Kafi-Pawat Family of Distributions with Applications to COVID-19, Labor Economics, Medical, and Environmental Data

International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5115 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 Kafi-Pawat Family of Distributions with Applications to COVID-19, Labor Economics, Medical, and Environmental Data Rahmat Al Kafi1, Pawat Paksaranuwat2 1Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok 16424, Indonesia 2Department of Statistics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand ABSTRACT: This article presents a novel and flexible family of continuous probability distributions, namely the Kafi-Pawat family of distributions. The Kafi-Pawat family is characterized by two parameters, playing an important role in controlling the shape of the hazard rate function, thereby enhancing its flexibility for modeling diverse data behaviors. We derive key distributional functions of the Kafi-Pawat family, including its hazard rate function. To demonstrate the flexibility and practical utility of the proposed family, we introduce and study several members of the Kafi-Pawat family. The hazard rate functions of all distributions within the Kafi-Pawat family can be monotone or non-monotone, highlighting their flexibility. Parameter estimation is conducted via the method of maximum likelihood. Since the maximum likelihood estimators cannot be obtained in closed form, we employ numerical optimization techniques to obtain the fitted parameter values. The final section is to apply the established distributions to the realworld datasets. Comparative analyses among the considered distributions are performed to exhibit their potential as flexible and effective tool for modeling uncertainty. KEYWORDS: Average estimate, Exponential distribution, Heavy-tailed, Inverse Burr, Maximum likelihood estimation, Quantile function INTRODUCTION In statistical theory, developing new probability distributions is a well-established and constantly evolving field. Developing and applying probability distributions to model uncertain events remains a dynamic and an expanding area of research. A common approach in this area involves extending existing distributions by introducing additional parameters, thereby enhancing the model's flexibility to capture a broader range of real-world phenomena. In continuous probability distributions, flexibility usually refers to the ability of a model to accommodate various shapes of the hazard rate function, which is crucial in survival analysis, reliability engineering, and related fields. One of the simplest and most widely used non-negative continuous distributions is exponential distribution, defined by a single parameter. Despite its mathematical simplicity and interpretability, the exponential distribution assumes a constant hazard rate, limiting its applicability to datasets characterized by increasing, decreasing, or non-monotonic hazard behaviors [1]. To address this limitation, more flexible models were introduced, such as Gama and Weibull distributions. These distributions generalize the exponential model and are able to capture increasing and decreasing hazard rate functions [2]. They also have heavier tails compared to the exponential distribution, allowing for better modeling of extreme events. However, distributions like Gamma, Weibull, and Pareto still have limitations. In particular, they are only able to have monotonic hazard rate functions, and thereby unable to capture data with non-monotonic (e.g., unimodal) hazard behaviors [2-3]. In this study, we introduce a new family of continuous distributions called the Kafi-Pawat (KP) family of distributions. While several distribution families have been proposed previously, such as [4-12], not all their members exhibit sufficient flexibility. The primary objective of this article is to present a new class of continuous distributions for positive random variables that offers greater flexibility than widely used models such as the gamma, Pareto, and Weibull distributions. All continuous distributions belong to the KP family are capable of generating data with both monotonic and non-monotonic hazard rate functions. This level of flexibility is not typically found in classic distributions as well as distributions in other families. This feature sets the distributions in KP family apart from widely used models, including the exponential [1], Weibull [2], Pareto [3], Lindley [13], Gompertz [14], Bilal [15], Rayleigh [16], and Muth [17] distributions, all of which are limited to monotonic hazard rate behaviors. International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5116 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 KAFI-PAWAT FAMILY OF DISTRIBUTIONS A. Formulation of the Kafi-Pawat Family The conception of the Kafi-Pawat (KP) family of distributions was initially motivated by an investigation of the following rational function defined on the positive real axis: 𝑔1(𝑥)=𝑥 1+𝑥, 𝑥≥0, (1) in which plot is given in Figure 1. Figure 1. The plot of function 𝒈𝟏 on the interval [0, 7]. It is evident that the function 𝑔1(𝑥) exhibits the characteristic shape of a cumulative distribution function (CDF) and, as such, may serve as a potential CDF for a positive real-valued random variable. However, since 𝑔1(𝑥) does not contain any parameters, it lacks the flexibility required to model a wide range of real datasets. Moreover, since KP is a family of distributions, it is necessary to generalize the Equation (1) to get the general form of CDF for a family of distributions. In this context, the function 𝑔1(𝑥) is modified to derive the CDF of the proposed family of continuous distributions, referred to as the Kafi-Pawat (KP) family of distributions, which is expected to produce flexible distributions. The CDF of the KP family is constructed as follows: 1. Consider a function defined in Equation (1). By adding parameter 𝛽>0, Equation (1) becomes 𝑔2(𝑥)=𝑥 𝛽 1+𝑥 𝛽, 𝑥≥0; 𝛽>0. (2) The value of 𝛽 must be positive in order to preserve the CDF curve as already formed by the function 𝑔1(𝑥). 2. By powering a positive real number 𝛼 to the term 𝑥𝛽, Equation (2) transforms into Equation (3). 𝑔3(𝑥)=(𝑥 𝛽)𝛼 1+(𝑥 𝛽)𝛼, 𝑥≥0;𝛼>0;𝛽>0. (3) The value of 𝛼 must also be positive in order to preserve the CDF curve as already formed by the function 𝑔2(𝑥). 3. By replacing the term 𝑥𝛽 with a function 𝐺(𝑥)=𝑓(𝑥) 𝛽, Equation (3) transforms into Equation (4). 𝑔4(𝑥)=(𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼, 𝑥>0;𝛼>0;𝛽>0. (4) However, the conditions on 𝑓(𝑥) must be explicitly specified to ensure that Equation (4) satisfies the properties of a CDF. The formal definition of the KP family, together with the characterization of 𝑓(𝑥), is presented in Definition 1. Definition 1. A continuous probability distribution for a positive random variable 𝑋 is said to belong to the Kafi-Pawat family of distributions if its cumulative distribution function (CDF) can be expressed as follows: 𝐹KP(𝑥)={0, 𝑥≤0 (𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼, 𝑥>0 , (5) where 𝛼>0, 𝛽>0, and 𝑓(𝑥) is a positive, non-decreasing, invertible, differentiable, and continuous function over 𝑥>0, such that lim 𝑥→0+𝑓(𝑥)=0 and lim 𝑥→∞𝑓(𝑥)=∞. International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5117 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 Proposition 1. The function presented in Equation (5) satisfies all the essential properties of a CDF, that is, 1. 0≤𝐹KP(𝑥)≤1, for all 𝑥∈ℝ. 2. 𝐹KP(𝑥) is a non-decreasing function over ℝ. 3. 𝐹KP(𝑥) is a right-continuous function for all 𝑥∈ℝ. 4. lim 𝑥→∞𝐹KP(𝑥)=1 and lim 𝑥→−∞𝐹KP(𝑥)=0. Proof. First, it will be shown that 0≤𝐹KP(𝑥)≤1, for all 𝑥∈ℝ. (i) For 𝑥≤0, 𝐹KP(𝑥)=0 and thereby belongs to [0,1]. (ii) Consider 𝐹KP(𝑥) for 𝑥>0. For any 𝛼>0 and 𝛽>0, the following equivalences hold. 1+(𝑓(𝑥) 𝛽)𝛼>1⇔0< 1 1+(𝑓(𝑥) 𝛽)𝛼<1⇔0< (𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼<1 The above expression implies that 𝐹KP(𝑥)∈(0,1) for all 𝑥>0. However, the interval (0,1)⊆[0,1], and thus 𝐹KP(𝑥) also belongs to the closed interval [0,1]. From case (i) and (ii), we conclude that 𝐹KP(𝑥)∈[0,1] for all 𝑥∈ℝ. Next, it will be shown that 𝐹KP(𝑥) is a non-decreasing function over ℝ. The function 𝐹KP(𝑥) is non-decreasing if 𝐹KP ′(𝑥)≥0. Differentiate 𝐹KP(𝑥) in Equation (5) with respect to 𝑥, we obtain: 𝐹KP ′(𝑥)={0, 𝑥≤0 𝛼(𝑓(𝑥))𝛼−1𝑓′(𝑥)𝛽𝛼 [(𝑓(𝑥))𝛼+𝛽𝛼]2, 𝑥>0 . Since 𝛼 and 𝛽 are positive, and 𝑓(𝑥) is non-decreasing function, the expression 𝛼(𝑓(𝑥))𝛼−1𝑓′(𝑥)𝛽𝛼 [(𝑓(𝑥))𝛼+𝛽𝛼]2 is positive. Hence, the function 𝐹KP ′(𝑥) is non-negative and this implies 𝐹KP(𝑥) is non-decreasing. Third, it will be shown that 𝐹KP(𝑥) is a right-continuous function for all 𝑥∈ℝ. Consider the limit from the right of 𝐹KP(𝑥) as 𝑥 tends to a given real number 𝑐. • For 𝑐<0, yields lim 𝑥→𝑐+𝐹KP(𝑥)=0=𝐹KP(𝑐). • For 𝑐=0, yields lim 𝑥→0+𝐹KP(𝑥)=lim 𝑥→0+((𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼) = (lim 𝑥→0+𝑓(𝑥) 𝛽)𝛼 1+(lim 𝑥→0+𝑓(𝑥) 𝛽)𝛼=0𝛼 1+0𝛼=0=𝐹KP(0). • For 𝑐>0, yields lim 𝑥→𝑐+𝐹KP(𝑥)=lim 𝑥→𝑐+((𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼) = (𝑓(𝑐) 𝛽)𝛼 1+(𝑓(𝑐) 𝛽)𝛼=𝐹KP(𝑐). Since lim 𝑥→𝑐+𝐹KP(𝑥)=𝐹KP(𝑐) for all 𝑐∈ℝ, then 𝐹KP(𝑥) is a right-continuous function. Fourth, it will be shown that lim 𝑥→∞𝐹KP(𝑥)=1 and lim 𝑥→−∞𝐹KP(𝑥)=0. We are given lim 𝑥→∞𝑓(𝑥)=∞, and thereby lim 𝑥→∞𝑓(𝑥) 𝛽=∞. Hence, lim 𝑥→∞𝐹KP(𝑥)=lim 𝑥→∞((𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼)=lim 𝑥→∞(1 (𝑓(𝑥) 𝛽)−𝛼+1)= 1 0+1=1, lim 𝑥→−∞𝐹KP(𝑥)= lim 𝑥→−∞0=0. As the function presented in Equation (5) satisfies the four defining properties of a CDF, it can be concluded that 𝐹KP(𝑥) constitutes the CDF of a random variable 𝑋. ∎ Given the CDF of the Kafi-Pawat (KP) family as defined in Equation (5), the corresponding probability density function (PDF) can be derived as follows: International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5118 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 𝑓KP(𝑥)=𝑑 𝑑𝑥[𝐹KP(𝑥)]=𝑑 𝑑𝑥[(𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼]=𝛼(𝑓(𝑥))𝛼−1𝑓′(𝑥)𝛽𝛼 [(𝑓(𝑥))𝛼+𝛽𝛼]2, 𝑥>0. (6) The survival function (SF) and hazard rate function (HRF) of KP family are given, respectively, by Equations (7) and (8). 𝑆KP(𝑥)=1−𝐹KP(𝑥)=1− (𝑓(𝑥) 𝛽)𝛼 1+(𝑓(𝑥) 𝛽)𝛼=1 1+(𝑓(𝑥) 𝛽)𝛼, 𝑥>0. (7) ℎKP(𝑥)=𝑓KP(𝑥) 𝑆KP(𝑥)=𝛼(𝑓(𝑥))𝛼−1𝑓′(𝑥)𝛽𝛼 [(𝑓(𝑥))𝛼+𝛽𝛼]2 1 1+(𝑓(𝑥) 𝛽)𝛼=𝛼(𝑓(𝑥))𝛼−1𝑓′(𝑥)𝛽𝛼 [(𝑓(𝑥))𝛼+𝛽𝛼]2 1 (𝑓(𝑥))𝛼+𝛽𝛼 𝛽𝛼=𝛼(𝑓(𝑥))𝛼−1𝑓′(𝑥) (𝑓(𝑥))𝛼+𝛽𝛼, 𝑥>0. (8) B. Some Members of Kafi-Pawat Family and Their Flexibility This subsection introduces several members of Kafi-Pawat (KP) family of distributions alongside their corresponding distributional functions. Specifically, four probability distributions are presented, three of which are new distributions. 1) Inverse Burr (IB) distribution [18], if 𝑓(𝑥)=𝑥 for 𝑥>0. CDF: 𝐹IB(𝑥)={0, 𝑥≤0 (𝑥 𝛽)𝛼 1+(𝑥 𝛽)𝛼, 𝑥>0 ; PDF: 𝑓IB(𝑥)=𝛼𝛽𝛼𝑥𝛼−1 [𝛽𝛼+𝑥𝛼]2,𝑥>0. SF: 𝑆IB(𝑥)=1 1+(𝑥 𝛽)𝛼,𝑥>0; HRF: ℎIB(𝑥)=𝛼𝑥𝛼−1 𝛽𝛼+𝑥𝛼,𝑥>0. 2) Kafi-Pawat shifted exponential (KPSE) distribution (New), if 𝑓(𝑥)=𝑒𝑥−1 for 𝑥>0. CDF: 𝐹KPSE(𝑥)={0, 𝑥≤0 (𝑒𝑥−1 𝛽)𝛼 1+(𝑒𝑥−1 𝛽)𝛼, 𝑥>0 ; PDF: 𝑓KPSE(𝑥)=𝛼𝑒𝑥(𝑒𝑥−1 𝛽)𝛼−1 𝛽[1+(𝑒𝑥−1 𝛽)𝛼]2,𝑥>0. SF: 𝑆KPSE(𝑥)=1 1+(𝑒𝑥−1 𝛽)𝛼,𝑥>0; HRF: ℎKPSE(𝑥)=𝛼𝑒𝑥(𝑒𝑥−1 𝛽)𝛼−1 𝛽[1+(𝑒𝑥−1 𝛽)𝛼],𝑥>0. 3) Kafi-Pawat shifted hyperbolic cosine (KPSHC) distribution (New), if 𝑓(𝑥)=cosh(𝑥)−1 for 𝑥>0. CDF: 𝐹KPSHC(𝑥)={0, 𝑥≤0 (cosh(𝑥)−1 𝛽)𝛼 1+(cosh(𝑥)−1 𝛽)𝛼, 𝑥>0 ; PDF: 𝑓KPSHC(𝑥)=𝛼sinh(𝑥)(cosh(𝑥)−1 𝛽)𝛼−1 𝛽[1+(cosh(𝑥)−1 𝛽)𝛼]2,𝑥>0. SF: 𝑆KPSHC(𝑥)=1 1+(cosh(𝑥)−1 𝛽)𝛼,𝑥>0; HRF: ℎKPSHC(𝑥)=𝛼sinh(𝑥)(cosh(𝑥)−1 𝛽)𝛼−1 𝛽[1+(cosh(𝑥)−1 𝛽)𝛼],𝑥>0. 4) Kafi-Pawat logarithmic (KPL) distribution (New), if 𝑓(𝑥)=ln(𝑥+1) for 𝑥>0. CDF: 𝐹KPL(𝑥)={0, 𝑥≤0 (ln(𝑥+1) 𝛽)𝛼 1+(ln(𝑥+1) 𝛽)𝛼, 𝑥>0 ; PDF: 𝑓KPL(𝑥)=𝛼 𝑥+1(ln(𝑥+1) 𝛽)𝛼−1 𝛽[1+(ln(𝑥+1) 𝛽)𝛼]2,𝑥>0. SF: 𝑆KPL(𝑥)=1 1+(ln(𝑥+1) 𝛽)𝛼,𝑥>0; HRF: ℎKPL(𝑥)=𝛼 𝑥+1(ln(𝑥+1) 𝛽)𝛼−1 𝛽[1+(ln(𝑥+1) 𝛽)𝛼],𝑥>0. FLEXIBILITY OF KAFI-PAWAT FAMILY As stated in Section 1, the primary objective of this study is to introduce a family of flexible continuous distributions. To this end, an analysis of the hazard rate functions (HRFs) of the distributions within the KP family is undertaken. If the HRFs of all members exhibit both monotonic and non-monotonic behaviors, the KP family is deemed flexible. Section 2 outlines several members of the International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5119 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 KP family, namely the Inverse Burr, Kafi-Pawat Shifted Exponential, Kafi-Pawat Shifted Hyperbolic Cosine, and Kafi-Pawat Logarithmic distributions. The corresponding hazard rate analyses are presented in separate subsections below. It is important to note that there are two methods that can be considered to demonstrate whether the HRF of a given distribution is monotone or non-monotone. The choice of method depends on the mathematical expression of the HRF itself. If the HRF has a closed-form expression and is sufficiently tractable, monotonicity can be established analytically using relevant definitions or theorems. Otherwise, a graphical approach is employed, whereby several parameter combinations are selected and the corresponding HRFs are plotted. C. Flexibility of Inverse Burr Distribution According to Section 2, the hazard rate function (HRF) of inverse Burr distribution is given as follows: ℎIB(𝑥)=𝛼𝑥𝛼−1 𝛽𝛼+𝑥𝛼,𝑥>0. (9) The HRF of the inverse Burr distribution, as defined in Equation (9), exhibits various forms depending on the values of its parameters. In this distribution, 𝛽 and 𝛼 are scale and shape parameters, respectively [18]. In particular, the shape parameter 𝛼 plays a crucial role in determining the overall behavior of the HRF. To illustrate these different behaviors, the analysis of the inverse Burr distribution’s hazard rate function is presented on a case-by-case basis, highlighting the influence of the shape parameter on the distribution’s failure rate characteristics. Case 1: 0<𝛼≤1 and 𝛽>0. In this case, the HRF given in Equation (9) is a decreasing function. The proof is given as follows: The first derivative of HRF in Equation (9) is ℎIB ′(𝑥)=𝛼(𝛼−1)𝑥𝛼−2(𝛽𝛼+𝑥𝛼)−𝛼2𝑥2𝛼−2 (𝛽𝛼+𝑥𝛼)2, 𝑥>0. It is obvious that denominator (𝛽𝛼+𝑥𝛼)2 is strictly positive. Meanwhile, since 0<𝛼≤1, 𝛽>0, and 𝑥>0, then 𝛽𝛼+𝑥𝛼>0 and −1<𝛼−1≤0. This implies 𝛼(𝛼−1)𝑥𝛼−2(𝛽𝛼+𝑥𝛼)≤0 and −𝛼2𝑥2𝛼−2<0, and thereby 𝛼(𝛼−1)𝑥𝛼−2(𝛽𝛼+𝑥𝛼)−𝛼2𝑥2𝛼−2<0. Therefore, ℎIB ′(𝑥)=𝛼(𝛼−1)𝑥𝛼−2(𝛽𝛼+𝑥𝛼)−𝛼2𝑥2𝛼−2 (𝛽𝛼+𝑥𝛼)2<0. This concludes that for 0<𝛼≤1, the HRF of the inverse Burr distribution is a decreasing function. ∎ The following is the asymptotic behavior of its hazard rate function. ℎIB(𝑥;𝛼,𝛽)={∞, if 𝑥→0 1𝛽,if 𝑥→0 and 𝛼=1 0, if 𝑥→∞ In particular if 𝛼=1, the HRF of IB(𝛼,𝛽) is bounded below and above by zero and 1 𝛽, respectively. Case 2: 𝛼>1 and 𝛽>0. In this case, the HRF given in Equation (9) is a unimodal function. The proof is given as follows: First, the stationary point of function ℎIB(𝑥) must be obtained. The stationary point 𝑥𝑠 of ℎIB(𝑥) is the solution of the following equation. ℎIB ′(𝑥𝑠)=0, 𝑥>0. ⟺𝛼(𝛼−1)𝑥𝑠𝛼−2(𝛽𝛼+𝑥𝑠𝛼)−𝛼2𝑥𝑠2𝛼−2 (𝛽𝛼+𝑥𝑠𝛼)2=0 ⟺(𝛼−1)(𝛽𝛼+𝑥𝑠𝛼)−𝛼𝑥𝑠𝛼=0 ⟺(𝛼−1)𝛽𝛼−𝑥𝑠𝛼=0⟺𝑥𝑠=[(𝛼−1)𝛽𝛼]1 𝛼=𝛽(𝛼−1)1 𝛼 International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5120 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 Hence, the stationary point of ℎIB(𝑥) is [(𝛼−1)𝛽𝛼]1 𝛼 and does maximize ℎIB(𝑥) on 𝑥>0. The next step is to show that the function ℎIB(𝑥) is increasing on the interval (0,𝛽(𝛼−1)1 𝛼) and is decreasing on the interval (𝛽(𝛼−1)1 𝛼,∞). Having 𝑥∈(0,𝛽(𝛼−1)1 𝛼), it implies 𝑥<[(𝛼−1)𝛽𝛼]1 𝛼⟺𝑥𝛼<(𝛼−1)𝛽𝛼 ⟺𝑥𝛼+(𝛼−1)𝑥𝛼<(𝛼−1)𝛽𝛼+(𝛼−1)𝑥𝛼 ⟺𝛼𝑥𝛼<(𝛼−1)(𝛽𝛼+𝑥𝛼) ⟺𝛼2𝑥2𝛼−2<𝛼(𝛼−1)(𝛽𝛼+𝑥𝛼)𝑥𝛼−2 ⟺𝛼(𝛼−1)(𝛽𝛼+𝑥𝛼)𝑥𝛼−2−𝛼2𝑥2𝛼−2>0 ⟺𝛼(𝛼−1)𝑥𝛼−2(𝛽𝛼+𝑥𝛼)−𝛼2𝑥2𝛼−2 (𝛽𝛼+𝑥𝛼)2>0⟺ℎIB ′(𝑥)>0 For 𝑥∈(0,𝛽(𝛼−1)1 𝛼), the first derivative of ℎIB(𝑥) is positive. This indicates that ℎIB(𝑥) is an increasing function on (0,𝛽(𝛼−1)1 𝛼). Next, having 𝑥∈(𝛽(𝛼−1)1 𝛼,∞), it implies 𝑥>[(𝛼−1)𝛽𝛼]1 𝛼⟺𝑥𝛼>(𝛼−1)𝛽𝛼 ⟺𝑥𝛼+(𝛼−1)𝑥𝛼>(𝛼−1)𝛽𝛼+(𝛼−1)𝑥𝛼 ⟺𝛼𝑥𝛼>(𝛼−1)(𝛽𝛼+𝑥𝛼) ⟺𝛼2𝑥2𝛼−2>𝛼(𝛼−1)(𝛽𝛼+𝑥𝛼)𝑥𝛼−2 ⟺𝛼(𝛼−1)(𝛽𝛼+𝑥𝛼)𝑥𝛼−2−𝛼2𝑥2𝛼−2<0 ⟺𝛼(𝛼−1)𝑥𝛼−2(𝛽𝛼+𝑥𝛼)−𝛼2𝑥2𝛼−2 (𝛽𝛼+𝑥𝛼)2<0⟺ℎIB ′(𝑥)<0 For 𝑥∈(𝛽(𝛼−1)1 𝛼,∞), the first derivative of ℎIB(𝑥) is negative. This indicates that ℎIB(𝑥) is a decreasing function on (𝛽(𝛼−1)1 𝛼,∞). Thus, since there is a point 𝑥𝑠=𝛽(𝛼−1)1 𝛼 such that ℎIB(𝑥) increases on (0,𝑥𝑠), ℎIB ′(𝑥𝑠)=0, and ℎIB(𝑥) decreases on (𝑥𝑠,∞), the HRF of the inverse Burr distribution is unimodal (upside-down bathtub). ∎ Therefore, based on the results from all considered cases, the inverse Burr distribution exhibits potential for effectively modeling and generating data with monotone (decreasing) and non-monotone (unimodal) hazard rate characteristics. D. Flexibility of Kafi-Pawat Shifted Exponential Distribution According to Section 2, the hazard rate function (HRF) of Kafi-Pawat shifted exponential (KPSE) distribution is given as follows: ℎKPSE(𝑥)=𝛼𝑒𝑥(𝑒𝑥−1 𝛽)𝛼−1 𝛽[1+(𝑒𝑥−1 𝛽)𝛼],𝑥>0. (10) The HRF of the KPSE distribution, as defined in Equation (10), exhibits various forms depending on the values of its parameters. In this distribution, both 𝛼 and 𝛽 are shape parameters. The analysis of the KPSE hazard rate function is presented on a case-bycase basis as follows: Case 1: 𝛼=1 and 𝛽=1. In this case, Equation (10) simplifies to ℎKPSE(𝑥)=𝑒𝑥(𝑒𝑥−1 1)1−1 [1+(𝑒𝑥−1 1)1]=𝑒𝑥 𝑒𝑥=1, 𝑥>0. Hence, in this case, the hazard rate function of KPSE distribution is only a constant function. Case 2: 𝛼=1 and 𝛽>0. In this case, Equation (10) simplifies to International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5121 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 ℎKPSE(𝑥;𝛽)=𝑒𝑥(𝑒𝑥−1 𝛽)1−1 𝛽[1+𝑒𝑥−1 𝛽]=𝑒𝑥 𝑒𝑥+𝛽−1. (11) If 0<𝛽≤1, the HRF given in Equation (11) is a monotonically decreasing function. The proof is given as follows: The first derivative of HRF in Equation (11) is ℎKPSE ′(𝑥;𝛽)=𝑒𝑥(𝛽−1) (𝑒𝑥+𝛽−1)2, 𝑥>0. It is obvious that denominator (𝑒𝑥+𝛽−1)2 is strictly positive. Meanwhile, since 0<𝛽≤1, then −1<𝛽−1≤0. This implies the numerator 𝑒𝑥(𝛽−1) is non-positive. Therefore, ℎKPSE ′(𝑥;𝛽)=𝑒𝑥(𝛽−1) (𝑒𝑥+𝛽−1)2≤0. This concludes that for 0<𝛽≤1, the HRF of the KPSE distribution is a monotonically decreasing function. ∎ If 𝛽>1, the HRF given in Equation (11) is an increasing function. The proof is given as follows: The first derivative of HRF in Equation (11) is ℎKPSE ′(𝑥;𝛽)=𝑒𝑥(𝛽−1) (𝑒𝑥+𝛽−1)2, 𝑥>0. It is obvious that denominator (𝑒𝑥+𝛽−1)2 is strictly positive. Meanwhile, since 𝛽>1, then 𝛽−1>0. This implies the numerator 𝑒𝑥(𝛽−1) is strictly positive as well. Therefore, ℎKPSE ′(𝑥;𝛽)=𝑒𝑥(𝛽−1) (𝑒𝑥+𝛽−1)2>0. This concludes that for 𝛽>1, the HRF of the KPSE distribution is an increasing function. ∎ Here is the asymptotic behavior of the HRF of KPSE(1,𝛽). ℎKPSE(𝑥;𝛽)={1, if 𝑥→∞ 1𝛽, if 𝑥→0. For 0<𝛽≤1, the HRF is a monotonically decreasing function, and it is bounded below and above, respectively, by one and 1 𝛽. For 𝛽>1, the HRF is an increasing function, and it is bounded below and above, respectively, by 1 𝛽 and one. Case 3: 𝛼>0 and 𝛽=1. In this case, Equation (10) simplifies to ℎKPSE(𝑥;𝛼)=𝛼𝑒𝑥(𝑒𝑥−1)𝛼−1 [1+(𝑒𝑥−1)𝛼]. (12) The HRF in Equation (12) can be bathtub and unimodal shapes. However, since Equation (12) is too complex, a graphical approach becomes relevant to be used to show the curves produced by this HRF. Figures 2 and 3 present the plot of the hazard rate function (HRF) for KPE(𝛼,1). Figure 2. The hazard rate of 𝐊𝐏𝐒𝐄(𝜶,𝟏) for 𝟎<𝜶≤𝟏. International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5122 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 Figure 3. The hazard rate of 𝐊𝐏𝐒𝐄(𝜶,𝟏) for 𝜶>𝟏. Here is the asymptotic behavior of its HRF ℎKPSE(𝑥;𝛼)={𝛼, if 𝑥→∞ ∞, if 𝑥→0 and 𝛼≤1 0, if 𝑥→0 and 𝛼>1. For 0<𝛼≤1, the HRF has a bathtub shape, and for 𝛼>1, the HRF has a unimodal shape. Case 4: 𝛼>0, 𝛽>0, and 𝛼,𝛽≠1. In this case, Equation (10) remains the same and the plots are given in Figures 4 and 5. The hazard rate of KPSE(𝛼,𝛽) can be increasing, monotonically decreasing, bathtub, and unimodal. Figure 4. The hazard rate of 𝐊𝐏𝐒𝐄(𝜶,𝜷) for 𝜶,𝜷>𝟏 and 𝜶,𝜷<𝟏. Figure 5. The hazard rate of 𝐊𝐏𝐒𝐄(𝜶,𝜷) for 𝜶<𝟏 and 𝜷>𝟏, and 𝜶>𝟏 and 𝜷<𝟏. International Journal of Current Science Research and Review ISSN: 2581-8341 Volume 08 Issue 10 October 2025 DOI: 10.47191/ijcsrr/V8-i10-24, Impact Factor: 8.048 IJCSRR @ 2025 www.ijcsrr.org 5123 *Corresponding Author: Rahmat Al Kafi Volume 08 Issue 10 October 2025 Available at: www.ijcsrr.org Page No. 5115-5129 Therefore, based on the results from all considered cases, the KPSE distribution exhibits potential for effectively modeling and generating data with monotone (constant, increasing, monotonically decreasing) and non-monotone (bathtub and unimodal) hazard rates. E. Flexibility of Kafi-Pawat Shifted Hyperbolic Cosine Distributions According to Section 2, the hazard rate function (HRF) of Kafi-Pawat shifted hyperbolic cosine (KPSHC) distribution is given as follows: ℎKPSHC(𝑥)=𝛼sinh(𝑥)(cosh(𝑥)−1 𝛽)𝛼−1 𝛽[1+(cosh(𝑥)−1 𝛽)𝛼],𝑥>0. (13) The HRF of the KPSHC distribution, as defined in Equation (13), exhibits various shapes depending on the values of its parameters. In this distribution, both 𝛼 and 𝛽 are shape parameters. The plots of KPSHC hazard rate function are given in Figures 6 and 7. The hazard rate of KPSHC(𝛼,𝛽) can be monotonically increasing, bathtub, and unimodal. Figure 6. The plots of hazard rate function of 𝐊𝐏𝐒𝐇𝐂(𝜶,𝜷) for 𝟎<𝜶≤𝟏 and 𝜷>𝟎. Figure 7. The plots of hazard rate function of 𝐊𝐏𝐒𝐇𝐂(𝜶,𝜷) for 𝜶>𝟏 and 𝜷>𝟎.