EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 5, 2017, 1005-1022 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Topp-Leone Inverse Weibull Distribution: Theory and Application Salman Abbas1,∗, Syed Ali Taqi2, Fakhar Mustafa3, MaryamMurtaza4, Muham- mad Qaiser Shahbaz5 1,4 Department of Mathematics, COMSATS Institute of Information Technology Wah Cantt, Pakistan 2 Department of Statistics, COMSATS Institute of Information Technology Lahore, Pakistan 3 Department of Mathematics, COMSATS Institute of Information Technology Sahiwal, Pakistan 5 Department of Statistics, King Abdulaziz University, Jeddah, Sudia Arabia Abstract. In this article, the discussion has been carried out through the generalization of In- verse Weibull distribution. We introduce a new three parameter life model called the Topp-Leone Inverse Weibull distribution. We provide comperhensive result of the mathematical characteristic, including moments, quantile function, random number generator, survival function, hazard rate function, and mode. Distributional properties of order statistics are analyzed. The parameters of the proposed model are estimated by the method of maximum likelihood. Simulation study is performed to investigate the performance of the maximum likelihood estimators. To assess the flexibility, empirical results of new model are obtained by modeling two real data sets. Key Words and Phrases: Reliability Analysis, Order Statistics, Maximum Likelihood, Simula- tion Study 1. Introduction Recently, a considerable number of authors are generalizing classical distributions to extended their form which are more flexible to model real data. Inverse Weibull distribu- tion has wider application in the field of reliability and biological studies due to its failure rate. Keller and Kanath [4] introduced the Inverse Weibull distribution to study the shape of the density and the failure rate function. The Inverse Weibull distribution provides a good fit of several data in terms of times to breakdown of an insulating fluid, the subject ∗Corresponding author. Email addresses: salmanabbas@ciitwah.edu.pk (S. Abbas), syedalitaqi@ciitlahore.edu.pk (S.A. Taqi) fakhar.m@ciitsahiwal.edu.pk (F. Mustafa), maryam murtaza@live.com (M. Murtaza) and qshahbaz@gmail.com (M.Q. Shahbaz) http://www.ejpam.com 1005 c© 2017 EJPAM All rights reserved. S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1006 leaded to the action of constant tension, see Nelson [23]. Calabria and Pulcini [16] dis- cussed the maximum likelihood and least squares estimation of its parameters. Calabria and Pulcini [17] considered Bayes 2-sample prediction of the distribution. Mahmoud et al. [10] discussed moments of order statistics of Inverse Weibull distribution and obtained BLUE (best linear unbiased estimator) for both location and scale parameters. Aleem and Pasha [11] derived single, product and ratio moment of Inverse Weibull Distribution. Aleem [12] worked on the product, ratio, and single moments of lower record values of Inverse Weibull distribution. Hanook et al. [21] derived Beta Inverse Weibull distribution. Shahbaz et al. [13] proposed the Kumaraswamy Inverse Weibull distribution using distri- bution function of kumaraswamy family of distributions. Ali et al. [1] proposed Topp-Leone family of distribution. The distribution and density function of proposed family is given by FTL−G(y) = [G(y)]α[2−G(y)]α = [1− (Ḡ(y))2]α :, x ∈ <, α > 0, (1) the density of Toop-Leone family is fTL−G(y) = 2αg(y)Ḡ(y)[G(y)]α−1[2−G(y)]α−1, α > 0, or fTL−G(y) = 2αg(y)Ḡ(y)[1− (Ḡ(y))2]α−1, α > 0, (2) where g(y) = G′(y) and Ḡ(y) = 1−G(y). This present article is designed as follows; Section 2, we derive three parameter life model called Topp-Leone Inverse Weibull distribution, the pdf and cdf expansion. The main mathematical properties of the proposed model including, moments, survival function, hazard rate function, quantile function, and mode are discussed in Section 3. Section 4 is based on the distributional properties of order statistics. Estimation of parameters is determined in Section 5. To analyse the flexibility of maximum likelihood estimators, simulation study is provided in Section 6. In Section 7, we prove empirically that the proposed distribution is a very competitive model to other classical models by means of two real data sets. Finally, extensive concluding remarks are offered in Section 8. 2. The Topp-Leone Inverse Weibull Distribution In this section, we derive three parameter Topp-Leone Inverse Weibull distribution. To construct the density and distribution function, consider pdf and cdf of Inverse Weibull distribution is given by F (y) = e − β yγ . (3) From above equation the density of Inverse Weibull distribution is given by f(y) = βγ yγ+1 e − β yγ , β, γ > 0, y ∈ <+. (4) S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1007 By inserting (3) and (4) into (1) and (2), we have cdf and pdf of the proposed model given by F (y) = [1− {1− e − β yγ }2]α, (5) and f(y) = 2αβγ yγ+1 e − β yγ (1− e − β yγ )[1− {1− e − β yγ }2]α−1, β, γ > 0 y ∈ <+. (6) The proposed model have two shape and one scale parameter. We will use the notation TLIW (α, β, γ) to denote the density (6). It is observed from (5) that the proposed model is a special case of exponentiated general- ized class of distribution derived by Cordeiro et al. [7]. The proposed model of exponen- tiated generalized class of distribution is given by F (x, α, β) = [1− (1−G(x))α]β, x ∈ <+ (7) If we replace α = 2, β = α and e − β yγ in (5), the above distribution converts to Topp-Leone Inverse Weibull distribution. Corderio et al. [7] used the method of adding parameter leads to the exponentiated type of distribution which was introduced by Lehmann [6] and studied by Nadarajah and Kotz [22]. Where Ali et al. [1] proposed Topp-Leone family of distribution by using survival function instead of distribution function. The proposed model provides some ideal sub models. For γ = 1 the proposed distribution in (5) converts to Topp-Leone Inverted Exponential distribution. For β = 1 and γ = 1 the distribution (5) reduces to Topp-Leone Standard Inverted Exponential distribution. 2.1. Shape For real value of α, using following series representation of Prudnikov et al. [3] (1 + x)α = ∞∑ j=0 (1)jΓ(α+ 1) j!Γ(α+ 1− j) xj . The cdf of TLIW distribution given in (5) is expressed as infinite sum given as follows F (y) = [1− {1− e − β yγ }2]α = ∞∑ j=0 (−1)j Γ(α+ 1) j!Γ(α+ 1− j) (1− e − β yγ )2j , or F (y) = ∞∑ j=0 2j∑ m=0 (−1)j+m Γ(α+ 1) j!Γ(α+ 1− j) ( 2j m ) [e − β yγ ]m = ∞∑ j=0 2j∑ m=0 a(j,m)[e − β yγ ]m, (8) S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1008 where a(j,m) = (−1)j+m Γ(α+ 1) j!Γ(α+ 1− j) ( 2j m ) . Again for (6), follows series representation the density function of TLIW distribution is written as follows f(y) = ∞∑ j=0 (−1)j 2Γ(α+ 1) j!Γ(α− j) βγ yγ+1 {e − β yγ }{1− e − β yγ }2j+1, or f(y) = ∞∑ j=0 2j+1∑ m=0 (−1)j+m 2Γ(α+ 1) j!Γ(α− j) ( 2j + 1 m ) βγ yγ+1 {e − β yγ }m+1 = ∞∑ j=0 2j+1∑ m=0 b(j,m)hm+1(y), (9) where b(j,m) = (−1)j+m Γ(α) j!Γ(α− j)(m+ 1) ( 2j+1 m ) and hm+1(y) = (m+1) βγ yγ+1 {e − β yγ }m+1 is exponentiated-G distribution with power function m. The density and distribution function of TLIW distribution are given (8) and (9) shows that the TLIW distribution is expressed as weighted sum of exponentiated family of dis- tribution. a = 1 a = 2 a = 3 a = 4 a = 5 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 Figure 1: Graph for pdf of TLIW for β = 1.5 , γ = 1.5 and for different values of α In Figure 1, we can see that for the lower values of α, peak are increased. For α > 1 a slow decrease are observed. In Figure 2, we can clearly see that at γ = 3.5 function shows high peak, but as the value of γ are decreasing a rapid change appears it starts decreasing but no change appears in the location of the curve. Figure 3 shows the graphs for the different values of β and fixed values of α and γ. Figure 4 shows the plot of probability density function of Topp-Leone Inverse Weibull dis- tribution for the different values of α and β when γ is fixed. This plot shows the different shapes for different values of parameters. For this plot it is surely clear that Topp-Leone Inverse Weibull distribution is unimodel distribution. S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1009 b = 1 b = 2 b = 3 b = 4 b = 5 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 2: Graph for pdf of TLIW for α = 2 , γ = 1.5 and for different values of β g = 3.5 g = 3 g = 2 g = 1.5 g = 1 0 1 2 3 4 0.0 0.5 1.0 1.5 Figure 3: Graph for pdf of TLIW for α = 2 , β = 2 and for different values of γ 3. Properties of TLIW Distribution In this section, we discuss important and useful statistical characteristics of the pro- posed distribution. 3.1. Quantile and Median The qth percentile of the distribution can be obtained by solving yq for variable Y. The qth percentile is obtained by solving Q(y) = F (y)−1 as: yq = −  β ln[1− √ 1− q 1 α ]  1 γ , q > 0. (10) The median of the TLIW distribution can be defined at q = 0.5. We can easily generate the random sample from (14) using q as uniform random number. S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1010 b = 1 b = 2 b = 3 b = 4 a = 3 0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 b = 2 b = 1 b = 4 b = 3 a = 2 0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 1.2 b = 1 b = 2 b = 3 b = 4 a = 1.5 0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 1.2 b = 0.5 b = 1 b = 1.5 b = 2 a = 0.5 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 Figure 4: Graph for pdf of TLIW for α = 0.5, 1.5, 2, 3 and β = 1, 2, 3, 4 and γ = 2 3.2. Moments The moments of Topp-Leone Inverse Weibull distribution is computed using following expression µ′r = ∫ ∞ 0 yrF (y)dy = ∫ ∞ 0 yr ∞∑ j=0 2j+1∑ m=0 (−1)j+m 2Γ(α+ 1) j!Γ(α− j) ( 2j + 1 m ) βγ yγ+1 {e − β yγ }m+1dy. (11) Making transformation as z = (m+1)β yr in above expression to solve the moment of Topp- Leone Inverse Weibull distribution and result are given as follows µ′r = ∞∑ j=0 2j+1∑ m=0 (−1)j+m 2Γ(α+ 1) j!Γ(α− j) ( 2j + 1 m )( 1 β(m+ 1) )1− r γ Γ(1− r γ ). (12) These moments are existing for r < γ. The coefficient of variation (CV ), coefficient of skewness (CS) and coefficient of kurtosis (CK) TLIW distribution are obtained as follows CV = √ µ2 µ1 − 1. CS = µ3 − 3µ2µ1 + 2µ31 (µ2 − µ1) 3 2 . CK = µ4 − 4µ3µ1 + 6µ2µ 2 1 (µ2 − µ21)2 . S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1011 3.3. Reliability Analysis The TLIW distribution is used for describing a random lifetime in reliability analysis. The reliability analysis of the TLIW distribution is denoted by R(y), also known as survival function and obtained as follows R(y) = 1− F (y). (13) The survival function of TLIW distribution is obtained by inserting (5) in to above ex- pression (15) to attain the following results R(y) = 1− [1− {1− e − β yγ }2]α. (14) Note that R(y) + F (y) = 1. Hazard rate function is another characteristics in reliability a = 0.5 a = 3 a = 2 a = 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 5: Graph for Survival function of TLIW for β = 1 & γ = 2 and various values of α analysis. It is denoted by h(y) and define a follows h(y) = 2αβγ yγ+1 e − β yγ (1− e − β yγ )[1− {1− e − β yγ }2]α−1 1− [1− {1− e − β yγ }2]α (15) The units for h(y) is the probability of failure per unit of distance and time. We define these failure rates at the different values of parameters. The cumulative hazard rate function of TLIW distribution is represented as H(y) and the result obtained are given as follows h(y) = −log|1− [1− {1− e − β yγ }2]α|. (16) h(y) unite is the cumulative probability of failure per unit of time, distance or cycles. The distribution has decreasing cumulative instantaneous failure rate for all choices of parameters. S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1012 a = 0.5 a = 1.5 a = 2 a = 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 Figure 6: Graph for Hazard rate function of TLIW for β = 1 & γ = 2 and various values of α 3.4. Mode We consider the density function of TLIW distribution given in (6) and solve ∂ ln f(y) ∂y = 0 for y, to obtain the mode of Topp-Leone Inverse Weibull distribution as follows ∂ ln f(y) ∂y = βγ y − e−βy −γ y−(1+γ)βγ 1− e−βy−γ + 2e−βy −γ (1− e−βy−γ )y−(1+γ)βγ 1(1− e−βy−γ )2 + 1 + γ y . By putting ∂ ln f(y) ∂y = 0, we have: βγ y − e−βy −γ y−(1+γ)βγ 1− e−βy−γ + 2e−βy −γ (1− e−βy−γ )y−(1+γ)βγ 1(1− e−βy−γ )2 + 1 + γ y = 0 (17) The maxima can be obtained by solving (17) iteratively. 4. Order Statistics Order statistics is used in the field of reliability and life testing widely. LetX1, X2, ..... , Xn be a simple random sample from TLIW (α, β, γ) with distribution and density functions given in (5) and (6). Let X(1:n) ≤ X(2:n)l........ ≤ X(n:n) denote the order statistics ob- tained from this sample. In reliability literature, X(j:n) is used to model the lifetime of an (ni + 1)-out-of-n system which consists of n independent and identically distributed components. The density function of X(i:n), 1 ≤ k ≤ n is given as follows: fi,n(x) = n! (i− 1)!(n− i)! [FEGWE(x)]i−1[1− FEGWE(x)]n−ifEGWE(x) The first order statistic is given by X(1) = min(X1, X2, .....Xn) and the last order statistics is given by X(n) = max(X1, X2, .....Xn). The distribution of first order statistics is given by f1:n(y) = 2nαβγ ∞∑ j=0 (−1)j Γ(n) j!Γ(n− j) [1− (1− e− β yγ )2]αj+α−1y−(γ+1)e − β yγ (1− e− β yγ ). (18) S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1013 The distribution of the nth order statistics is given by fn:n(y) = 2nαβγy−(γ+1)e − β yγ (1− e− β yγ )[1− (1− e− β yγ )2]n+α−2 (19) 5. Parameters Estimation and Fisher Information Matrix In this section, we derive the maximum likelihood estimates (MLE) and inference for unknown parameters of Topp-Leone Inverse Weibull distribution. Let y1, y2........yn be a realization of a random sample of size n from TLIW distribution than the likelihood function is written as follows LF = L(α, β, γ|yi) = n∏ i=0 F (yi), the log-likelihood function is given as follows ln(LF ) =n ln(2) + n ln(α) + n ln(β) + n ln(γ)− (γ + 1) n∑ j=1 ln(yj)− β n∑ j=1 ln(y−γj )+ n∑ j=1 ln(1− e − β y γ j ) + (α− 1) n∑ j=1 ln(1− {1− e − β y γ j }2), (20) differentiating (22) w.r.t α, β, γ, and equating them 0, we have n α + n∑ j=1 ln ( 1− w2 ) = 0, (21) n β + n∑ j=1 y−γj e − β y γ j w − n∑ j=1 ln(y−γj )− (α+ 1) n∑ j=1 2y−γj (w)e − β y γ j 1− w2 = 0, (22) n γ + β n∑ j=1 ln(yj)− n∑ j=1 ln(yj)− n∑ j=1 y−γj β ln(yj)e − β y γ j w + (α+ 1) n∑ j=1 2y−γj β ln(yj)we − β y γ j 1− w2 = 0, (23) where (1 − e − β y γ j ) = w. The maximum likelihood estimate of α, β, and γ are obtained iteratively solving (21), (22), and (23), simultaneously. The fisher information matrix for the parameters of the TLIW distribution is obtained as follows α̂ β̂ γ̂  ∼ N  α β γ  .  Ĵαα Ĵαβ Ĵαγ Ĵββ Ĵβγ Ĵγγ  , S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1014 1 J = −E  Jαα Jαβ Jαγ Jββ Jβγ Jγγ  . By determining the inverse dispersion matrix, the asymptotic variances and covariances of the ML estimators for α, β, and γ may be obtained. Using above, approximate 100(1−λ)% confidence intervals for α, β, and γ are determined respectively as follows α̂± Zλ 2 √ Ĵαα, β̂ ± Zλ 2 √ Ĵββ , γ̂ ± Zλ 2 √ Ĵγγ , (24) where Zγ is demonstrated the upper 100γth quantile of the standard normal distribution. 6. Simulation Study In this section of article, we discuss some simulations for different sample size to de- termine the efficiency of MLEs. The different methods have been derived for simulating a random variable like the inversion method, the rejection, acceptance sampling techniques, and many more from different probability distributions in the field of computational statis- tics. The Inversion method is considered the most powerful technique. We can simulate random variable Y given by y = −  β ln[1− √ 1− U 1 α ]  1 γ , where U is uniform random number in (0,1). We generate sample of size n = 50, 100, 200, 500, 1000 from TLIW distribution for some selected combination of parameters. This process is repeated N = 1000 time to calculate mean estimate and means squared error. Obtained results are given in following tables. It is observed that when sample size increases the mean squared error decreases. There- fore, the maximum likelihood method works very well to estimate the parameters of TLIW distribution. 7. Application In this section, we provide the application with real data sets to assess the flexibility of TLIW distribution. The parameters are estimated using maximum likelihood method and R software is used for computation. We describe data sets to find the MLEs of the parameters. To assess the fitness of the real data for proposed distribution, we apply goodness-of-fit tests. The log-likelihood values for different models are obtained to decide the best model. S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1015 Table 1: Estimated Mean and MSEs of TLIW distribution α=2.5 β=2.0 γ=1.5 n α̂ β̂ γ̂ 50 2.5073 2.0288 1.5741 0.6507 0.4327 0.3810 100 2.5033 2.0118 1.8671 0.3684 0.4463 0.3544 200 2.5018 2.1605 1.8339 0.3582 0.3758 0.3673 500 2.5014 2.3003 1.7115 0.2661 0.3688 0.3637 1000 2.5009 2.0421 1.9818 0.2373 0.2884 0.2906 Table 2: Estimated Mean and MSEs of TLIW distribution α=1.5 β=2.5 γ=2.0 n α̂ β̂ γ̂ 50 1.5111 2.8992 2.2547 0.5547 0.4081 0.3601 100 1.5017 2.9065 2.1330 0.4977 0.4028 0.4072 200 1.5033 2.8129 2.1991 0.4018 0.4181 0.3456 500 1.5006 2.7396 2.1248 0.3475 0.2794 0.3011 1000 1.5006 2.6807 2.1041 0.2536 0.2679 0.2137 Data Set 1: For getting the performance of the proposed model a data related to influence of physiographic and historical factors on species richness of native and non-native vascular plants on 22 coastal islands is selected. Different variables are effecting on the richness. We select variable area (hectares) having values 3, 4, 4, 8, 10, 34, 40, 46, 47, 61, 128, 140, 350, 1190, 1350, 1900, 2300, 2707, 10900, 13600, 13600 and 26668. It is depicted from the results of Table 7 that our proposed model provide best fit than recent developed models. It is be more reliable with these types of data. From Figure. 7, we see that the data provides best fitting for proposed distribution. Data Set 2: This data set consists of the waiting times(in seconds), between 65 successive eruptions of the Kiama Blowhole. These values were recorded with the aid of digital watch on July S. Abbas et al. / Eur. J. Pure Appl. Math, 10 (5) (2017), 1005-1022 1016 Table 3: Parameter Estimation for Various Distributions Model parameters LL α β γ TLIW 4.597689 0.4233699 0.20622405 -73.86899 EE 0.328589 0.02146033 -76.14519 EW 2.07528 1.533359 0.2841065 -96.15165 IE 4.597689 0.42333699 -249.7032 Figure 7: Goodness of Fit 12, 1998 by Jim Irish and has been referenced by several authors including da Silva, Thiago, Maciel, Campos and Cordeiro [8] and Pinho, Cordeiro and Nobre [9]. The actual data are: 83, 51, 87, 60, 28, 95, 8, 27, 15, 10, 18, 16, 29, 54, 91, 8, 17, 55, 10, 35, 47, 77, 36, 17, 21, 36, 18, 40, 10, 7, 34, 27, 28, 56, 8, 25, 68, 146, 89, 18, 73, 69, 9, 37, 10, 82, 29, 8, 60, 61, 61, 18, 169, 25, 8, 26, 11, 83, 11, 42, 17, 14, 9, 12. Table 4: Parameter Estimation for Various Distributions Model parameters LL α β γ TLIW 4.597689 0.4233699 0.2062405 -54.82303 EW 2.075585 1.532594 0.291697 -166.880 In Table 8, the value of log-likelihood of TLIW distribution is minimum than other existing distributions, which indicates that new model is better. The data of waiting time of customers are also provides better fit to follow the curve. 8. Conclusion We derive a three parameter Topp-Leone Inverse Weibull distribution. Some of desir- able properties are computed. We study the distributional properties of order statistics. The parameters are estimated by method of maximum likelihood. Performance of MLE’s are tested through simulation study. Finally, two real data applications are analysed to assess the flexibility of new model over existing distribution. It is significantly observed that the proposed model provides better result than derived models. REFERENCES 1017 Figure 8: Goodness of Fit References [1] A. Al-Shomrani, O. 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Appendix REFERENCES 1019 Table 5: Mean of Toop-Leone Inverse Weibull distribution for different values of parameters β α γ 4 5 6 7 8 9 1 1 0.125 0.221556731 0.281270921 0.320461669 0.347920893 0.368165662 1 2 0.08 0.158533092 0.208887035 0.242459148 0.266188013 0.283779878 1 3 0.055555556 0.120600209 0.163808521 0.193046537 0.213880406 0.229405316 1 4 0.040816327 0.095703512 0.133374899 0.159213001 0.177760331 0.191645616 1 5 0.03125 0.078332134 0.111622439 0.134737534 0.151441365 0.163999158 1 6 0.024691358 0.065646439 0.095399949 0.116291492 0.131480543 0.142943258 2 1 0.097222222 0.228051309 0.316474846 0.376617996 0.419571752 0.45161973 2 2 0.062222222 0.163180233 0.235031378 0.284946648 0.321006795 0.348105772 2 3 0.043209877 0.124135409 0.184310829 0.226875184 0.257926955 0.281405839 2 4 0.031746032 0.098508906 0.150068128 0.187112804 0.214368309 0.235086947 2 5 0.024305556 0.080628313 0.125593126 0.158348361 0.182629212 0.201173719 2 6 0.01920439 0.067570758 0.107340226 0.136669914 0.158557657 0.175344966 3 1 0.087916667 0.238778166 0.347213665 0.422774371 0.477437645 0.518548753 3 2 0.056266667 0.170855748 0.257859691 0.319868251 0.365278948 0.399694261 3 3 0.039074074 0.129974371 0.202212717 0.254679846 0.293499353 0.323109548 3 4 0.028707483 0.103142473 0.164644064 0.210044392 0.243933249 0.269926302 3 5 0.021979167 0.08442083 0.137791834 0.177754726 0.207816806 0.23098721 3 5 0.017366255 0.070749086 0.117766053 0.153419479 0.180425384 0.201330694 4 1 0.083129252 0.248977245 0.373566076 0.461917367 0.526433444 0.575236864 4 2 0.053202721 0.178153614 0.27743042 0.349483579 0.402764752 0.443389116 4 3 0.036946334 0.135526046 0.217560017 0.278259639 0.323618963 0.358432109 4 4 0.027144245 0.107548061 0.177140024 0.229491566 0.268966266 0.299434833 4 5 0.020782313 0.088026749 0.148249795 0.194212329 0.229143466 0.256238892 4 6 0.016420593 0.073771036 0.126704121 0.167623978 0.198941071 0.223340305 5 1 0.080157943 0.258247777 0.39666873 0.496177192 0.569372114 0.624992261 5 2 0.051301083 0.184787067 0.294587704 0.375404333 0.435616356 0.481740277 5 3 0.035625752 0.140572285 0.231014702 0.298897803 0.350015021 0.389434872 5 4 0.026174022 0.111552554 0.188094993 0.246512665 0.290904564 0.325334597 5 5 0.020039486 0.091304377 0.15741809 0.208616811 0.247833607 0.278402402 5 6 0.015833668 0.07651786 0.134539954 0.180056436 0.215167746 0.242658235 6 1 0.078108153 0.26666609 0.417325949 0.526846126 0.607887428 0.669699166 6 2 0.049989218 0.190810721 0.309928874 0.398608242 0.465083729 0.51620009 6 3 0.034714735 0.145154634 0.243045198 0.31737281 0.373691871 0.417291901 6 4 0.025504703 0.115188924 0.197890369 0.261749722 0.310582873 0.34860641 6 5 0.019527038 0.0942807 0.165615913 0.221511509 0.264598371 0.298317064 6 6 0.015428771 0.079012175 0.141546358 0.191185805 0.229722821 0.260016048 REFERENCES 1020 Table 6: Variance of Toop-Leone Inverse Weibull distribution for different values of parameters β α γ 4 5 6 7 8 9 1 1 0.001953125 0.004661601 0.005216654 0.004891093 0.004335786 0.003775348 1 2 0.001152 0.003083723 0.003646001 0.003521791 0.003180363 0.002804469 1 3 0.000685871 0.002115403 0.002640853 0.002624235 0.002411365 0.00215147 1 4 0.000424993 0.001506302 0.001977551 0.002016728 0.001882668 0.001697738 1 5 0.000274658 0.001108364 0.001523567 0.001590785 0.001506476 0.00137166 1 6 0.000184404 0.000838659 0.001202209 0.001282461 0.00123043 0.001130183 2 1 0.002492311 0.008558125 0.010981446 0.01103603 0.010200011 0.00913207 2 2 0.00118062 0.00509626 0.007082722 0.00741353 0.007023159 0.006392909 2 3 0.000623835 0.003274471 0.004870687 0.005277731 0.00510592 0.00471372 2 4 0.000358556 0.002228641 0.003515642 0.003925325 0.003867254 0.0036141 2 5 0.000220038 0.001586127 0.002634618 0.003020203 0.003023556 0.002856213 2 6 0.000142254 0.001169653 0.002034192 0.002387347 0.00242439 0.002312322 3 1 0.002475099 0.011379596 0.016066296 0.016927302 0.016091374 0.014677856 3 2 0.001117531 0.006565329 0.010091408 0.011102363 0.010834666 0.010058523 3 3 0.000573636 0.004131432 0.006816692 0.007775728 0.007756672 0.007308022 3 4 0.000323372 0.002770284 0.004856429 0.00571401 0.005808386 0.005542175 3 5 0.000195709 0.001949481 0.003603018 0.004355608 0.004501103 0.004342689 3 6 0.000125211 0.001424863 0.002759653 0.003417197 0.003583404 0.003491525 4 1 0.002426513 0.013752778 0.02079772 0.022671748 0.021994874 0.020335553 4 2 0.001072231 0.007806363 0.012879095 0.014676978 0.014627679 0.013771144 4 3 0.000543067 0.00485898 0.008615332 0.010186439 0.010382128 0.009922473 4 4 0.000303361 0.003232387 0.00609377 0.007435109 0.007724326 0.00747801 4 5 0.000182388 0.002260907 0.004495791 0.005637689 0.00595556 0.005830823 4 6 0.000116105 0.001644537 0.003427988 0.004404187 0.00472185 0.004669276 5 1 0.002383433 0.015857995 0.02528833 0.028305203 0.027897746 0.026065817 5 2 0.001040135 0.008909254 0.015516512 0.018167471 0.018402466 0.017513043 5 3 0.000522688 0.005506887 0.010313572 0.012533485 0.012986384 0.012548475 5 4 0.000290405 0.003644747 0.007260406 0.009107116 0.00962012 0.009417354 5 5 0.000173912 0.002539342 0.005336683 0.006881104 0.007391947 0.007318602 5 6 0.000110377 0.001841286 0.004056991 0.005360118 0.00584439 0.005844781 6 1 0.0023482 0.017777048 0.029595667 0.033846826 0.033792679 0.03184612 6 2 0.001016344 0.009915348 0.018039936 0.021589905 0.022158922 0.021273907 6 3 0.000508067 0.006098486 0.011935698 0.01482955 0.015571548 0.015180957 6 4 0.000281262 0.00402163 0.008373418 0.010740034 0.011498479 0.011357662 6 5 0.000167991 0.002794055 0.006138195 0.008093839 0.008813009 0.008804798 6 6 0.000106404 0.002021425 0.004656108 0.006291462 0.006953607 0.007017534 REFERENCES 1021 Table 7: Coefficient Skewness Table of Toop-Leone Inverse Weibull distribution for different values of parameters β α γ 4 5 6 7 8 9 1 1 0.595170064 0.44020366 0.355459867 0.296998208 0.254493126 0.222361967 1 2 0.582377652 0.458810019 0.373560513 0.312841916 0.26825751 0.234420311 1 3 0.595170064 0.47859545 0.391051237 0.327786601 0.281104898 0.245608469 1 4 0.611011148 0.496551293 0.406805639 0.341251432 0.292686152 0.25569677 1 5 0.625259785 0.512279767 0.420789994 0.353286558 0.303076239 0.264766849 1 6 0.63718253 0.525966374 0.433203744 0.364068159 0.312429005 0.272954236 2 1 0.61181318 0.491730312 0.399990748 0.334114492 0.285803679 0.249242975 2 2 0.636540479 0.519239851 0.424383364 0.355018962 0.303788457 0.264901376 2 3 0.653449639 0.540633294 0.444183098 0.372308778 0.318809681 0.278054474 2 4 0.665160849 0.557618288 0.460579618 0.386891791 0.331601108 0.289318072 2 5 0.673507548 0.571399058 0.474418518 0.399413524 0.342683707 0.299128606 2 6 0.679630911 0.582797364 0.48629096 0.410328964 0.352425876 0.307795142 3 1 0.64455463 0.525932188 0.42920009 0.358551554 0.306481958 0.267030449 3 2 0.663187691 0.551736925 0.453570335 0.379982246 0.32515735 0.283407843 3 3 0.674746479 0.570998667 0.472801624 0.397306103 0.340444782 0.296913327 3 4 0.682333473 0.585910543 0.488451014 0.41171273 0.353303161 0.308348995 3 5 0.687557619 0.597799548 0.501499074 0.423962495 0.364349609 0.318232934 3 6 0.691298849 0.607505692 0.512591088 0.434563263 0.373999243 0.326915003 4 1 0.661850384 0.547910128 0.449100662 0.375623929 0.321119419 0.279720092 4 2 0.6760347 0.571771624 0.472867786 0.396987389 0.339943143 0.296332247 4 3 0.684527968 0.589268286 0.491389767 0.414082734 0.355216045 0.309920793 4 4 0.689984409 0.602654923 0.506337079 0.428204846 0.367988327 0.321366816 4 5 0.693687521 0.613237061 0.518724385 0.440154911 0.37891557 0.331223044 4 6 0.696311928 0.621819793 0.529205403 0.450458034 0.388431041 0.339856478 5 1 0.672058453 0.563383103 0.463827394 0.388529415 0.332308138 0.2894842 5 2 0.683344122 0.585604157 0.486944539 0.409687845 0.351123296 0.306176831 5 3 0.689975891 0.601727657 0.504826258 0.426517997 0.366310149 0.31976694 5 4 0.694186506 0.613975473 0.519183129 0.440364459 0.378966181 0.331178398 5 5 0.697021075 0.623606163 0.531035751 0.452045938 0.389766266 0.340982479 5 6 0.699018113 0.631385074 0.541034307 0.462093722 0.399152327 0.349555166 6 1 0.678684577 0.575005194 0.475356676 0.398813552 0.341307364 0.297380969 6 2 0.687993054 0.595871432 0.497868403 0.419734508 0.360056827 0.31409033 6 3 0.69339867 0.610904142 0.51519317 0.436308022 0.375137626 0.327651048 6 4 0.696805022 0.622266808 0.529053126 0.449904702 0.387674997 0.339013353 6 5 0.699086216 0.631168414 0.540464558 0.461351002 0.39835462 0.34875972 6 6 0.700687231 0.638337579 0.55007028 0.471179891 0.407622908 0.357271411 REFERENCES 1022 Table 8: Coefficient Kurtosis Table of Toop-Leone Inverse Weibull distribution for different values of parameters β α γ 4 5 6 7 8 9 1 1 2.21799308 1.966578604 1.862381523 1.80009056 1.760174038 1.73327196 1 2 2.16901906 1.989904972 1.883432354 1.816402864 1.772737797 1.74310779 1 3 2.181110946 2.016869168 1.90481036 1.832527085 1.785022917 1.752667989 1 4 2.202955207 2.042152468 1.924734817 1.847602602 1.796534824 1.761637457 1 5 2.223602742 2.064704401 1.942880501 1.861487726 1.807202265 1.769977724 1 6 2.24096953 2.084540107 1.959311759 1.874237491 1.817070246 1.777725965 2 1 2.204489811 2.036104942 1.916580368 1.839858533 1.78981241 1.755946971 2 2 2.240606729 2.075608857 1.94820648 1.863880822 1.808153183 1.770216939 2 3 2.264802891 2.10672518 1.974688289 1.884564185 1.824177496 1.782790477 2 4 2.281082808 2.131525557 1.997085128 1.902538595 1.838305036 1.79396941 2 5 2.29236257 2.151632244 2.016268719 1.918330963 1.850887744 1.804006191 2 6 2.300429861 2.168208738 2.032900341 1.932349445 1.862200357 1.813098363 3 1 2.252468424 2.085733006 1.954876814 1.868222783 1.811079219 1.772270322 3 2 2.278579891 2.123307968 1.987756083 1.894151879 1.831246757 1.788122103 3 3 2.294145059 2.151374944 2.014300699 1.915844774 1.848442191 1.801788923 3 4 2.304018064 2.173008331 2.036215819 1.934338592 1.863357444 1.81376549 3 5 2.310626055 2.190142298 2.054660701 1.95036039 1.876483991 1.824405257 3 6 2.315248396 2.204025169 2.070437485 1.964427808 1.888176302 1.833964757 4 1 2.276880262 2.11795912 1.981834288 1.888905862 1.826872113 1.784520163 4 2 2.295923215 2.152696196 2.014564254 1.91555608 1.847940813 1.801233494 4 3 2.306864739 2.178027155 2.040520928 1.937535073 1.865686459 1.815487319 4 4 2.313668839 2.197244192 2.061686867 1.956085192 1.88094743 1.827883598 4 5 2.318168782 2.212292909 2.079337196 1.972033938 1.894291198 1.838833043 4 6 2.321292188 2.224382876 2.094325414 1.985952959 1.906115698 1.848625904 5 1 2.290740331 2.140641649 2.00214852 1.904969444 1.83933291 1.794274901 5 2 2.305401566 2.172858021 2.034373807 1.931912495 1.86092724 1.811540645 5 3 2.313681404 2.196018321 2.059645978 1.953931371 1.878974432 1.826163993 5 4 2.318782032 2.213421623 2.080091359 1.972393941 1.894407978 1.838818611 5 5 2.322136141 2.226956821 2.097039316 1.988188021 1.907844608 1.849953668 5 6 2.324455544 2.237774744 2.111363531 2.001916881 1.919710249 1.859882055 6 1 2.299464409 2.157619068 2.018233649 1.918014776 1.849588667 1.802366837 6 2 2.311255591 2.187713111 2.049862918 1.945056629 1.871517707 1.82001981 6 3 2.317850054 2.209140615 2.074472692 1.967012738 1.889742551 1.834897154 6 4 2.321890778 2.22513783 2.094270298 1.985335727 1.905264668 1.847725164 6 5 2.324539269 2.237521501 2.110611256 2.000953224 1.918735791 1.858981232 6 6 2.326366829 2.247384254 2.124375815 2.014488774 1.930601569 1.868994819