Int. J. Anal. Appl. (2023), 21:61 Powered Inverse Rayleigh Distribution Using DUS Transformation M. I. Khan1, Abdelfattah Mustafa1,2,∗ 1Mathematics Department, Faculty of Science, Islamic University of Madinah, Madinah 42351, KSA 2Mathematics Department, Faculty of Science, Mansoura University, Mnasoura 35516, Egypt ∗Corresponding author: amelsayed@mans.edu.eg Abstract. This article reports an extension of powered inverse Rayleigh distribution via DUS trans- formation, named DUS-Powered Inverse Rayleigh (DUS-PIR) distribution. Some statistical properties of suggested distribution in particular, moments, mode, quantiles, order statistics, entropy and , in- equality measures have been investigated extensively. To estimate the parameters, maximum likelihood estimation (MLE) is discussed. The model flexibility is validated by two real data. 1. Introduction The accuracy and consistency of statistical analysis are extremely affected by the assumed probability model or distribution. As a result of this verity, in recent decades formulating new distributions becomes a basic conception in statistical theory; this is generally done by adding an extra parameter to the baseline distribution. For example, [1–5] and many more. The different transformation techniques have been used by the several authors. For example, DUS, sine, and MG transformations are reported by [6–8]. In all transformation exponential distribution is deemed as baseline distribution. If g(x) and G(x) denote the probability density function (PDF) and cumulative density function (CDF) of a baseline lifetime distribution, then the PDF and CDF of a DUS-transformation are given as f (x) = 1 e − 1 g(x)eG(x), x > 0, (1.1) F (x) = eG(x) − 1 e − 1 , x > 0. (1.2) Received: Mar. 17, 2023. 2020 Mathematics Subject Classification. 62G05, 62G20. Key words and phrases. DUS transformation; lifetime distribution; entropy; maximum likelihood estimation. https://doi.org/10.28924/2291-8639-21-2023-61 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-61 2 Int. J. Anal. Appl. (2023), 21:61 Hazard rate function (HRF) is. h(x) = g(x) e−[G(x)−1] − 1 . (1.3) Inverse Rayleigh (IR) distribution was introduced by [9]. IR distribution finds enormous applications in survival analysis. Various properties of IR distribution have been studied by [10]. Powered IR distribution was proposed by [11] through the powered transformation to increase its flexibility and applicability. [5] established and studied in detail the length powered IR distribution. [12] established the several recurrence relations from powered IR distribution. A random variable (r.v.) X follows powered IR distribution, if its PDF and CDF are given, respec- tively by: g(x; α,θ) = 2α θx2α+1 e − 1 θx2α , α,θ > 0, x > 0. (1.4) G(x; α,θ) = e − 1 θx2α , α,θ > 0, x > 0. (1.5) To modelling all kinds of data sets, no single distribution can be speculated as the best fit. Despite of existence many distributions in the literature. We are therefore induced to establish a new distribution via DUS transformation and named as DUS-Powered IR distribution. The paper is framed as follows: In Section 2, DUS-Powered IR distribution is derived and graphically depicted. Several mathematical and statistical properties are established in Section 3. Also, the entropies and measures of inequality are addressed in Sections 3. The parameters estimation is obtained in Section 4. The model superiority is shown through two real data in Section 5. Section 6 reports the concluding remarks. 2. DUS- Powered IR Distribution Now utilizing (1.4) and (1.5) into (1.1) and (1.2) respectively. We can obtain the CDF, PDF and HRF for DUS-PIR distribution as follows: F (x; α,θ) = exp ( e− 1 θ x−2α ) − 1 e − 1 , x > 0; α,θ > 0, (2.1) f (x; α,θ) = 2αx−(2α+1) (e − 1)θ exp ( e− 1 θ x−2α ) e− 1 θ x−2α, (2.2) h(x,α,θ) = 2αx−(2α+1)e− 1 θ x−2α θ [ exp ( 1 −e− 1 θ x−2α ) − 1 ] (2.3) respectively. The depiction of plots are shown in the following figures for fix parameters. Int. J. Anal. Appl. (2023), 21:61 3 Figure 1. f (x) for fix parameters. Figure 2. h(x) for fix values of α and θ. The depiction from Figures 1 and 2 are. (i) The DUS-PIR distribution has unimodal, (ii) The failure rate is increasing, then decreasing for fix values of parameters α and θ. 3. Some Statistical Properties Some statistical properties of DUS-PIR distribution, including rth moments, quantile function, skewness, kurtosis, and order statistics are studied. 3.1. The moments: Let X ∼DUS-PIR distribution with parameters (α,θ), then the rth moment is given in Theorem 3.1. 4 Int. J. Anal. Appl. (2023), 21:61 Theorem 3.1. The moments of DUS-PIR distribution is given as µ ′ r = ( 1 e − 1 ) ∞∑ k=0 1 (k + 1)! ( k + 1 θ ) r 2α Γ ( 1 − r 2α ) . (3.1) Proof: The rth moment of the r.v. X is µ ′ r = ∫ ∞ 0 xrf (x; α,θ)dx. From (2.2), we have µ ′ r = ∫ ∞ 0 2α (e − 1)θ xr−(2α+1)ee − 1 θ x2α e− 1 θ x2αdx, Since θ > 0, we have ee − 1 θ x2α = ∑∞ k=0 e − k θx2α k! , so µ ′ r = ∞∑ k=0 2α k!(e − 1)θ ∫ ∞ 0 xr−(2α+1)e− (k+1) θ x−2αdx (3.2) Let u = ( k+1 θ ) x−2α, then Equation (3.2) reduces as µ ′ r = ( 1 e − 1 ) ∞∑ k=0 1 (k + 1)! ( k + 1 θ ) r 2α Γ ( 1 − r 2α ) . 3.2. Mode: Setting first derivative of (2.2) as follows. f ′ (x) = 2α (e − 1)θ2 x−2(2α+1) exp ( e− 1 θ x−2α − 1 θ x−2α )[ 2α−θ(1 + 2α)x2α + 2αe− 1 θ x−2α ] = 0. (3.3) Above equation does not possess analytic solution in x. For a quick graphical solution of the mode, we sketch the plot of left-hand side of (3.3) at different values of α,θ as depicted in Figure 3. Figure 3. f ′ (x) for selected values of α and θ. Int. J. Anal. Appl. (2023), 21:61 5 It confirms from these plots that DUS-PIR distribution has one mode based on selected values of α and θ. 3.3. Quantiles and Random Number Generation: The quantile xq,(0 < q < 1), of DUS-PIR(α,θ) distribution can be attained, by employing the CDF in (2.1), in the given simple form. xq = { − 1 θ ln [ln(q(e − 1) + 1)] } 1 2α . (3.4) One of the good characteristics of the suggested distribution is that we can smoothly calculate its quantiles in simple as well as an explicit form. To generate random sample with size (n ≥ 1) form DUS-PIR(α,θ) distribution, we can use (3.4) by generating n random values for q, where q ∼ U(0, 1). To find the median, using the above equation for q = 0.50, Med. = { − 1 θ ln [ln(0.5(e + 1))] } 1 2α . The shapes of DUS-PIR distribution can be viewed by skewness and kurtosis. Utilizing the concept of quantiles, skewness and kurtosis are as follows, [13]. Bowley’ skewness: sk = x0.75 − 2x0.50 + x0.25 x0.75 −x0.25 . Moors’ kurtosis: ku = x0.875 + x0.375 − (x0.625 + x0.125) x0.75 −x0.25 3.4. Order Statistics: The rth order statistic (O.S.) X(r) based on ordered sample (X1 < X2 < · · · < Xn) from a continuous distribution having CDF FX(x) and PDF fX(x) is. fX(r) (x) = n! (r − 1)!(n− r)! fx (x)[Fx (x)] r−1[1 −Fx (x)]n−r, r = 1, 2, · · · ,n. (3.5) So, the rth order statistic from DUS-PIR distribution is fX(r) (x) = n! (r − 1)!(n− r)! 2αx−(2α+1) (e − 1)nθ e ( e − 1 θ x−2α−1 θ x−2α ) [ ee − 1 θ x−2α − 1 ]r−1 [ e −ee − 1 θ x−2α ]n−r (3.6) Putting r = 1 and r = n in (3.6), we can obtain PDF of smallest and largest (O.S.). 6 Int. J. Anal. Appl. (2023), 21:61 3.5. Entropy: Entropy helps to measure the uncertainty of the r.v. X. Some notable entropies are defined as follows. Rényi entropy: Rδ(x) = 1 1 −δ log [∫ ∞ 0 f δ(x)dx ] , δ > 0 and δ 6= 1. (3.7) Tsallis entropy: Tδ(x) = 1 1 −δ [∫ ∞ 0 f δ(x)dx − 1 ] , δ > 0 and δ 6= 1. (3.8) Havrda and Charvat entropy (H-C) HCδ(x) = 1 21−δ − 1 [∫ ∞ 0 f δ(x)dx − 1 ] . (3.9) Theorem 3.2. If X ∼DUS-PIRD, then the Rényi Entropy of X is given as Rδ(x) = 1 1 −δ log   1 2α ( 2α θ(e − 1) )δ Γ ( δ(2α + 1) − 1 2α ) ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α   . (3.10) Proof: From (2.2) into (3.7), we have f δ(x) = ( 2α θ(e − 1)x2α+1 )δ ∞∑ k=0 δke− (k+δ) θ x−2α k! and ∫ ∞ 0 f δ(x)dx = ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ∫ ∞ 0 x−δ(2α+1)e− (k+δ) θ x−2αdx Let u = k+δ θ x−2δ, then x = ( θ k+δ )− 1 2α u− 1 2α and ∫ ∞ 0 f δ(x)dx = ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α 1 2α ∫ ∞ 0 u δ(2α+1)−1 2α−1 −1e−udu = 1 2α ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α Γ ( δ(2α + 1) − 1 2α ) . Therefore, the Renyi entropy is Rδ(x) = 1 1 −δ log   1 2α ( 2α θ(e − 1) )δ Γ ( δ(2α + 1) − 1 2α ) ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α   . Theorem 3.3. If X ∼ DUS-PIRD(α,θ), the Tsallis entropy of X is Tδ(x) = 1 1 −δ   1 2δ ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α Γ ( δ(2α + 1) − 1 2α ) − 1   . (3.11) Proof: Proof is easy. Int. J. Anal. Appl. (2023), 21:61 7 Theorem 3.4. If X ∼DUS-PIRD(α,θ), the Havrda and Charvat entropy of X is HCδ(x) = 1 21−δ − 1   1 2α ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α Γ ( δ(2α + 1) − 1 2α ) − 1   . (3.12) Proof: Proof is easy. 3.6. Bonferroni and Lorenz curves: A model for inequality of wealth distribution was proposed by [14] and to measure the income inequality introduced by [15]. Both models are used in financial mathematics, insurance, and population studies. Bonferroni and Lorenz’s curves are defined as: B(p) = 1 pµ ∫ q 0 xf (x)dx, L(p) = 1 µ ∫ q 0 xf (x)dx. (3.13) From (2.2), ∫ q 0 xf (x)dx = ∞∑ k=0 2α k!(e − 1)θ ∫ ∞ 0 x−2αe− (k+1) θ x−2αdx Let u = ( k+1 θ ) x−2α, then ∫ q 0 xf (x)dx = ∞∑ k=0 1 k!(e − 1)θ ( θ k + 1 )1− 1 2α ∫ ∞ ( k+1θ )q −2α u− 1 2αe−udx = 1 e − 1 ( 1 θ ) 1 2α ∞∑ k=0 1 (k + 1)! (k + 1) 1 2α Γ ( 1 − 1 2α , (k + 1) θ q−2α ) . (3.14) From equations (3.1), when r = 1 and (3.14) into (3.13), then the Bonferroni curve is given by B(p) = 1 pµ ∫ q 0 xf (x)dx = ∑∞ k=0 (k+1) 1 2α (k+1)! Γ ( 1 − 1 2α , (k+1) θ q−2α) ) p ∑∞ k=0 (k+1) 1 2α (k+1)! Γ ( 1 − 1 2α ) . (3.15) The Lorenz curve is obtained as L(p) = 1 µ ∫ q 0 xf (x)dx = ∑∞ k=0 (k+1) 1 2α (k+1)! Γ ( 1 − 1 2α , (k+1) θ q−2α) ) ∑∞ k=0 (k+1) 1 2α (k+1)! Γ ( 1 − 1 2α ) . (3.16) 4. Estimation of Parameters To understand the probabilistic model fully, estimating the unknown parameters for designated sample is a main procedure. Various estimation approaches under classical and Bayesian model are reported in literature. This section considers the estimation of DUS-PIR distribution via maximum likelihood approach based on complete data. 8 Int. J. Anal. Appl. (2023), 21:61 4.1. Maximum Likelihood Estimation: Let x1,x2, · · · ,xn random sample follows the DUS-PIR dis- tribution. The likelihood function (L.F.) of (2.2) is L(α,θ) = n∏ i=1 f (xi,α,θ) = n∏ i=1 [ 2αx−(2α+1) (e − 1)θ exp ( e− 1 θ x−2α ) e− 1 θ x−2α ] . (4.1) The log-L.F.is. given by LogL(α,θ) = −n ln(e−1) +n ln(2α)−n ln(θ)−(2α+ 1) n∑ i=1 ln(xi )− 1 θ n∑ i=1 x−2α i + n∑ i=1 e− 1 θ x−2α i . (4.2) The partial derivatives of (4.2) are as follows. ∂ ∂α LogL(α,θ) = n α − 2 n∑ i=1 ln(xi ) + 2 θ n∑ i=1 x−2α i ln(xi ) + 2 θ n∑ i=1 e− 1 θ x−2α i x−2α i ln(xi ), ∂ ∂θ LogL(α,θ) = − n θ + 1 θ2 n∑ i=1 x−2α i + 1 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i . The MLEs of α and θ can be derived as follows. n α − 2 n∑ i=1 ln(xi ) + 2 θ n∑ i=1 x−2α i ln(xi ) + 2 θ n∑ i=1 e− 1 θ x−2α i x−2α i ln(xi ) = 0, (4.3) − n θ + 1 θ2 n∑ i=1 x−2α i + 1 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i = 0. (4.4) Equations (4.3) and (4.4) has no closed form. So, we shall use a numerical program system to find its solution with respect to α and θ. 4.2. Asymptotic Confidence Interval: We derive asymptotic confidence intervals of unknown pa- rameters using variance- covariance matrix VVV , which is the inverse Fisher information matrix. The ML estimators are asymptotically normally distributed with multivariate normal distribution, see, [16]. (α̂, θ̂) ∼ N2(ΘΘΘ,VVV ), where ΘΘΘ = (α,θ) and VVV is given as follows VVV = ( −∂ 2LogL ∂α2 −∂ 2LogL ∂α∂θ −∂ 2LogL ∂α∂θ −∂ 2LogL ∂θ2 )−1 Θ→Θ̂ , Int. J. Anal. Appl. (2023), 21:61 9 where, ∂2 ∂α2 LogL(α,θ) = − n α2 − 4 θ n∑ i=1 x−2α i [ln(xi )] 2 + 4 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i [ln(xi )] 2 (4.5) − 4 θ n∑ i=1 e− 1 θ x−2α i x−2α i [ln(xi )] 2, ∂2 ∂α∂θ LogL(α,θ) = − 2 θ2 n∑ i=1 x−2α i ln(xi ) + 2 θ3 n∑ i=1 e− 1 θ x−2α i x−4α i ln(xi ) (4.6) − 2 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i ln(xi ), ∂2 ∂θ2 LogL(α,θ) = n θ2 − 2 θ3 n∑ i=1 x−2α i + 1 θ4 n∑ i=1 e− 1 θ x−2α i x−4α i − 2 θ3 n∑ i=1 e− 1 θ x−2α i x−2α i . (4.7) A 100(1 −δ)% confidence interval for ΘΘΘ = (α,θ), can be approximated by α̂±zδ 2 √ var(α̂), and θ̂±zδ 2 √ var(θ̂) where zδ 2 is upper 100 δ 2 -th percentile of N(0, 1), and var(Θ̂i ) is the diagonal i-th element in VVV . 5. Practical Illustration The main objective of any new distribution is to increase its adaptability and applicability, which makes it useful in several field of studies, particularly, in the fields concerning with lifetime analysis. This section depicts the usefulness of DUS-PIR distribution and compare with the Powered IR, Exponential transformed IR, Transmuted IR, Exponentiated IR, IR and Rayleigh distribution using two sets of data. For comparison some criteria such as, • K-S. (Kolmogorov Smirnov) statistic, K −S = sup x |Fm(x) − F̂ (x)| • R2: the determination coefficient, R2 = ∑m i=1 ( F̂ (xi ) −F )2∑m i=1 ( F̂ (xi ) −F )2 + ∑m i=1 ( Fm(xi ) − F̂ (xi ) )2 , • RMSE: the root mean square error RMSE = [ 1 m m∑ i=1 ( Fm(xi ) − F̂ (xi ) )2]1/2 , • A.I.C. (Akaike Information Criterion), [17]. AIC = 2k − 2`. 10 Int. J. Anal. Appl. (2023), 21:61 • A. I.C.C. (Akaike Information Criterion with Correction), [18]. AAIC = AIC + 2k(k + 1) m−k + 1 , • B.I.C. (Bayesian Information Criterion), [19]. BIC = k ln(m) − 2`, • and H.Q.I.C. (Hannan-Quinn Information Criterion) HQIC = 2k ln[ln(m)] − 2`, have been used, where k and m stands for number of parameters and observed data, ` = LogL, F̂ (x) is estimated CDF and Fm(x) is the empirical DF. F̄ (x) = 1 m m∑ i=1 F̂ (xi ), Fm(x) = 1 m m∑ i=1 I ( x(i) ≤ x ) and I ( x(i) ≤ x ) = { 1, if x(i) ≤ x 0, otherwise According to prevailing knowledge, the model with the lowest AIC, AAIC, BIC, HQIC and K-S value is considered as best fit for the data. Dataset 1: The following data reported by [20]. It comprises thirty consecutive March precipitation (in inches) observations. 0.77 1.74 0.81 1.20 1.95 1.20 0.47 1.43 3.37 2.20 3.00 3.09 1.51 2.10 0.52 1.62 1.31 0.32 0.59 0.81 2.81 1.87 1.18 1.35 4.75 2.48 0.96 1.89 0.90 2.05 For the above considered data, we have extracted the values of MLEs of parameters, K-S test, and p-values in below table. Table 1. MLEs, K-S statistics and p-value. Models α̂ θ̂ λ̂ K-S p-value DUS-PIR 0.860 1.332 – 0.14557 0.52380 PIR 0.775 0.975 – 0.15223 0.462057 ETIR – 1.454 – 0.18935 0.207305 TIR – 1.591 -0.67 0.18176 0.247695 EIR 0.731 1.456 – 0.19818 0.166981 IR – 1.164 – 0.23956 0.053115 Rayleigh – 3.773 – 0.35059 0.000843 The log–likelihood (`), information criteria, RMSE and R2 are reported below. Int. J. Anal. Appl. (2023), 21:61 11 Table 2. The `, Information Criteria, RMSE and R2. Models ` AIC AICC BIC HQIC RMSE R2 DUS-PIR -41.238 86.4760 86.9210 89.2790 87.3730 0.054453 0.96024 PIR -41.917 87.8340 88.2780 90.6360 88.7310 0.059373 0.95182 ETIR -42.026 86.0530 86.1950 87.4540 86.5010 0.075709 0.93570 TER -42.101 88.2020 88.6470 91.0050 89.0990 0.073716 0.94105 EIR -136.04 276.081 276.525 278.883 276.977 0.078017 0.92284 IR -44.137 90.2730 90.4160 91.6740 90.7210 0.107514 0.88381 Rayleigh -38.924 79.8490 79.9910 81.2500 80.2970 0.201452 0.48597 Listed values in the Tables 1-2. It has been noticed that DUS-PIR distribution interprets a better fit among all lifetime distributions. The variance-covariance matrix is given as VVV = ( 0.012 0.008 0.008 0.076 ) . Then the 95% confidence interval for α and θ for DUS-PIR distribution are (0.648, 1.073) and (0.790, 1.874), respectively. It is shown that the LF has a unique solution by Figure 4. Figure 4. The profile of the log-LF of α and θ. Dataset 2: The given data set is reported by [21]. It represents the survival times (in days) of 72 guinea pigs injected with different doses of tubercle bacilli. 2 24 34 44 54 57 60 61 65 70 76 84 95 109 129 146 233 297 15 32 38 48 54 58 60 62 67 72 76 85 96 110 131 175 258 341 22 32 38 52 55 58 60 63 68 73 81 87 98 121 143 175 258 341 24 33 43 53 56 59 60 65 70 75 83 91 99 127 146 211 263 376 Estimated values of parameters, test statistic and criterion are provided in the following table. 12 Int. J. Anal. Appl. (2023), 21:61 Table 3. MLEs, K-S statistics and p-value. Models α̂ θ̂ λ̂ K-S p-value DUS-PIR 0.782 0.003 – 0.18446 0.01290686 PIR 0.797 0.004 – 0.19755 5.1528E-24 ETIR – 5.715×10−4 – 0.20597 0.00371676 TIR – 6.503×10−4 -0.781 0.17999 0.01642093 EIR 0.616 6.555×10−4 – 0.20997 0.00290489 IR – 4.571×10−4 – 0.25083 0.00017822 Rayleigh – 1.628×104 – 0.97964 3.3351×10−62 Table 4. The ` , Information Criteria, RMSE and R2. Models ` AIC AICC BIC HQIC RMSE R2 DUS-PIR -394.466 792.932 793.106 797.485 794.744 0.068685 0.931008 PIR -395.649 795.298 795.472 799.852 797.111 0.076096 0.913825 ETIR -400.074 802.149 802.206 804.426 803.055 0.092092 0.899998 TIR -398.920 801.839 802.013 806.392 803.652 0.078811 0.929423 EIR -614.106 1232.00 1232.00 1237.00 1234.00 0.083047 0.899951 IR -406.736 815.472 815.529 817.749 816.378 0.126351 0.831767 Rayleigh -408.300 818.600 818.657 820.877 819.506 0.576828 5.726×10−05 From Tables 3-4. It has been observed that DUS-PIR distribution suggests a better fit among all lifetime distributions for considered data. The variance-covariance matrix is given as VVV = ( 0.004 −7.853 × 10−5 −7.853 × 10−5 1.680 × 10−6 ) . Then the 95% confidence interval for α and θ for DUS-PIR distribution are (0.659, 0.905) and (1.589R×10−4, 0.005.), respectively. It is shown that the LF has a unique solution by Figure 5. Figure 5. The profile of the log-LF of α and θ. Int. J. Anal. Appl. 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Epidemiol. 72 (1960), 130-148. https://doi.org/10.1093/oxfordjournals.aje.a120129. https://doi.org/10.1109/tac.1974.1100705 https://doi.org/10.1093/biomet/76.2.297 https://www.jstor.org/stable/2958889 https://www.jstor.org/stable/2958889 https://doi.org/10.2307/2346869 https://doi.org/10.2307/2346869 https://doi.org/10.1093/oxfordjournals.aje.a120129 1. Introduction 2. DUS- Powered IR Distribution 3. Some Statistical Properties 3.1. The moments: 3.2. Mode: 3.3. Quantiles and Random Number Generation: 3.4. Order Statistics: 3.5. Entropy: 3.6. Bonferroni and Lorenz curves: 4. Estimation of Parameters 4.1. Maximum Likelihood Estimation: 4.2. Asymptotic Confidence Interval: 5. Practical Illustration 6. Conclusion References