The modied beta Weibull distribution

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume , The modied beta Weibull distribution Muhammad Nauman Khan Abstract A new ve-parameter model called the modied beta Weibull probability distribution is being introduced in this paper. This model turns out to be quite eible for analyzing positive data and has bathtub and upside down bathtub hazard rate function. Our main objectives are to obtain representations of certain statistical functions and to estimate the parameters of the proposed distribution. As an application, the probability density function is utilized to model two actual data sets. The new distribution is shown to provide a better t than related distributions. The proposed distribution may serve as a viable alternative to other distributions available in the literature for modeling positive data arising in various elds of scientic investigation such as reliability theory, hydrology, medicine, meteorology, survival analysis and engineering. Keywords: Weibull distribution, Modied beta distribution, Goodnessoft statistics, Lifetime data. 2 AMS Classication: 6E5, 62E5 Received : Accepted : Doi :.5672/HJMS Introduction The Weibull distribution is a very popular distribution named after Waloddi Weibull, a Swedish physicist. He used it in 939 to analyze the breaking strength of materials. Ever since, it has been widely used for analyzing lifetime data. However, this distribution does not have a bathtub or upsidedown bathtub shaped hazard rate function, that is why it cannot be utilized to model the life time of certain systems. To overcome this shortcoming, several generalizations of the classical Weibull distribution have been discussed by dierent authors in recent years. Many authors introduced eible distributions for modeling comple data and obtaining a better t. Etensions of Weibull distribution arise in dierent areas of research as discussed for instance in Department of Mathematics, Kohat University of Science & Technology, Kohat, Pakistan 26, zaybasdf@gmail.com Corresponding Author.

2 554 [, 2, 3, 4, 5, 6, 7, 8, 9,,, 2, 9, 2, 2, 24] and [27]. Many etended Weibull models have an upsidedown bath tub shaped hazard rate, which is the case of the etensions discussed by [4], [4], [8] and [25], among others. Adding parameters to an eisting distribution enables one to obtain classes of more eible distributions. Nadarajah et al. [7] introduced an interesting method for adding three new parameters to an eisting distribution. The new distribution provides more eibility to model various types of data. The baseline distribution has the cdf G, then the new distribution is. F = Ba, b cg } c G+ a b d. The Modied beta Weibull probability density function obtained from. can be epressed in the following form:.2 f = ca gg} a G} b Ba, b cg} a+b. The cdf and pdf of Weibull distribution are dened as follows:.3 and.4 G = e λ k, λ >, k >, > g = k k e λ k. λ λ We further generalize this model by applying the modied beta technique [7], which results in what we are referring to as the modied beta Weibull MBW distribution. The cdf, survival function, pdf and hazard rate function of the modied beta Weibull distribution, for which G is the baseline function, are respectively given by.5.6 F = Ba, b B c + c e λ k ; a, b, c S = Ba, b B c + c e λ k ; a, b, c.7 and.8 f = c a k +k e λ k k b e λ k k +a c e λ k k} a b λ k Ba, b h = k c a +k λ k e k λ k b e k λ k +a c e k λ k} a b Ba, b B c c c + e λ k ; a, b, >, where Ba, b = ta t b dt, Rea >, Reb > and Bz; a, b = z ta t b dt.

3 555 Also λ >, k >, a >, b >, c >. Equations.5 to.8 can be easily evaluated numerically using computational packages such as Mathematica, Maple, MATLAB and R. The following Mathematica code can be used for integration purposes: Integrate[f,,, Innity}]. Further, Figure shows the correctness of the dened cdf Figure. The MBW cdf. λ =.8, k =.6, a =.4, b =.5, c =.8, dotted line, λ = 4.8, k = 2.6, a = 3.4, b = 2.5, c =.8, dashed line, λ =, k = 6, a = 4, b = 5, c = 5, solid line, λ =, k =.2, a =, b = 2, c =., thick line. Note that on making use of the identity.9 z τ = n= Γτ + n Γτ n! z n, z <, τ >, one has the following series representations of the pdf specied by.7. f = ca +k k λ k Ba, b n= m= e λ k k m+b. Γa + b + n c n Γa + bn! Γ a n + m Γ a nm!

4 556 Moreover, the rst derivative of h, which is used to study the shapes of hazard rate functions as eplained in [3] is d d h = c+a k 2 +k λ k e k λ e k λ k +a e k λ k b c c c e λ k } 2 Ba, b B +a c + c e λ k c c + c e λ k c e k λ k} a b c c c + e λ +b k ; a, b + k 2 2+2k λ 2k a + b cc a e k λ k +b e k λ k +a c e k λ k} a b Ba, b B c c c + e λ k ; a, b + c a k 2 λ 2k a 2+2k e k λ k +b e k λ k 2+a c e k λ k} a b Ba, b B c c c + e λ k ; a, b b c a e k λ k b e k λ k +a k 2 2+2k λ 2k c e k λ k} a b Ba, b B c c c + e λ k ; a, b + c a k k λ k 2+k e k λ k b e k λ k +a c e k λ k} a b. Ba, b B c c c + e λ k ; a, b Fig. 2, 3 and 4 plots some MBW curves for dierent choices of parameters for pdf and hazard rate function. Figures 2 and 3 indicate how the new parameters a, b and c aect the MBW density. These graphs illustrate the versatility of the MBW distribution. As can be seen from left panel of Figure 2 that a is a scale parameter and from the right panel of Figure 2 and left panel of Figure 3 that b and c are shape parameters. Similarly 2

5 Figure 2. The MBW pdf. Left panel: λ =.5, k = 3.5, b = 2.8, c = 2. and a = 3 dotted line a = 5 dashed line, a = 7 solid line, a = thick line. Right panel: λ =.5, k =, a =.5, c =.5 and b = dotted line b = 2 dashed line, b = 3 solid line, b = 4 thick line Figure 3. Left panel: The MBW pdf. λ =.8, k =.6, a =.4, b =.5 and c =.8 dotted line, c = 2 dashed line, c = 4 solid line, c = 6 thick line. Right panel: The MBW hazard rate function. λ =.7, k =.2, b =.5, c = 3.5 and a =.2 dotted line a =.6 short dashes, a = 2 long dashes, a = 2.5 solid line, a = 3 thick line. right panel of Figure 3 and left and right panels of Figure 4 represent bathtub shaped and upside down bathtub shaped hazard rate function. The rest of the paper is organized as follows. Representations of certain statistical functions are provided in Section 2. The parameter estimation technique described in Section 3 is utilized in connection with the modeling of two actual data sets originating from the engineering and biological sciences in Section 4, where the new model is compared with several related distributions.

6 Figure 4. Left panel: The MBW hazard rate function. λ =.7, k =.2, a =.5, c = 3.5 and b = dotted line, b =.5 dashed line, b =.9 long dashes, b = 2.3 solid line, b = 2.8 thick line. Right panel: The MBW hazard rate function. λ = 2, k = 4, a = 2, b =.5 and c =.6 dashed line, c = 2 long dashes, c = 2.4 solid line, c = 2.8 thick line. 2. Statistical Functions of the MBW Distribution Here, we derive computable representations of some statistical functions associated with the MBW distribution whose probability density function can be represented by.. The resulting epressions can be evaluated eactly or numerically with symbolic computational packages such as Mathematica, MATLAB or Maple. In numerical applications, innite sum can be truncated whenever convergence is observed. 2.. Moments. We now derive closed form representations of the positive, negative and factorial moments of a MBW random variable. The r th raw moment of the MBW distribution is 2. Which gives 2.2 EX r c a k = λ k Ba, b EX r = c a k λ k Ba, b n= m= Γa + b + n c n Γa + bn! r +k e λ k k m+b d. n= m= b + mλ k k+r k k Γa + b + n c n Γa + bn! Γ k+r k. Γ a n + m Γ a nm! Γ a n + m Γ a nm! The h th order negative moment can readily be determined by replacing r with h in 2.: EX h c a k Γa + b + n c n Γ a n + m = λ k Ba, b Γa + bn! Γ a nm! n= m= h +k e k /λ k m+b d

7 559 Which gives, EX h = c a k λ k Ba, b n= m= The factorial moments of X are Γa + b + n c n Γa + bn! b + mλ k + h k Γ h k EXX X 2 X γ + k. γ m= Γ a n + m Γ a nm! φ m j EX γ m, where EX γ m can be evaluated by replacing r by γ m in Moment Generating Function. The moment generating function of the MBW distribution whose density function is specied by. will be derived here. First, we consider a result developed in [23]: 2.5 η e θ k e s d = 2π q+p/2 q /2 p η /2 s η G q,p p,q p p q θ i+η, i =,,..., p p, s q j/q, j =,,..., q where Rη, Rθ, Rs < and k is rational number such that k = p/q, where p and q are integers. The moment generating function of the MBW distribution whose density function is specied by. is Mt = ca Ba, b n= m= Γa + b + n Γ a n + m c n Γa + b Γ a n n! m! k e λ k m+b k e t d On replacing η with k, θ with λ k m + b and s with t. In the integrand of integral and making use of 2.5, we have the following representation of the moment generating function when k = p/q: 2.6 Mt = c a k λ k Ba, b n= m= Γa + b + n c n Γa + bn! 2π q+p/2 q /2 p k /2 t k G q,p p p λ k q m + b p,q t q Γ a n + m Γ a nm! i+k, p i =,,..., p j/q, j =,,..., q 2.3. Entropy. Entropy is a concept encountered in Physics and Engineering. An etension of Shannon's entropy for the continuous case can be dened as follows: 2.7 Hf = f logf d.

8 56 Combining. with 2.7, one has the following representation: Hf = ca k λ k Ba, b n= m= c a k log λ k Ba, b Γa + b + n c n Γa + bn! n= m= +k e λ k k m+b d ca k + k λ k Ba, b + ca k m + b λ 2k Ba, b n= m= Γa + b + n c n Γa + bn! +k e λ k k m+b log d n= m= +2k e λ k k m+b d. Γ a n + m Γ a nm! Γ a n + m Γ a nm! Γa + b + n Γ a n + m c n Γa + b Γ a n n! m! Γa + b + n Γ a n + m c n Γa + b Γ a n n! m! 2.8 c a Hf = m + bba, b n= m= c a k log λ k Ba, b n= m= ca k + k λ k Ba, b + n= m= Γa + b + n c n Γa + bn! Γa + b + n c n Γa + bn! Γ a n + m Γ a nm! Γ a n + m Γ a nm! Γa + b + n Γ a n + m c n Γa + b Γ a n n! m! +k e λ k k m+b log d c a m + bba, b n= m= Γa + b + n Γ a n + m c n Γa + b Γ a n n! m!. Note that, the integral on the righthand side of 2.8 can be evaluated by numerical integration Mean Residue Life Function. The mean residue life function is dened as K = S = S = S y fy dy y fy dy ] y fy dy, [ EY

9 56 where fy, S and EY are as given in.,.6 and 2.2, respectively and 2.9 c a k yfydy = λ k Ba, b = ca k λ k Ba, b = ca k λ k Ba, b n= m= y k e λ k y k m+b dy. n= m= e m+b λ k y k y k dy n= m= y k G,, Γa + b + nγ a n + m c n Γa + b Γ a nn!m! Γa + b + n Γ a n + m c n Γa + b Γ a n n! m! Γa + b + n Γ a n + m c n Γa + b Γ a n n! m! m + b λ k y p/q dy, where e g = G,, g, k = p/q, p, q are natural co-prime numbers and y t G,, m + b β y p/q dy t+ q p m + b λ k q p t = Gq,p p2 π q /2 p,p+q, t,..., p t, p p p 2. q q, t, t,..., p t 2. p p p Equation 2. is obtained by making use of Equation 3 of [5] Mean Deviation. The mean deviation about the mean is dened by δx = = EX EX f d EX f d + EX f d. EX where EX can be evaluated by letting r = in 2.2. The mean deviation can easily be evaluated by numerical integration. 3. Parameter Estimation In this section, we will make use of the MBW, TransmutedWeibullTW [], Kumaraswamy modied Weibull KwMW [9], Etended Weibull EtW [2], Eponential Weibull EW [5], GammaWeibull GW [22], Generalized modied Weibull GMW [4], Modied Weibull MW [5], Generalized gamma GG [26], Two parameter Weibull Weibull and Two parameter gamma Gamma distributions to model two wellknown real data sets, namely the `Carbon bres' [9] and the `Cancer patients' [6] data sets. The parameters of the MBW distribution can be estimated from the loglikelihood of the samples in conjunction with the NMaimize command in the symbolic computational package Mathematica. Additionally, three goodness-of-t measures are proposed to compare the density estimates.

10 Maimum Likelihood Estimation. In order to estimate the parameters of the proposed MBW density function as dened in Equation.7, the loglikelihood of the sample is maimized with respect to the parameters. Given the data i, i =,..., n, the loglikelihood function is 3. lλ, k, a, b, c = n a logc + logk k logλ logba, b} + k log i + b log e i k /λ k + a a + b log e i k /λ k log c e i k /λ k} where f is as given in.7. The associated nonlinear loglikehood system lθ = θ for MLE estimator derivation reads as follows: 3.2 lθ λ lθ k = k n n λ + b k λ k k i + a c e λ k k i k λ k k i a + b = c e λ k k i } = n k logλ + log i + b } λ k logλ k i λ k log i k i k k e λ i k λ k k i e λ k k i e λ k k } i λ k logλ k i λ k log i k i a e λ k k i ce λ k k } i λ k logλ k i λ k log i k i a + b = c e λ k k i lθ a = nlogc ψ a + ψ a + b} + lθ b lθ c log c e λ k k i = n ψ b + ψ a + b} + = a n c log e λ k k i } = log e λ k k i } log c e λ k k i = n a + b e λ k k i =. c e λ k k i Where ψ is the polygamma function. The above equations cannot be solved analytically and statistical software can be used to solve them numerically.

11 Goodness-of-Fit Statistics. To verify the goodness-of-t of certain statistical models, some goodness-of-t statistics shall be used. They are computed using the symbolic computation package Mathematica. The following goodness-of-t statistics are considered: the Anderson-Darling, Cramér-von Mises and Akaike Information Criterion AIC statistics for comparison purposes. The Anderson-Darling and Cramér-von Mises statistics are widely utilized to determine how closely a specic distribution whose associated cumulative distribution function denoted by cdf ts the empirical distribution associated with a given data set. Upper tail percentiles of the asymptotic distributions of Anderson-Darling and Cramér-von Mises statistics were tabulated in [9]. The distribution having the better t will be the one whose goodness-of-t statistic is the smallest. 4. Empirical illustrations In this section, we present two applications where the MBW model is compared with other related models, namely TransmutedWeibullTW [], Kumaraswamy modied Weibull KwMW [9] Etended Weibull EtW [2], EponentialWeibull EW [5], GammaWeibull GW [22], Generalized modied Weibull GMW [4], Modied Weibull MW [5], Generalized gamma GG [26], Two parameter Weibull Weibull and Two parameter gamma Gamma distributions. We make use of two data sets: rst, the Carbon bres data set [9] and, secondly, the Cancer patients data set [6]. The classical gamma Gamma distribution with density function f = ξ e /φ, >, φ, ξ >. φ ξ Γξ The classical Weibull Weibull distribution with density function f = k k e /λ k, >, k, λ >. λ λ The generalize gamma GG distribution [26] with density function f = k λ ξ ξ e λ k k, >, ξ, k, λ >. Γξ/k The modied Weibull MW distribution [5] with density function f = α γ γ + λ e λ α γ e λ, >, γ, α >, λ. The generalized modied Weibull GMW distribution [4] with density function f = ϕ α γ γ + λ e λ αγ e λ e αγ e λ } ϕ, >, γ, α, ϕ >, λ. The gammaweibull distribution [22] with density function f = k λ k ξ ξ+k e λ k k, >, ξ + k >, λ >. Γ + ξ/k The eponentialweibull EW distribution [5] with density function f = λ + β k k e λ β k, >, λ, β, k >. The TransmutedWeibullTW [] with density function ηe σ η η 2λ e } σ η λ + σ f =, >, σ, η, λ >. σ

12 564 The etended Weibull EtW distribution [2] with density function f = a c + b 2+b e c/ ab e c/, >, a, b >, c. The Kumaraswamy modied Weibull KwMW distribution [9] with density function f = a b α γ γ + λ e λ α γ e λ } e α γ e λ a [ } a] e α γ e λ b, >, a, b, α, γ >, λ Figure 5. Left panel: The MBW density estimate superimposed on the histogram for Carbon bres data. Right panel: The MBW cdf estimates and empirical cdf. Note that: The empirical cdf can be plotted using the following code in mathematica. ListPlot[Table[data[[i]], i/n-/2n},i,,n}]]. 4.. The Carbon Fibres Data Set. The rst data set represents the uncensored real data set on the breaking stress of carbon bres in Gba as reported in [5]. The data are n = 66: 3.7, 2.74, 2.73, 2.5, 3.6, 3., 3.27, 2.87,.47, 3., 3.56, 4.42, 2.4, 3.9, 3.22,.69, 3.28, 3.9,.87, 3.5, 4.9,.57, 2.67, 2.93, 3.22, 3.39, 2.8, 4.2, 3.33, 2.55, 3.3, 3.3, 2.85,.25, 4.38,.84,.39, 3.68, 2.48,.85,.6, 2.79, 4.7, 2.3,.89, 2.88, 2.82, 2.5, 3.65, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.35, 2.55, 2.59, 2.3,.6, 2.2, 3.5,.8, 2.56,.8, The Cancer Patients Data Set. The second data set represents the remission times in months of a random sample of 28 bladder cancer patients as reported in [6]. The data are.8, 2.9, 3.48, 4.87, 6.94, 8.66, 3., 23.63,.2, 2.23, 3.52, 4.98, 6.97, 9.2, 3.29,.4, 2.26, 3.57, 5.6, 7.9, 9.22, 3.8, 25.74,.5, 2.46, 3.64, 5.9, 7.26, 9.47, 4.24, 25.82,.5, 2.54, 3.7, 5.7, 7.28, 9.74, 4.76, 26.3,.8, 2.62, 3.82, 5.32, 7.32,.6, 4.77, 32.5, 2.64, 3.88, 5.32, 7.39,.34, 4.83, 34.26,.9, 2.69, 4.8, 5.34, 7.59,.66, 5.96, 36.66,.5, 2.69, 4.23, 5.4, 7.62,.75, 6.62, 43.,.9, 2.75, 4.26, 5.4, 7.63, 7.2, 46.2,.26, 2.83, 4.33, 5.49, 7.66,.25, 7.4, 79.5,.35, 2.87, 5.62, 7.87,.64, 7.36,.4, 3.2, 4.34, 5.7, 7.93,.79, 8.,.46, 4.4, 5.85, 8.26,.98, 9.3,.76, 3.25, 4.5, 6.25, 8.37, 2.2, 2.2, 3.3, 4.5, 6.54, 8.53, 2.3, 2.28, 2.2, 3.36, 6.76, 2.7, 2.73, 2.7, 3.36, 6.93, 8.65, 2.63, The pdf and cdf estimates of the MBW distribution are plotted in Figures 5 and 6 for the Carbon bres and Cancer patients data, respectively. The estimated hazard

13 565 Table. Estimates of the Parameters, Goodness-of-Fit Statistics and Loglikelihood for the Carbon Fibres Data Distributions Estimates Gamma ξ, φ Weibull k, λ GG k, λ, ξ MWα, γ, λ GMWϕ, α, γ, λ GW k, ξ, λ EW k, λ, β TW η, σ, λ EtW a, b, c KwMWα, γ, λ, a, b MBW λ, k, a, b, c Distributions A W AIC l ˆΘ Gamma ξ, φ Weibull k, λ GG k, λ, ξ MWα, γ, λ GMWϕ, α, γ, λ GW k, ξ, λ EW k, λ, β TW η, σ, λ EtW a, b, c KwMWα, γ, λ, a, b MBW λ, k, a, b, c Figure 6. Left panel: The MBW density estimate superimposed on the histogram for Cancer patients data. Right panel: The MBW cdf estimates and empirical cdf. rate function of MBW distribution are plotted in Figure 7. It can be seen that both shapes of hazard rate function, for carbon bers and cancer patients data sets are like bathtub shaped hazard rate function. The estimates of the parameters and the values of AIC, Anderson-Darling and Cramér-von Mises goodnessoft statistics are given in Tables and 2 for the Carbon bres and Cancer patients data, respectively. It is seen that the proposed MBW model provides the best t for both data sets when considering Anderson-Darling and Cramér-von Mises goodnessoft statistics and is a competitive model when considering AIC.

14 566 Table 2. Estimates of the Parameters, Goodness-of-Fit Statistics and Loglikelihood for the Cancer Patients Data Distributions Estimates Gamma ξ, φ Weibull k, λ GG k, λ, ξ MWα, γ, λ GMWϕ, α, γ, λ GW k, ξ, λ EW k, λ, β TW η, σ, λ EtW a, b, c KwMWα, γ, λ, a, b MBW λ, k, a, b, c Distributions A W AIC l ˆΘ Gamma ξ, φ Weibull k, λ GG k, λ, ξ MWα, γ, λ GMWϕ, α, γ, λ GW k, ξ, λ EW k, λ, β TW η, σ, λ EtW a, b, c KwMWα, γ, λ, a, b MBW λ, k, a, b, c Figure 7. Left panel: The estimated MBW hazard rate function for carbon data. Right panel: The estimated MBW hazard rate function for cancer patients data. 5. Discussion There has been a growing interest among statisticians and applied researchers in constructing eible lifetime models in order to improve the modeling of survival data. As a result, signicant progress has been made towards the generalization of some well known lifetime models, which have been successfully applied to problems arising in several areas of research. In particular, several authors have proposed new distributions which are based on the traditional Weibull model. In this paper, we introduce a veparameter distribution which is obtained by applying the modied beta technique to the Weibull model. Interestingly, our proposed model has bathtub and up side down bathtub shaped hazard rate function.we studied some of its statistical properties. We also provided

15 567 computable representations of the positive and negative moments, the factorial moments, the moment generating function, the mean residue life function, the mean deviation and the associated Shannon's entropy. The proposed distribution was applied to two data sets and shown to provide a better t than other related models. The distributional results developed in this article should nd numerous applications in the physical and biological sciences, reliability theory, hydrology, medicine, meteorology, engineering and survival analysis. Acknowledgements The author would like to epress thanks to the editor and two anonymous referees for useful suggestions and comments which have improved the rst version of the manuscript. References [] Aryal, G.R. and Tsokos, C.P. Transmuted Weibull distribution: a generalization of the Weibull probability distribution, Eur J Pure Appl Math 4, 892, 2. [2] Badmus, N.I., Ikegwu. and Emmanuel M. The beta-weighted Weibull distribution: some properties and application to bladder cancer data, J Appl Computat Math 2 45 [cited 23 Nov 25]. Available from: DOI:.472/ [3] Barreto-Souza, W., Morais, A.L. and Cordeiro G.M. The Weibull-Geometric distribution, J Statist Comput Simulation 6, 3542,2. [4] Carrasco, J.M.F., Edwin, M.M. Ortega and Cordeiro, G.M. A generalized modied Weibull distribution for lifetime modeling, Comput Stat Data Anal 53, 45462, 28. [5] Cordeiro, G.M., Edwin, M.M. Ortega and Lemonte, A.J. The eponential-weibull lifetime distribution, J Statist Comput Simulation, 84, ,24. [6] Cordeiro, G.M., Hashimoto, E.M. and Edwin, M.M. Ortega. The McDonald Weibull model Statistics: A Journal of Theoretical and Applied Statistics 48, [7] Cordeiro, G.M. and Lemonte, A.J. On the MarshallOlkin etended Weibull distribution, Stat Papers 54, , 23. [8] Cordeiro, G.M., Edwin, M.M. Ortega and Nadarajah, S. The Kumaraswamy Weibull distribution with application to failure data J Franklin Inst 347, , 2. [9] Cordeiro, G.M., Edwin, M.M. Ortega., and Silva, G.O. The Kumaraswamy modied Weibull distribution: theory and applications J Statist Comput Simulation 84, 3874, 24. [] Cordeiro, G.M., Gomes, A.E., da-silva, C.Q. and Edwin, M.M. Ortega. The beta eponentiated Weibull distribution J Statist Comput Simulation 83, 438, 23. [] Cordeiro, G.M., Silva, G.O. and Edwin, M.M. Ortega. The betaweibull geometric distribution Statistics: A Journal of Theoretical and Applied Statistics 47, 87834, 23. [2] Ghitany, M.E., Al-Hussaini, E.K. and Al-Jarallah, R.A. MarshallOlkin etended Weibull distribution and its application to censored data, J Appl Statist 32, 2534, 25. [3] Glaser, R.E., Bathtub and related failure rate charecterizations, Journal of American statistical association 75, , 98. [4] Jiang, R. and Murthy, D.N.P. Miture of Weibull distributionsparametric characterization of failure rate function, Appl Stoch Model D A 4, 4765, 998. [5] Lai, C. D., Xie, M.and Murthy, D. N. P. A modied Weibull distribution, IEEE Transactions on Reliability 52, [6] Lee, E. and Wang, J. Statistical Methods for Survival Data Analysis, Wiley & Sons, New York 23. [7] Nadarajah, S. Teimouri, M. and Shih, S.H. Modied Beta Distributions, Sankhyā. Ser. B 76-B, 948, 24. [8] Nadarajah, S. Cordeiro, G.M. and Edwin, M.M. Ortega. General results for the beta modied Weibull distribution, J Statist Comput Simulation 8, 232, 2. [9] Nichols, M.D. and Padgett, W.J. A bootstrap control chart for Weibull percentiles, Qual Reliab Eng Int. 22, 45, 26.

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