 0.7340, 0.1093, 0.5567, 0.9387, 1.8573, 0.2471, 4.7900, 3.3970, 0.5157, 0.0906, 0.5231, 6.8012, 3.6290, 0.5311, 1.2831, 0.5055, 1.9178, 0.3403, 0.8380, 1.1789, 0.9154, 2.6150, 2.7315, 11.6818, 2.2646, 2.4233, 0.1811, 2.3978, 1.1695, 1.0499, 0.5909, 0.2941, 4.9680, 0.2384, 1.3019, 2.6077, 0.1240, 8.2503, 2.9467, 3.0594, 0.1872, 1.1923, 0.0963, 3.4258, 1.1326, 0.4119, 1.3306, 4.7766, 5.7625, 0.2337, 0.3317, 0.2655, 1.5516, 1.2505, 0.6846, 0.1453, 0.5547, 2.8622, 1.0275, 2.4804, 1.5594, 0.5888, 0.1658, 0.8130, 6.7805, 3.9116, 5.4941, 1.9930, 0.6470, 0.2995, 1.0672, 0.5859, 0.1239, 0.3557, 0.1683. APPENDIX 3972 3- Breaking Strength of Fluid Data X1 0.66552, 1.32920, 1.99020, 2.64780, 3.30510, 3.96520, 4.61540, 5.26260, 5.91770, 6.55930, 7.20630, 7.85490, 8.48920, 9.12330, 9.75220, 10.39500, 11.01200, 11.65400, 12.27500, 12.90500, 13.51400, 14.14600, 14.77300, 15.39700, 16.00400, 16.62100, 17.23300, 17.84900, 18.46800, 19.04800, 19.68000, 20.24500, 20.84200, 21.45700, 22.04300, 22.65200, 23.21200, 23.82100, 24.43700, 25.02500, 25.58700, 26.19300, 26.74800, 27.31500, 27.88500, 28.51000, 29.07300, 29.61700, 30.21500, 30.74700, 31.33800, 33.04800, 34.72200, 36.38200, 38.03300, 39.60300, 41.23000, 42.84500, 44.44800, 45.98700, 47.54100, 49.14000, 50.65300, 52.27000, 53.63200, 55.23800, 56.51000, 58.14300, 59.55000, 60.92400. X2 1.433193, 1.921320, 2.669056, 3.279862, 3.703085, 4.175902, 5.180407, 5.311075, 5.480077, 5.944436, 6.644375, 6.870545, 7.567605, 7.363139, 7.708288, 7.861320, 8.041136, 8.567537, 8.816040, 8.680506, 9.233999, 9.491558, 9.536301, 9.699463, 9.991789, 10.379586, 10.684267, 11.283192, 11.077667, 11.295088, 11.256025, 12.181274, 12.205218, 11.998618, 12.517958, 12.407828, 13.409593, 14.055641, 13.446431, 14.335534, 14.528933, 15.351420, 15.256486, 15.450883, 15.000152, 16.024559, 16.133539, 16.790022, 17.621796, 17.252878, 17.397621, 18.850729, 18.921950, 20.292019, 21.790637, 22.214351, 23.902455, 24.655257, 26.207445, 27.813584, 27.866845, 29.506678, 30.753878, 32.763854, 34.007137, 35.547121, 37.161858,38.220492, 39.666229, 41.468994.