EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6295 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bayesian Analysis of the Lomax-Power Rayleigh (T-X) Distribution Muhammad Ijaz1, Naeem Khan1, M. I. Khan2,∗, Ghulam Mustafa3, Hana N. Alqifari4 1 Department of Mathematics and Statistics, The University of Haripur, Haripur, Pakistan 2 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, 42351, Saudi Arabia 3 Department of Statistics, Islamia College Peshawar, Peshawar, Pakistan 4 Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia Abstract. This study presents a Bayesian analysis of the Lomax-Power Rayleigh (T-X) distri- bution using both uniform and Jeffery priors. Bayes estimators are derived under four different loss functions: squared error loss, quadratic loss, weighted loss, and precautionary loss. The per- formance of the proposed estimators is demonstrated through simulation studies and a real-world dataset. Results indicate that the quadratic loss function yields the most reliable parameter esti- mates. 2020 Mathematics Subject Classifications: 47N30, 62C10, 62C12, 62XX Key Words and Phrases: Uniform and Jeffery priors, Loss functions, application, Bayes theo- rem, simulation 1. Introduction Probability distributions are central to modeling uncertainty in a variety of fields, in- cluding reliability analysis, life testing problems, engineering, agriculture sciences, and econometrics etc. Recently, many advance probability distributions have been derived by researchers to model some more complex and large data structures better than the existing probability distributions. For example, recently a lot of new versions of the Lomax dis- tribution have been derived, e.g [1] derived X-Gamma Lomax distribution, [2] presented Inverse Power Lomax distribution, [3] discuss Lomax Power Rayleigh distribution. For more study, we refer to see [4–8]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6295 Email addresses: m.ijaz@uoh.edu.pk (M. Ijaz), prof.naeemkhan1@gmail.com (N. Khan), izhar.stats@gmail.com, khanizhar@iu.edu.sa (M. I. Khan), ghulam29@gmail.com (G. Mustafa), hn.alqifari@qu.edu.sa (H. N. Alqifari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 2 of 12 Bayesian analysis is always preferable when we are given some prior information. Bayesian analyses are found to be better than classical statistics when there are extreme values in data. A large number of problems have been tackle using Bayesian analysis under different probability distributions, for example, [9] discussed Bayesian analysis of for skew symmet- ric distribution, [10] explore Bayesian and non-Bayesian inference for Truncated Cauchy power Weibull-G class of distributions, [11] considered Lomax distribution with four loss functions, [12] discussed the Bayesian analysis of the Extended Lomax distribution. Although the Lomax-Power Rayleigh (LPR) distribution is highly flexible for modeling real-world phenomena, it has received little attention within a Bayesian framework. This study aims to bridge this gap by providing a comprehensive Bayesian analysis of the LPR distribution. By deriving Bayes estimators under different priors and loss functions, we offer researchers and practitioners a novel and practical approach for modeling and ana- lyzing data that follows this distribution. This research is motivated by the growing need for accurate and reliable statistical methods in fields such as reliability analysis, survival analysis, and actuarial science. 1.1. Motivation Bayesian estimation combines prior information with new data to inform statistical in- ference. [3] introduced the Lomax-Power Rayleigh Distribution (LPRD), a four-parameter distribution with various applications in real-world datasets. The LPRD performance in fitting real-life data sets, particularly in bioscience, motivated us to explore its Bayesian estimation. This paper aims to estimate the parameters of the LPRD using a Bayesian approach and compare the results with the classical approach proposed by [3]. The LPRD (T-X) flexibility and performance make it suitable for applications in fields like engineer- ing, survival analysis, hydrology, economics, and bioscience. In this study, we considered uniform and Jeffery priors which are commonly used because of the non-informative nature. These priors contribute little subjective information, making them suitable when there is no strong information about the parameter values. Secondly, prior information is not available and hence these priors can be used as benchmarks or as reference priors. Thirdly, Jeffery priors are dominant over others because of their in- variance property. The paper is structured as follows: Section 2 introduces the LPRD (T-X)and explores its likelihood-based parameter estimation. Section 3 derives Bayesian estimation methods for the LPRD parameters. Section 4 discusses the simulation setup used in the study. Section 5 presents the analysis and results, while Section 6 concludes the study, outlining future research directions and contributions. Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 3 of 12 2. Methods and Material 2.1. Lomax-Power Rayleigh (T-X) distribution Lomax-Power Rayleigh (T-X) distribution introduced by [3] is a statistical distribu- tion combining Lomax and Power Rayleigh distributions using Transformed-Transformer (T-X) family method. It is a flexible distribution for modeling various data types, par- ticularly those with heavy tails and positive skewness. Let a random variable X follow a Lomax-Power Rayleigh (T-X) distribution with the following density function: f (x) = αθλ β2 x2α−1 ( 1 + λx2α 2β2 )−(θ+1) , α, β, λ, θ ≥ 0, x ≥ 0 (1) where, α, λ are shape parameters and β and θ are scale parameters. The corresponding CDF is given by F (x) = ( 2β2 2β2 + λx2α )θ , α, β, λ, θ ≥ 0, x ≥ 0 The hazard rate function of LPRD (T-X) is defined as h (x) = 2αθλ ( 2β2x 2α x2α + λ )−1 , α, β, λ, θ ≥ 0, x ≥ 0 1 defines the plot of PDF and CDF while 2 explains different shapes of the hazard rate functions. , Figure 1: Pdf and CDF Plots of LPRD (T-X) Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 4 of 12 Figure 2: Hazard rate function of LPRD (T-X) 2.2. The classical estimation The likelihood function of (1) which introduced by [3] becomes n∏ i=0 f (x) = ( αθλ β2 )n n∏ i=0 x2α−1e ∑n i=0 ( 1+λx2α 2β2 )−(θ+1) (2) L = n∏ i=1 f(x) = ( αθλ β2 )n n∏ i=0 x2α−1 ( 1 + λx2α 2β2 )−(θ+1) The log- likelihood function logL = nlogα+ nlogθ + nlogλ − 2nlogβ + (2α− 1) n∑ i=1 log(xi) − (θ + 1) n∑ i=1 log ( 2β2 + λx2α ) + (θ + 1) n∑ i=1 log ( 2β2 ) By partially differentiating the above equation with respect to α, θ and λ respectively and setting it equal to zero, we obtain the following estimators for the parameters: α̂ = n 2 ∑n i=1 xi [(θ + 1)λ− 1] , θ̂ = n 2αλ ∑n i=1 xi , λ̂ = n 2α (θ + 1) ∑n i=1 xi . These estimators are also presented by [3]. Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 5 of 12 3. Bayesian estimation 3.1. Posterior distribution under Uniform prior The posterior distribution of LPR under a uniform prior is defined by f(θ /xi) = ∏n i=0 f(xi /θ)g(θ)∫∞ 0 ∏n i=0 f(xi /θ)g(θ)dθ (3) where g (θ) ∝ 1 is the prior probability function and ∏n i=0 f (xi/θ) is the MLE of the Lomax-Power Rayleigh (T-X) distribution. Using Eq (2), we get f (θ /xi) = C(n+1) Γ (n+ 1) θne−θC . This indicates that f (θ/xi), Γ(n+1, C) and C= ∑n i=0 log ( 1+λx2α 2β2 ) . 3.2. Posterior distribution under Jeffery prior The posterior distribution of LPR under Jeffery prior is defined by f(θ /xi) = ∏n i=0 f(xi /θ)g(θ)∫∞ 0 ∏n i=0 f(xi /θ)g(θ)dθ (4) Where g (θ) = E [ ∂2logL(xi,θ) ∂λ2 ] = 1 θ is the Jeffery prior and ∏n i=0 f(xi/θ) is the MLE of the Lomax-Power Rayleigh (T-X) distribution. Using Eq (2) in (4), we get f (xi/θ) = C(n) Γn θn−1e−θC This indicates that f (θ/xi), Γ (n+ 1, C) and C = ∑n i=0 log ( 1 + λx2α 2β2 ) . 3.3. Bayesian estimates under Uniform prior In this section, Bayesian estimates are derived under four loss functions by using a unifor prior 3.3.1. Bayesian Estimates under Squared error loss function (SELF) The SELF is defined by θ̂SELF = (θ̂−θ) 2 The above expression yields θ̂ ∫ C(n+1) Γ (n+ 1) θne−θCdθ− ∫ θ C(n+1) Γ (n+ 1) θne−θCdλ= 0. The result for θ is given below θ̂SELF = (n+ 1) C (5) Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 6 of 12 3.3.2. Bayesian Estimates under Quadratic error loss function (QELF) The QELF is defined by θ̂QELF = ( θ̂ − θ θ )2 Simplifying the above equation for θ, we get θ̂ ∫ 1 θ2 C(n+1) Γ (n+ 1) θne−θCdθ− ∫ 1 θ C(n+1) Γ (n+ 1) θne−θCdθ= 0 Simplifying the above expression, we get θ̂QELF = (n− 1) C (6) 3.3.3. Bayesian Estimates under Weighted error loss function (WELF) The WELF is given by θ̂WELF = ( θ̂ − θ )2 θ Simplifying the above equation for θ, we get θ̂ ∫ 1 θ C(n+1) Γ (n+ 1) θne−θCdλ− ∫ C(n+1) Γ (n+ 1) θne−θCdλ = 0 Simplifying the above expression, we get θ̂WELF = n C . (7) 3.3.4. Bayesian Estimates under Precautionary error loss function (PELF) The PELF [13] can be defined as θ̂PELF = ( θ̂ − θ )2 θ̂ Simplifying further the above equation for λ θ̂ ∫ C(n+1) Γ (n+ 1) θne−θCdθ − 1 θ̂2 ∫ θ2 C(n+1) Γ (n+ 1) θne−θCdθ = 0 Hence, we obtained the following expression θ̂PELF = √ {(n+ 1) (n+ 2)} C . (8) Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 7 of 12 3.4. Bayesian estimates under Jeffery prior This section elloborates Bayesian estimates under Jeffery priors by using four loss functions. 3.4.1. Bayesian estimates under Squared error loss function (SELF) The SELF is defined by θ̂SELF = ( θ̂ − θ )2 The above expression produced θ̂ ∫ C(n) Γ (n) θ(n−1)e−θCdθ − ∫ θ C(n) Γ (n) θ(n−1)e−θCdθ = 0 We obtained the following result θ̂SELF = n C (9) 3.4.2. Bayesian estimates under Quadratic error loss function (QELF) The QELF is given by θ̂QELF = ( θ̂ − θ θ )2 After simplification, the following expression is obtained θ̂ ∫ 1 θ2 C(n) Γ (n) θ(n−1)e−θCdθ − ∫ 1 θ C(n) Γ (n) θ(n−1)e−θCdθ = 0 Finally, we obtainedthe result θ̂QELF = (n− 2) C . (10) 3.4.3. Bayesian estimates under Weighted error loss function (WELF) The WELF can be explained by θ̂WELF = ( θ̂ − θ )2 θ Simplifying the above equation, we get θ̂ ∫ 1 θ C(n) Γ (n) θ(n−1)e−θCdθ − ∫ C(n) Γ (n) θ(n−1)e−θCdλ = 0 Lastly, we obtained the following result θ̂WELF = n C . (11) Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 8 of 12 3.4.4. Bayesian estimates under Precautionary error loss function (PELF) The PELF [13] is defined by θ̂PELF = ( θ̂ − θ )2 θ̂ The Bayes estimator θ̂ can be defined as θ̂ ∫ C(n) Γ (n) θ(n−1)e−θCdλ− 1 θ̂2 ∫ θ2 C(n) Γ (n) θ(n−1)e−θCdθ = 0 Hence, we determined the result θ̂PELF = √ {n (n+ 2)} C (12) 4. Practical Applications Bayesian Analysis Application This section illustrates the practical application of the Bayesian analysis of the Lomax- Power Rayleigh (T-X) distribution developed in this study. By employing both uniform and Jeffery priors, we evaluate the performance of the Bayesian estimators using real and simulated datasets. A real dataset, sourced from existing literature [14], is analyzed to demonstrate the effectiveness of the proposed approach. 4.1. Real Data Analysis The real data set defines the failure time of 84 windshields for a particular model of aircraft (the unit for measurement is 1000 hours) with the following values: 0.040, 1.866, 2.385, 3.443, 0.301, 1.876, 2.481, 3.467, 0.309, 1.899, 2.610, 3.478, 0.557,1.911, 2.625, 3.578, 0.943, 1.912, 2.632, 3.595, 1.070, 1.914, 2.646, 3.699, 1.124, 1.981,2.661, 3.779, 1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.82,3, 4.035, 1.281, 2.085, 2.890,4.121, 1.303, 2.089, 2.902, 4.167, 1.432, 2.097, 2.934, 4.240, 1.480, 2.135, 2.962, 4.255,1.505, 2.154, 2.964, 4.278, 1.506, 2.190, 3.000, 4.305, 1.568, 2.194, 3.103, 4.376, 1.615,2.223, 3.114, 4.449, 1.619, 2.224, 3.117, 4.485, 1.652, 2.229, 3.166, 4.570, 1.652, 2.300,3.344, 4.602, 1.757, 2.324, 3.376, 4.663. Table 1 demonstrates that with the fixed values of all other parameters rather than θ, the result of QELF perform better than others when the uniform and Jeffery priors are considered. Similarly, Table 2 declared that as the sample size grows, the MSE decreases. Figure 3 clearly demonstrates that with the increase in sample size, the MSE of Bayesian estimates are decreased in both Uniform and Jeffery priors. Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 9 of 12 Table 1: Estimated value and MSE under uniform and Jeffery Priors Uniform Prior, when Jeffery’s Prior, when n = 50, α = 3.5, β = 0.05, λ = 1.5 n = 50, α = 1.5, β = 1.0, λ = 0.5 θ SELF QELF WELF PELF θ SELF QELF WELF PELF 1.5 0.0681 1.4998 1.5145 1.5385 1.5 0.0001 0.5969 0.6096 0.6284 2.0503 0.0227 0.0238 0.0258 0.2500 0.0104 0.0130 0.0176 1 2.0377 1.9935 2.0186 2.0434 2.5 0.6216 0.5975 0.6091 0.6281 0.3305 0.2830 0.3109 0.3378 0.0159 0.0105 0.0129 0.0175 2.5 2.5481 2.5007 2.5263 2.5604 3.5 0.6213 0.5968 0.6095 0.6282 1.1682 1.0641 1.1191 1.1907 0.0158 0.0103 0.0130 0.0175 3 3.0657 3.0060 3.0342 3.0753 4.5 0.6224 0.5977 0.6092 0.6279 2.5454 2.3598 2.4488 2.5758 0.0161 0.0106 0.0129 0.0175 3.5 3.5619 3.5101 0.0685 3.5855 5.5 0.6227 0.5970 0.6103 0.6282 4.4103 4.1661 2.0494 4.4808 0.0161 0.0104 0.0132 0.0175 Figure 3: (a) MSE under a Uniform Prior; (b) MSE under Jeffery Prior 5. Simulation Analysis In this section, a simulation study has been carried by choosing different parameter values with replication of 5000 times. The following quantile function was used to simulate data x = [ 2β2 λ { (1− U) −1 θ − 1 }] 1 2α Table 3, and Table 4 conclude that with different parameter values of θ and n, the MSE of QELF is lower than the MSE of all other error loss functions. Figure 4 declares that the smaller MSE are attained in both under a uniform and Jeffery priors when the QELF is considered. As we increase the sample of size n, the results of WELF becomes closure to QELF. Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 10 of 12 Table 2: Estimated value and MSE under uniform and Jeffery Priors Uniform Prior, when Jeffery’s Prior, when α = 1.5, β = 1.0, λ = 0.5, θ = 0.8 α = 1.5, β = 1.0, λ = 0.5, θ = 0.5 n SELF QELF WELF PELF n SELF QELF WELF PELF 10 0.7028 0.5753 0.6379 0.7336 10 0.6402 0.5124 0.5755 0.6714 0.0588 0.0176 0.0331 0.0733 0.0341 0.0095 0.0173 0.0451 20 0.6576 0.5982 0.6290 0.6744 20 0.6288 0.5653 0.5966 0.6447 0.0306 0.0147 0.0222 0.0365 0.0220 0.0086 0.0142 0.0266 30 0.6455 0.6040 0.6252 0.6558 30 0.6239 0.5836 0.6049 0.6355 0.0243 0.0136 0.0187 0.0276 0.0182 0.0096 0.0137 0.0214 40 0.6390 0.6074 0.6237 0.6457 40 0.6225 0.5911 0.6066 0.6298 0.0212 0.0132 0.0171 0.0231 0.0167 0.0099 0.0131 0.0186 50 0.6343 0.6095 0.6220 0.6406 50 0.6221 0.5965 0.6097 0.6280 0.0191 0.0130 0.0160 0.0209 0.0160 0.0103 0.0131 0.0175 Table 3: Estimated value and MSE under uniform and Jeffery Priors Uniform Prior, when Jeffery’s Prior, when n = 50, α = 1.5, β = 1.0, λ = 0.5 n = 50, α = 1.5, β = 1.0, λ = 0.5 θ SELF QELF WELF PELF θ SELF QELF WELF PELF 0.5 0.5092 0.5006 0.5060 0.5124 1.5 1.5143 1.4853 1.4985 1.5247 0.0027 0.0025 0.0026 0.0029 1.0515 0.9929 1.0196 1.0732 1.5 1.5303 1.4993 1.5149 1.5360 2.5 2.5214 2.4706 2.5057 2.5343 1.0852 1.0212 1.0541 1.0968 4.1497 3.9441 4.0864 4.2042 2.5 2.5473 2.5050 2.5202 2.5605 3.5 3.5350 3.4562 3.5074 3.5508 4.2569 4.0851 4.1461 4.3124 9.3372 8.8572 9.1747 9.4351 3.5 3.5702 3.4924 3.5343 3.5895 4.5 4.5520 4.4474 4.5057 4.5817 9.5584 9.0796 9.3410 9.6821 16.6341 15.7817 16.2528 16.8798 4.5 4.5915 4.5126 0.5050 4.6059 5.5 5.5583 5.4544 5.5171 5.5846 16.9504 16.3146 0.0026 17.0719 25.8990 24.8465 25.4738 26.1739 6. Conclusion In this paper, we developed a Bayesian framework for estimating LPRD(T-X) distribu- tion parameters under various loss functions. Simulation and real data set results confirm dominancy of the quadratic loss function over others. Moreover, when the sample size in- creases, the MSE of the WELF tends to the MSE of QELF. Hence, it can be claimed that when we choose such priors for the LPRD(T-X) distribution, the estimates of QELF are preferable as compared to others. Future research could explore Bayesian predictive mod- eling and real-time inference using this distribution in engineering and medical survival datasets. Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 11 of 12 Table 4: Estimated value and MSE under uniform and Jeffery Priors Uniform Prior, when Jeffery’s Prior, when α = 1.5, β = 1.0, λ = 0.5, θ = 0.8 α = 1.5, β = 1.0, λ = 0.5, θ = 0.5 n SELF QELF WELF PELF n SELF QELF WELF PELF 10 0.9814 0.7967 0.8827 1.0219 10 0.5553 0.4460 0.4979 0.5888 0.3482 0.1689 0.2427 0.4050 0.0400 0.0277 0.0320 0.0529 20 0.8835 0.8008 0.8417 0.9091 20 0.5257 0.4748 0.5012 0.5385 0.1911 0.1244 0.1568 0.2144 0.0160 0.0131 0.0137 0.0174 30 0.8540 0.7981 0.8297 0.8644 30 0.5184 0.4827 0.4990 0.5270 0.1516 0.1122 0.1338 0.1587 0.0101 0.0086 0.0088 0.0106 40 0.8406 0.8034 0.8221 0.8550 40 0.5129 0.4859 0.4988 0.5184 0.1342 0.1098 0.1214 0.1465 0.0071 0.0064 0.0064 0.0074 50 0.8320 0.7993 0.8156 0.8394 50 0.5111 0.4904 0.5008 0.5146 0.1244 0.1025 0.1133 0.1297 0.0057 0.0052 0.0053 0.0058 Figure 4: (a) MSE under a Uniform Prior; (b) MSE under Jeffery Prior Author Contributions Conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing-original draft preparation, writing-review and editing, visualization, supervision, project administration, funding acquisition, have done by M.I. and N.K., M.I.K., G.M., and H.N.A. All authors have read and agreed to the published version of the manuscript. Acknowledgements The authors are grateful to anonymous reviewers whose constructive comments led to improved presentation of the article. Muhammad Ijaz et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6295 12 of 12 References [1] E. M. Almetwally, M. Kilai, and R. Aldallal. The evolution of fuzzy rules in two-player games. Southern Economic Journal, 1(1):129–140, 2022. [2] A. S. Hassan and M. Abd-Allah. On the inverse power lomax distribution. Annals of Data Science, 6:259–278, 2019. [3] S. Q. U. Ain, K. U. I. Rather, and R. Tripathi. Lomax-power rayleigh (tx) distribu- tion, structural properties and applications in biological sciences. Pakistan Journal of Statistics, 39(1):29–44, 2023. [4] G.P. Dhungana and V. Kumar. Exponentiated odd lomax exponential distribution with application to covid-19 death cases of nepal. PLOS ONE, 17(6):e0269450, 2022. [5] M.S. Hamed, G.M. Cordeiro, and H.M. Yousof. 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