8_shuaib.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 1, 2013, 66-88 ISSN 1307-5543 – www.ejpam.com Transmuted Modified Weibull Distribution: A Generalization of the Modified Weibull Probability Distribution Muhammad Shuaib Khan∗, Robert King School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia Abstract. This paper introduces a transmuted modified Weibull distribution as an important compet- itive model which contains eleven life time distributions as special cases. We generalized the three parameter modified Weibull distribution using the quadratic rank transmutation map studied by Shaw et al. [12] to develop a transmuted modified Weibull distribution. The properties of the transmuted modified Weibull distribution are discussed. Least square estimation is used to evaluate the parame- ters. Explicit expressions are derived for the quantiles. We derive the moments and examine the order statistics. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. This model is capable of modeling of various shapes of aging and failure criteria. 2010 Mathematics Subject Classifications: 90B25; 62N05 Key Words and Phrases: Reliability functions, moment estimation, moment generating function, least square estimation, order statistics, maximum likelihood estimation. 1. Introduction The Weibull distribution is a life time probability distribution used in the reliability engi- neering discipline. We introduce the transmuted modified Weibull distribution which extends recent development on transmuted Weibull distribution by Aryal et al. [1]. More recently Aryal et al. [2] introduced the transmuted extreme value distribution. In this article, we introduce and study several mathematical properties of new reliability model referred to as the transmuted modified Weibull distribution. This paper focuses on all the properties of this model and presents the graphical analysis of the subject distribution. This paper presents the relationship between shape parameter and other properties such as non-reliability function, reliability function, instantaneous failure rate, cumulative instantaneous failure rate models. Recently Ammar et al. [10] proposed the modified Weibull distribution. FMW (t) = 1− exp(−αt −ηtβ ). (1) ∗Corresponding author. Email addresses: Shuaib.stat�gmail. om (M. Shuaib), robert.king�new astle.edu.au (R. King) http://www.ejpam.com 66 c© 2013 EJPAM All rights reserved. M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 67 This article defined the family of transmuted modified Weibull distributions. The main fea- ture of this model is that a transmuted parameter λ is introduced in the subject distribution which provides greater flexibility in the form of new distributions. Using the quadratic rank transmutation map studied by Shaw et al. [12], we develop the four parameter transmuted modified Weibull distribution T MW D(α,β ,η,λ, t). We provide a comprehensive description of mathematical properties of the subject distribution with the hope that it will attract wider applications in reliability, engineering and in other areas of research. These distributions have several attractive properties for more details we refer to [5, 8, 4, 6, 7, 13]. The article is organized as follows, In Section 2, we present the flexibility of the subject distribution and some special sub-models. In Section 3, we demonstrate the reliability func- tions of the subject model. A range of mathematical properties are considered in Section 4-5. These include quantile functions, moment estimation, moment generating function and least square estimation. In Section 6, the minimum, maximum and median order statistics models are discussed. We also demonstrate the joint density functions g(t1, tn) of the trans- muted modified Weibull distribution. In Section 7, we demonstrate the maximum likelihood estimates (M LES) of the unknown parameters and the asymptotic confidence intervals of the unknown parameters. However, some of these quantities could not be evaluated in closed form and therefore special cases were used to express them. In Section 8, we fit the TMW model to illustrate its usefulness. In Section 9, concluding remarks are addressed. 2. Transmuted Modified Weibull Distribution A random variable T is said to have transmuted Modified Weibull probability distribu- tion with parameters α,β ,η > 0 and −1 ≤ λ ≤ 1 . It can be used to represent the failure probability density function is given by fT MW (t) = (α+ βηtβ−1)exp(−αt −ηtβ )(1−λ+ 2λexp(−αt −ηtβ )) t > 0 (2) Where β and η are the shape parameters representing the different patterns of the trans- muted modified Weibull distribution and are positive, α is a scale parameter representing the characteristic life and is also positive, λ is the transmuted parameter. The restrictions in equa- tion (2) on the values of α,β ,η and λ are always the same. The transmuted modified Weibull distribution is very flexible model that approaches to different distributions when its parameters are changed. The flexibility of the transmuted modified Weibull distribution is explained in Table 1. The subject distribution includes as special cases the transmuted modified Exponential (TME), transmuted Linear Failure Rate (TLFR), transmuted Weibull (TW), transmuted Rayleigh (TR) and transmuted Exponential distributions. Figure 1 shows the transmuted modified Weibull distribution that approaches to eleven different lifetime distributions when its parameters are changed. The cumulative distribution function of the transmuted modified Weibull distribution is denoted by FT MW (t) and is defined as FT MW (t) = (1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ )) (3) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 68 Figure 1: Sub Models of Transmuted Modified Weibull Distribution Table 1: Modified and Transmuted Modified type distributions: T=Transmuted; M=Modified; W=Weibull; E=Exponential; R=Rayleigh S.N o Distr ibution T MW T MW T MW T MW α β η λ 1 T M E α 1 η λ 2 T MR α 2 η λ 3 MW α β η 0 4 MR α 2 η 0 5 M E α 1 η 0 6 TW 0 β η λ 7 TR 0 2 η λ 8 T E 0 1 η λ 9 W 0 β η 0 10 R 0 2 η 0 11 E 0 1 η 0 Figure 2a shows the diverse shape of the transmuted modified Weibull PDF with different choice of parameters. The beauty of the subject distribution and its sub models are explained in Table 1. When the CDF of the T MW D(α,β ,η,λ, t) distribution has zero value then it represents no failure components. It is clear from the Figure 2b that two curves intersect at the point of (0.4, 0.574997) and two curves approximately intersect at the point (1.1, 0.885533) the characteristic point for the transmuted modified Weibull CDF. M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 69 (a) Probability Density Function (b) Cumulative Distribution Function Figure 2: Transmuted Modified Weibull PDF & CDF 3. Reliability Analysis The transmuted modified Weibull distribution can be a useful characterization of life time data analysis. The reliability function (RF) of the transmuted modified Weibull distribution is denoted by RT MW (t) also known as the survivor function and is defined as RT MW (t) = 1− FT MW (t) RT MW (t) = 1− (1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ )) (4) It is important to note that RT MW (t) + FT MW (t) = 1 . Figure 3a illustrates the reliability pattern of the transmuted modified Weibull distribution with different choice of parameters. One of the characteristic in reliability analysis is the hazard rate function defined by hT MW (t) = fT MW (t) 1− FT MW (t) (5) The hazard function (HF) of the transmuted modified Weibull distribution also known as instantaneous failure rate denoted by hT MW (t) and is defined as fT MW (t) RT MW (t) hT MW (t) = (α+ βηtβ−1)exp(−αt −ηtβ )(1−λ+ 2λexp(−αt −ηtβ )) 1− (1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ )) (6) It is important to note that the units for hT MW (t) is the probability of failure per unit of time, distance or cycles. Figures 3b and 3c shows the transmuted modified Weibull instantaneous failure rate pat- terns. These failure rates are defined with different choices of parameters. M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 70 (a) Reliability Function (b) Hazard Function (wear out period) (c) Hazard Function (useful life period) (d) Cumulative Hazard Function Figure 3: Transmuted Modified Weibull Reliability and Hazard Functions In both cases when β = 5 the distribution has the strictly increasing HR. when β = 1 , the HR is steadily increasing in Figure 3b which represents failure life between early failures and wear out periods. When β = 1 , the HR is constant in Figure 3c which represents failure life in useful life period. When β > 1 , the HF is continually increasing which represents failures occurs after the useful life periods and approaches to the wear-out failures. The HR of the TMWD as given in equation (6) becomes identical with the HR of transmuted modified Rayleigh distribution for β = 2 and for β = 1 it coincides with the transmuted modified Exponential distribution. So the transmuted modified Weibull distribution is a very flexible reliability model. The Cumulative hazard function of the transmuted modified Weibull distribution is de- noted by HT MW (t) and is defined as HT MW (t) = −ln|(1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ ))| (7) It is important to note that the units for HT MW (t) is the cumulative probability of failure per unit of time, distance or cycles. Figure 3d shows the transmuted modified Weibull cumulative M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 71 hazard failure rates with different choices of parameters. For all choice of parameters the distribution has the decreasing patterns of cumulative instantaneous failure rates. Theorem 1. The hazard rate function of the transmuted modified Weibull distribution has the following properties (i) If β = 1 the failure rate is same as the T M ED(α,η,λ, t) (ii) If β = 2 the failure rate is same as the T LFRD(α,η,λ, t) (iii) If α = 0 the failure rate is same as the TW D(β ,η,λ, t) (iv) If α = 0,β = 1 the failure rate is same as the T ED(η,λ, t) (v) If α = 0,β = 2 the failure rate is same as the TRD(η,λ, t) (vi) If λ= 0 the failure rate is same as the MW D(α,β ,η, t) (vii) If λ= 0,β = 1 the failure rate is same as the M ED(α,η, t) (viii) If λ= 0,β = 2 the failure rate is same as the MRD(α,η, t) Proof. The hazard function (HF) of the transmuted modified Weibull distribution is given in equation (6) has the special cases with different choice of parameters (i) If β = 1 the failure rate is same as the T M ED(α,η,λ, t) hT M E(t) = (α+η)exp(−αt −ηt)(1−λ+ 2λexp(−αt −ηt)) 1− (1− exp(−αt −ηt))(1+λexp(−αt −ηt)) (ii) If β = 2 the failure rate is same as the T LFRD(α,η,λ, t) hT LFR(t) = (α+ 2ηt)exp(−αt −ηt2)(1−λ+ 2λexp(−αt −ηt2)) 1− (1− exp(−αt −ηt2))(1+λexp(−αt −ηt2)) (iii) If α= 0 the failure rate is same as the TW D(β ,η,λ, t) hTW (t) = (βηtβ−1)exp(−ηtβ )(1−λ+ 2λexp(−ηtβ )) 1− (1− exp(−ηtβ ))(1+λexp(−ηtβ )) (iv) If α= 0,β = 1 the failure rate is same as the T ED(η,λ, t) hT E(t) = (η)exp(−ηt)(1−λ+ 2λexp(−ηt)) 1− (1− exp(−ηt))(1+λexp(−ηt)) (v) If α= 0,β = 2 the failure rate is same as the TRD(η,λ, t) hTR(t) = (2ηt)exp(−ηt2)(1−λ+ 2λexp(−ηt2)) 1− (1− exp(−ηt2))(1+λexp(−ηt2)) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 72 (vi) If λ= 0 the failure rate is same as the MW D(α,β ,η, t) hMW (t) = (α+ βηtβ−1)exp(−αt −ηtβ ) 1− (1− exp(−αt −ηtβ )) (vii) If λ= 0,β = 1 the failure rate is same as the M ED(α,η, t) hM E(t) = (α+η)exp(−αt −ηt) 1− (1− exp(−αt −ηt)) (viii) If λ= 0,β = 2 the failure rate is same as the MRD(α,η, t) hMR(t) = (α+ 2ηt)exp(−αt −ηt2) 1− (1− exp(−αt −ηt2)) 4. Statistical Properties This section explain the statistical properties of the T MW D(α,β ,η,λ, t) 4.1. Quantile and Median The quantile tq of the T MW D(α,β ,η,λ, t) is the real solution of the following equation ηtβq +αtq + ln  1− (1+λ)− p (1+λ)2− 4λq 2λ   = 0 (8) The above equation (8) has no closed-form solution in tq , so we have different cases by sub- stituting the parametric values in the above quantile equation. So the derived special cases are (1) The q-th quantile of the T LFRD(α,η,λ, t) by substituting β = 2 tq = −α+ Ç α2 − 4η ln � 1− (1+λ)− p (1+λ)2−4λq 2λ � 2η (2) The q-th quantile of the TW D(β ,η,λ, t) by substituting α= 0 tq =  − 1 η ln  1− (1+λ)− p (1+λ)2 − 4λq 2λ     1 β M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 73 (3) The q-th quantile of the TRD(η,λ, t) by substituting α = 0,β = 2 tq = √ √ √ √  − 1 η ln  1− (1+λ)− p (1+λ)2− 4λq 2λ     (4) The q-th quantile of the T ED(η,λ, t) by substituting α= 0,β = 1 tq =  − 1 η ln  1− (1+λ)− p (1+λ)2 − 4λq 2λ     By putting q = 0.5 in equation (8) we can get the median of T MW D(α,β ,η,λ, t). The median life of the subject distribution is the 50th percentile. In practice, this is the life by which 50 percent of the units will be expected to have failed and so it is the life at which 50 percent of the units would be expected to still survive. Figure 4a shows the trans- muted modified Weibull median life with different choice of parameters are 0.1 ≤ β ≤ 5, λ = 0.3,0.5,0.7,1 and the value of η = 2. It is important to note that as the λ increases the pattern of the median life increases. Figure 4b shows the multiple patterns of the sub- ject distribution for B-life with different choice of parameters. Here the B − 0.1 life shows the maximum values of percentiles life and B − 0.00001 life shows the minimum values of percentiles life. So as the percentile decreases the pattern of B-lives decreases. All of these B-lives are of increasing patterns. Here as β →∞ then these B-lives are also increasing. The subject distribution for percentile life is applying in those situations where the B-lives are of increasing order. (a) Median (b) B-lives Figure 4: Transmuted Modified Weibull Quantiles 4.2. Random Number Generation The random number as t of the T MW D(α,β ,η,λ, t) is defined by the following equation (1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ )) = ξ, where ξ∼ U(0,1) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 74 ηtβ +αt + ln  1− (1+λ)− p (1+λ)2 − 4λξ 2λ   = 0 (9) The above equation is not in closed form solution in t, Using ξ a random number uni- formly distributed from zero to one, we have solved the above equation F(t) = ξ to obtain a random number in t. 4.3. Moments The following theorem gives the kth moment of the T MW D(α,β ,η,λ, t) Theorem 2. If T has the T MW D(α,β ,η,λ, t) with |λ| ≥ 1 , then the kth moment of T say µk is given as follows µk =          ∑∞ i=0 (−1)iηi i! h (1−λ) � Γ(iβ+k+1) αiβ+k + βηΓ(β(i+1)+k) αβ(i+1)+k �i if α,β ,η > 0 +2λ ∑∞ i=0 (−1)i (2η)i i! h� αΓ(iβ+k+1) (2α)iβ+k+1 + βηΓ(β(i+1)+k) (2α)β(i+1)+k �i η −k β Γ � 1+ k β � � (1−λ) +λ2 −k β � if α= 0 α−kΓ(1+ k)((1−λ) +λ2−k) if β = 0 (10) Based on the above results given in Theorem 2, the coefficient of variation, coefficient of skewness and coefficient of kurtosis of T MW D can be obtained according to the following relation CVT MW = r µ2 µ1 − 1 (11) CST MW = µ3− 3µ2µ1+ 2µ3 1 (µ2−µ2 1) 3 2 (12) CKT MW = µ4− 4µ3µ1+ 6µ2µ 2 1− 3µ4 1 (µ2−µ2 1) 2 (13) The relationship between β and the mean life is shown in Figure 5. From this analysis it is clear that as β → ∞ the mean life of the subject distribution is also increasing. So β and the mean life has the positive proportion. The relationship between β and the variance life is shown in Figure 6a. From our calculation it is clear that as β →∞ the variance life of the subject distribution is decreasing. So β and the variance life has the negative proportion. The relationship between β and the C .VT MW life is shown in Figure 6b. M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 75 Figure 5: β vs Mean (a) β vs VarT MW (b) β vs CVT MW (c) β vs CST MW (d) β vs CKT MW Figure 6: Transmuted Modified Weibull β vs. Coefficients This relationship shows that it becomes asymptotic decreasing as β → ∞. The C .VT MW life is used to measure the consistency of the data. Here C .ST MW is the quantity used to M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 76 measure the skewness of the distribution. The relationship between β and C .ST MW is shown in Figure 6c. From our calculations it is clear that C .ST MW becomes asymptotically decreasing as β → ∞ . Here C .KT MW is the quantity used to measure the kurtosis of the distribution. The relationship between β and C .KT MW is shown in Figure 6d. From our calculations it is clear that as β →∞ the value of C .KT MW becomes asymptotically decreasing. 4.4. Moment Generating Function The following theorem gives the moment generating function (mgf) of T MW D(α,β ,η,λ, t) Theorem 3. If T has the T MW D(α,β ,η,λ, t) with |λ| ≥ 1 , then the moment generating function of T say Mx(t) is given as follows Mx (t) =            ∑∞ i=0 (−1)iηi i! h (1−λ) � αΓ(iβ+1) (α−t)iβ+1 + βηΓ(β(i+1)) (α−t)β(i+1) �i if α,β ,η > 0 +2λ ∑∞ i=0 (−1)i (2η)i i! h� αΓ(iβ+1) (2α−t)iβ+1 + βηΓ(β(i+1)+k) (2α−t)β(i+1) �i ∑∞ i=0 t i i! η −i β Γ � 1+ i β � � (1−λ) +λ2 −i β � if α= 0 α � 1−λ α−t + 2λ 2α−t � if β = 0 (14) The proof of this theorem is provided in Appendix. Based on the above results given in Theorem 3, the measure of central tendency, measure of dispersion, coefficient of variation, coefficient of skewness and coefficient of kurtosis of T MW D(α,β ,η,λ, t) can be obtained according to the above relation in Theorem 2. 5. Least Square Estimation Let T1, T2, . . . , Tn be a random sample of T MW D(α,β ,η,λ, t) transmuted modified Weibull distribution with cdf FT MW (t), and suppose that T(i), i = 1,2, . . . , n denote the ordered sam- ple. For sample of size n, we have E(F(T(i))) = i n+ 1 The least square estimators (LSES) are obtained by minimizing Q(α,β ,η,λ) = n ∑ i=0 � F(T(i))− i n+ 1 �2 (15) By using (3) and (15) we have the following equation Q(α,β ,η,λ) = n ∑ i=0 � (1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ ))− i n+ 1 �2 (16) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 77 To minimize equation (16) with respect to α,β ,η and λ , we differentiate with respect to these parameters, which leads to the following equations n ∑ i=0 ((1− exp(−αt −ηtβ )) � 1+λexp(−αt −ηtβ ))− i n+ 1 � (t) × ((1+λ)exp(−αt −ηtβ ))− 2λ(1− exp(−αt −ηtβ )) = 0 n ∑ i=0 ((1− exp(−αt −ηtβ )) � 1+λexp(−αt −ηtβ ))− i n+ 1 � ηtβ ln(t) × ((1+λ)exp(−αt −ηtβ ))− 2λ(1− exp(−αt −ηtβ )) = 0 n ∑ i=0 ((1− exp(−αt −ηtβ )) � 1+λexp(−αt −ηtβ ))− i n+ 1 � tβ × ((1+λ)exp(−αt −ηtβ ))− 2λ(1− exp(−αt −ηtβ )) = 0 n ∑ i=0 ((1− exp(−αt −ηtβ )) � 1+λexp(−αt −ηtβ ))− i n+ 1 � × (exp(−αt −ηtβ ))− 2(1− exp(−αt −ηtβ )) = 0 6. Order Statistics Let T1:n ≤ T2:n ≤ . . . ≤ Tn:n be the order statistics from the continuous distribution, then the pdf of Tr:n 1≤ r ≤ n is given by fr:n(t) = Cr:n(F(t)) r−1(1− F(t))n−r f (t), t > 0 (17) The joint pdf of Tr:n and Ts:n 1≤ r ≤ s ≤ n, is given by fr:s:n(t,u) = Cr:s:n(F(t)) r−1(F(u)− F(t))s−r−1(1− F(t))n−s f (t) f (u), (18) for 0≤ t ≤ u≤∞ and where Cr:n = n! (r−1)!(n−r)! and Cr:s:n = n! (r−1)!(s−r−1)!(n−s)! 6.1. Distribution of Minimum and Maximum Let t1, t2, . . . tn be n given random variables. Here we define T1 = Min(t1, t2, . . . tn) and Tn = Max(t1, t2, . . . tn) ordered random variables. We find the transmuted modified Weibull distribution for the minimum and maximum observations and its sub models when its param- eters are changed [9, 11]. M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 78 Theorem 4. Let t1, t2, . . . tn are independently identically distributed ordered random variables from the transmuted modified Weibull distribution having Ist order and nth order probability density function is given by f1:n(t) =n(1− (1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ )))n−1(α + βηtβ−1)× exp(−αt −ηtβ )(1−λ+ 2λexp(−αt −ηtβ )) (19) fn:n(t) =n((1− exp(−αt −ηtβ ))(1+λexp(−αt −ηtβ )))n−1(α+ βηtβ−1) × exp(−αt −ηtβ )(1−λ+ 2λexp(−αt −ηtβ )) (20) Proof. For the minimum and maximum order statistic of the four parameters transmuted modified Weibull distribution have different life time distributions when its parameters are changed. Case A: Minimum order statistic 1. The minimum order statistic of the T LFRD(α,η,λ, t) by substituting β = 2 f1:n(t) =n(1− (1− exp(−αt −ηt2))(1+λexp(−αt −ηt2)))n−1(α+ 2ηt) × exp(−αt −ηt2)(1−λ+ 2λexp(−αt −ηt2)) 2. The minimum order statistic of the T M ED(α,η,λ, t) by substituting β = 1 f1:n(t) =n(1− (1− exp(−αt −ηt))(1+λexp(−αt −ηt)))n−1(α+η) × exp(−αt −ηt)(1−λ+ 2λexp(−αt −ηt)) 3. The minimum order statistic of the TW D(β ,η,λ, t) by substituting α= 0 f1:n(t) =n(1− (1− exp(−ηtβ ))(1+λexp(−ηtβ )))n−1(βηtβ−1) × exp(−ηtβ )(1−λ+ 2λexp(−ηtβ )) 4. The minimum order statistic of the TRD(η,λ, t) by substituting α = 0,β = 2 f1:n(t) =n(1− (1− exp(−ηt2))(1+λexp(−ηt2)))n−1(2ηt) × exp(−ηt2)(1−λ+ 2λexp(−ηt2)) 5. The minimum order statistic of the T ED(η,λ, t) by substituting α= 0,β = 1 f1:n(t) = n(1−(1−exp(−ηt))(1+λexp(−ηt)))n−1(η)×exp(−ηt)(1−λ+2λexp(−ηt)) 6. The minimum order statistic of the MW D(α,β ,η, t) by substituting λ = 0, f1:n(t) = n(1− (1− exp(−αt −ηtβ )))n−1(α+ βηtβ−1)exp(−αt −ηtβ ) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 79 7. The minimum order statistic of the MRD(α,η, t) by substituting β = 2,λ= 0, f1:n(t) = n(1− (1− exp(−αt −ηt2)))n−1(α+ 2ηt)exp(−αt −ηt2) 8. The minimum order statistic of the M ED(α,η, t) by substituting β = 1,λ= 0, f1:n(t) = n(1− (1− exp(−αt −ηt)))n−1(α+η)exp(−αt −ηt) Case B: Maximum order statistic 1. The maximum order statistic of the T LFRD(α,η,λ, t) by substituting β = 2 fn:n(t) =n((1− exp(−αt −ηt2))(1+λexp(−αt −ηt2)))n−1(α+ 2ηt) × exp(−αt −ηt2)(1−λ+ 2λexp(−αt −ηt2)) 2. The maximum order statistic of the T M ED(α,η,λ, t) by substituting β = 1 fn:n(t) =n((1− exp(−αt −ηt))(1+λexp(−αt −ηt)))n−1(α+η) × exp(−αt −ηt)(1−λ+ 2λexp(−αt −ηt)) 3. The maximum order statistic of the TW D(β ,η,λ, t) by substituting α= 0 fn:n(t) =n((1− exp(−ηtβ ))(1+λexp(−ηtβ )))n−1(βηtβ−1) × exp(−ηtβ )(1−λ+ 2λexp(−ηtβ )) 4. The maximum order statistic of the TRD(η,λ, t) by substituting α= 0,β = 2 fn:n(t) =n((1− exp(−ηt2))(1+λexp(−ηt2)))n−1(2ηt) × exp(−ηt2)(1−λ+ 2λexp(−ηt2)) 5. The maximum order statistic of the T ED(η,λ, t) by substituting α= 0,β = 1 fn:n(t) = n((1− exp(−ηt))(1+λexp(−ηt)))n−1(η)× exp(−ηt)(1−λ+ 2λexp(−ηt)) 6. The maximum order statistic of the MW D(α,β ,η, t) by substituting λ= 0, fn:n(t) = n((1− exp(−αt −ηtβ )))n−1(α+ βηtβ−1)exp(−αt −ηtβ ) 7. The maximum order statistic of the MRD(α,η, t) by substituting β = 2,λ = 0, fn:n(t) = n((1− exp(−αt −ηt2)))n−1(α+ 2ηt)exp(−αt −ηt2) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 80 8. The maximum order statistic of the M ED(α,η, t) by substituting β = 1,λ= 0, fn:n(t) = n((1− exp(−αt −ηt)))n−1(α+η)exp(−αt −ηt) Theorem 5. Let t1, t2, . . . tn are independently identically distributed ordered random variables from the transmuted modified Weibull distribution having median order Tm+1 probability density function is given by g( t̃) = (2m+ 1)! m!m! (F( t̃))m(1− F( t̃))m f ( t̃), 0≤ t̃ ≤∞ (21) Proof. Using (21) the median order statistic of the four parameters transmuted modified Weibull distribution is given below g( t̃) = (2m+ 1)! m!m! ((1− exp(−α t̃ −η t̃β ))(1+λexp(−α t̃ −η t̃β )))m(α+ βη t̃β−1) × (1− (1− exp(−α t̃ −η t̃β ))(1+λexp(−α t̃ −η t̃β )))m × exp(−α t̃ −η t̃β )(1−λ+ 2λexp(−α t̃ −η t̃β )) (22) Using (22) we have different life time distributions of median order statistic when its param- eters are changed 1. The median order statistic of the T LFRD(α,η,λ, t) by substituting β = 2 g( t̃) = (2m+ 1)! m!m! ((1− exp(−α t̃ −η t̃2))(1+λexp(−α t̃ −η t̃2)))m(α+ 2η t̃) × (1− (1− exp(−α t̃ −η t̃2))(1+λexp(−α t̃ −η t̃2)))m × exp(−α t̃ −η t̃2)(1−λ+ 2λexp(−α t̃ −η t̃2)) 2. The median order statistic of the T M ED(α,η,λ, t) by substituting β = 1 g( t̃) = (2m+ 1)! m!m! ((1− exp(−α t̃ −η t̃))(1+λexp(−α t̃ −η t̃)))m(α+η) × (1− (1− exp(−α t̃ −η t̃))(1+λexp(−α t̃ −η t̃)))m × exp(−α t̃ −η t̃)(1−λ+ 2λexp(−α t̃ −η t̃)) 3. The median order statistic of the TW D(β ,η,λ, t) by substituting α= 0 g( t̃) = (2m+ 1)! m!m! ((1− exp(−η t̃β ))(1+λexp(−η t̃β )))m(βη t̃β−1) × (1− (1− exp(−η t̃β ))(1+λexp(−η t̃β )))m × exp(−η t̃β )(1−λ+ 2λexp(−η t̃β )) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 81 4. The median order statistic of the TRD(η,λ, t) by substituting α= 0,β = 2 g( t̃) = (2m+ 1)! m!m! ((1− exp(−η t̃2))(1+λexp(−η t̃2)))m(2η t̃) × (1− (1− exp(−η t̃2))(1+λexp(−η t̃2)))m × exp(−η t̃2)(1−λ+ 2λexp(−η t̃2)) 5. The median order statistic of the T ED(η,λ, t) by substituting α= 0,β = 1 g( t̃) = (2m+ 1)! m!m! ((1− exp(−η t̃))(1+λexp(−η t̃)))m(η t̃) × (1− (1− exp(−η t̃))(1+λexp(−η t̃)))m × exp(−η t̃)(1−λ+ 2λexp(−η t̃)) 6. The median order statistic of the MW D(α,β ,η, t) by substituting λ= 0, g( t̃) = (2m+ 1)! m!m! ((1− exp(−α t̃ −η t̃β ))m(α+ βη t̃β−1) × (1− (1− exp(−α t̃ −η t̃β ))m× exp(−α t̃ −η t̃β ) 7. The median order statistic of the MRD(α,η, t) by substituting β = 2,λ = 0, g( t̃) = (2m+ 1)! m!m! ((1− exp(−α t̃ −η t̃2))m(α+ 2η t̃) × (1− (1− exp(−α t̃ −η t̃2))m× exp(−α t̃ −η t̃2) 8. The median order statistic of the M ED(α,η, t) by substituting β = 1,λ= 0, g( t̃) = (2m+ 1)! m!m! ((1− exp(−α t̃ −η t̃))m(α+η) × (1− (1− exp(−α t̃ −η t̃))m× exp(−α t̃ −η t̃) 6.2. Joint Distribution of rth Order Statistic Tr and sth Order Statistic Ts The joint pdf of Tr and Ts with Tr = t and Ts = u, (1≤ r ≤ s ≤ n) in (18) by taking r = 1 and s = n in (17) the minimum and maximum joint density can be written as g(t1, tn) = n(n− 1)(F(tn)− F(t1)) n−2 f (t1) f (t2) (23) Theorem 6. Let t1, t2, . . . , tn be independently identically distributed ordered random variables from the transmuted modified Weibull distribution having joint probability density function using (2) and (3) in (23) is given by g(t1, tn) =n(n− 1)(α+βηt β−1 1 )exp(−αt1 −ηt β 1 )(1−λ+ 2λexp(−αt1 −ηt β 1 )) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 82 × (α+ βηtβ−1 n )exp(−αtn −ηtβn )(1−λ+ 2λexp(−αtn −ηtβn )) × [(1− exp(−αtn −ηtβn ))(1+λexp(−αtn−ηtβn )) − (1− exp(−αt1 −ηt β 1 ))(1+λexp(−αt1 −ηt β 1 ))] n−2 (24) Proof. Using (24), the joint probability density function g(t1, tn) of the subject distribution has different joint distributions when its parameters are changed 1. The min and max order statistic of the T LFRD(α,η,λ, t) by substituting β = 2 g(t1, tn) =n(n− 1)(α+ 2ηt1)exp(−αt1 −ηt2 1)(1−λ+ 2λexp(−αt1 −ηt2 1)) × (α+ 2ηtn)exp(−αtn−ηt2 n)(1−λ+ 2λexp(−αtn −ηt2 n)) × [(1− exp(−αtn −ηt2 n))(1+λexp(−αtn −ηt2 n)) − (1− exp(−αt1 −ηt2 1))(1+λexp(−αt1 −ηt2 1))] n−2 2. The min and max order statistic of the T M ED(α,η,λ, t) by substituting β = 1 g(t1, tn) =n(n− 1)(α+η)exp(−αt1 −ηt1)(1−λ+ 2λexp(−αt1 −ηt1)) × (α+η)exp(−αtn −ηtn)(1−λ+ 2λexp(−αtn−ηtn)) × [(1− exp(−αtn −ηtn))(1+λexp(−αtn −ηtn)) − (1− exp(−αt1 −ηt1))(1+λexp(−αt1 −ηt1))] n−2 3. The min and mix order statistic of the TW D(β ,η,λ, t) by substituting α = 0 g(t1, tn) =n(n− 1)(βηt β−1 1 )exp(−ηt β 1 )(1−λ+ 2λexp(−ηt β 1 )) × (βηtβ−1 n )exp(−ηtβn )(1−λ+ 2λexp(−ηtβn )) × [(1−ηtβn ))(1+λexp(−ηtβn )) − (1− exp(−ηt β 1 ))(1+λexp(−ηt β 1 ))] n−2 4. The min and max order statistic of the TRD(η,λ, t) by substituting α= 0,β = 2 g(t1, tn) =n(n− 1)(2ηt1)exp(−ηt2 1)(1−λ+ 2λexp(−ηt2 1)) × (2ηtn)exp(−ηt2 n)(1−λ+ 2λexp(−ηt2 n))× [(1−ηt2 n))(1+λexp(−ηt2 n)) − (1− exp(−ηt2 1))(1+λexp(−ηt2 1))] n−2 5. The min and max order statistic of the T ED(η,λ, t) by substituting α = 0,β = 1 g(t1, tn) =n(n− 1)(η)exp(−ηt1)(1−λ+ 2λexp(−ηt1)) × (η)exp(−ηtn)(1−λ+ 2λexp(−ηtn))× [(1−ηtn))(1+λexp(−ηtn)) − (1− exp(−ηt1))(1+λexp(−ηt1))] n−2 M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 83 6. The min and max order statistic of the MW D(α,β ,η, t) by substituting λ= 0, g(t1, tn) =n(n− 1)(β2η2 t β−1 1 tβ−1 n )exp(−ηt β 1 −ηtβn ) [(1− exp(−ηtβn ))− (1− exp(−ηt β 1 ))] n−2 7. The min and max order statistic of the MRD(α,η, t) by substituting β = 2,λ= 0, g(t1, tn) =n(n− 1)(4η2t1 tn)exp(−ηt2 1 −ηt2 n) [(1− exp(−ηt2 n))− (1− exp(−ηt2 1))] n−2 8. The min and max order statistic of the M ED(α,η, t) by substituting β = 1,λ= 0, g(t1, tn) =n(n− 1)(η2)exp(−ηt1 −ηtn) [(1− exp(−ηtn))− (1− exp(−ηt1))] n−2 7. Maximum Likelihood Estimation Consider the random samples t1, t2, . . . tn consisting of n observations from the transmuted modified Weibull distribution T MW D(α,β ,η,λ, t) having probability density function. The likelihood function of equation (2) is given by L(t1, . . . tn,α,β ,η,λ, ) = ∏ (α+ βηtβ−1)exp(−αt −ηtβ )(1−λ+ 2λexp(−αt −ηtβ )) (25) By accumulation taking logarithm of equation (25), we find the log-likelihood function Ł = ln L , differentiating equation (26) with respect to α,β ,η and λ then equating it to zero, we obtain the estimating equations are L(t1, t2, . . . tn,α,β ,η,λ, ) = n ∑ i=0 ln(α+ βηt β−1 i )−α n ∑ i=0 t i −η n ∑ i=0 t β i + n ∑ i=0 ln(1−λ+ 2λexp(−αt i −ηt β i )) (26) ∂ Ł ∂ α = n ∑ i=0 1 (α+ βηt β−1 i ) − n ∑ i=0 t i − n ∑ i=0 2λt i exp(−αt i −ηt β i ) (1−λ+ 2λexp(−αt i −ηt β i )) (27) ∂ Ł ∂ β = n ∑ i=0 t β−1 i (1+ β ln(t i)) (α+ βηt β−1 i ) − n ∑ i=0 t β i ln(t i)− n ∑ i=0 2λexp(−αt i −ηt β i )t β i ln(t i) (1−λ+ 2λexp(−αt i −ηt β i )) = 0 (28) ∂ Ł ∂ η = n ∑ i=0 β t β−1 i (α+ βηt β−1 i ) − n ∑ i=0 t β i − n ∑ i=0 2λexp(−αt i −ηt β i )t β i (1−λ+ 2λexp(−αt i −ηt β i )) (29) M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 84 ∂ Ł ∂ λ = n ∑ i=0 2 exp(−αt i −ηt β i )− 1 (1−λ+ 2λexp(−αt i −ηt β i )) = 0 (30) By solving this nonlinear system of equations (27) -(30), these solutions will yield the ML estimators α̂, β̂ , η̂ and λ̂. For the four parameters transmuted modified Weibull distribution T MW D(α,β ,η,λ, t) pdf all the second order derivatives exist. Thus we have the inverse dispersion matrix is      α̂ β̂ η̂ λ̂      ∼ N           α β η λ      ,      V̂11 V̂12 V̂13 V̂14 V̂21 V̂22 V̂23 V̂24 V̂31 V̂32 V̂33 V̂34 V̂41 V̂42 V̂43 V̂44           V−1 = −E        V11 ... V14 ... ... ... V41 ... V44    = −E     ∂ 2Ł ∂ α2 ... ∂ 2Ł ∂ α∂ λ ... ... ... ∂ 2Ł ∂ α∂ λ ... ∂ 2Ł ∂ λ2         (31) Equation (31) is the variance covariance matrix of the T MW D(α,β ,η,λ, t) V11 = ∂ 2Ł ∂ α2 V12 = ∂ 2Ł ∂ α∂ β V22 = ∂ 2Ł ∂ β2 V13 = ∂ 2Ł ∂ α∂ η V33 = ∂ 2Ł ∂ η2 V14 = ∂ 2Ł ∂ α∂ λ V44 = ∂ 2Ł ∂ λ2 V23 = ∂ 2Ł ∂ β∂ η V24 = ∂ 2Ł ∂ β∂ λ V34 = ∂ 2Ł ∂ η∂ λ By solving this inverse dispersion matrix, these solutions will yield the asymptotic variance and co-variances of these ML estimators for α̂, β̂ , η̂ and λ̂. By using (31), approximately 100(1−α)% confidence intervals for α, β , η and λ can be determined as α̂± Z α 2 p V̂11 β̂ ± Z α 2 p V̂22 η̂± Z α 2 p V̂33 λ̂± Z α 2 p V̂44 Where Z α 2 is the upper αth percentile of the standard normal distribution. 8. Numerical Example In this section we provide a data analysis in order to assess the goodness-of-fit of a model with respect to a maximum flood levels data to see how the new model works in practice. The M. Shuaib, R. King / Eur. J. Pure Appl. Math, 6 (2013), 66-88 85 data have been obtained from Dumonceaux and Antle [3]. The analysis of least square estimates for the unknown parameters in these distributions namely: Transmuted Modified Weibull (TMW), Transmuted Weibull (TW), Transmuted Mod- ified exponential (TME), Modified Weibull (MW) and Weibull (W) distributions by using the method of least squares are defined. The LSE(s) of the unknown parameter(s), coefficient of determination (R2) and the corresponding Mean square error for transmuted modified Weibull families of distributions are given in Table 2. We have provided the parametric estimate of the cumulative distribution function and the fitted functions in Figure 7. It is clear that the transmuted modified Weibull (TMW) distribution provides better fit than the other distributions. In this analysis some estimated values are negative which is not good in the LSE. Another check is to compare the respective coefficients of determination for these regression lines. We have supporting evidence that the coefficient of determination of (TMW) is 0.984714, which is higher than the coefficient of determination of (TME), (TW), (MW) and (W) distributions. Hence the data point from the transmuted modified Weibull (TMW) has better relationship and hence this distribution is good model for life time data. Table 2: Estimated parameters of TMW, TME, TW, MW and W Distributions Parameters T MW T M E TW MW W α 0.035028 -0.45467 - 0.03065 - β 0.1 1 0.458922 0.05 0.496531 η 0.384726 0.494294 0.254649 0.219493 0.183961 λ -0.57901 0.380212 -0.32547 - - R2 0.984714 0.953394 0.978145 0.984598 0.977871 MSE 0.003663 0.01051 0.004929 0.003473 0.004713 Figure 7: Empirical CDF for the fitted models REFERENCES 86 9. Concluding Remarks In this paper we introduce a new generalization of the Weibull distribution called trans- muted modified Weibull distribution and presented its theoretical properties. The new dis- tribution is very flexible model that approaches to different life time distributions when its parameters are changed. From the instantaneous failure rate analysis it is observed that it has increasing and decreasing failure rate pattern for life time data. This model has the capability to provide consistent results from all estimation methods. References [1] G. R. Aryall and C. P. Tsokos. Transmuted Weibull distribution: A Generalization of the Weibull Probability Distribution. European Journal of Pure and Applied Mathematics, Vol. 4, No. 2, 89-102. 2011. [2] G. R. Aryall and C. P. Tsokos. On the transmuted extreme value distribution with ap- plications. Nonlinear Analysis: Theory, Methods and applications, Vol. 71, 1401-1407. 2009. [3] R. Dumonceaux and C. Antle. Discrimination between the log-normal and the Weibull distributions, Technometrics, 15(4), 923-926. 1973. [4] C.-C. Liu. A Comparison between the Weibull and Lognormal Models used to Analyze Reliability Data. PhD thesis University of Nottingham, UK. 1997. [5] G. Mudholkar, D. Srivastava, and G. Kollia. A generalization of the Weibull distribu- tion with application to the analysis of survival data. Journal of the American Statistical Association, 91(436):1575-1583, 1996. [6] M. M. Nassar and F. H. Eissa. On the Exponentiated Weibull Distribution. Communica- tions in Statistics - Theory and Methods, 32(7):1317-1336, 2003. [7] M. Pal, M. M. Ali, and J. Woo. On the exponentiated Weibull distribution. Statistica. (2):139-147, 2006. [8] H. Pham and C.-D. Lai. On Recent Generalizations of the Weibull Distribution. IEEE Transactions on Reliability, 56(3):454-458, 2007. [9] M. Z. Raqab. Inferences for generalized exponential distribution based on record Statis- tics. Journal of Statistical Planning and Inference. Vol. 104, 2, pp. 339-350. 2002. [10] A. M. Sarhan and M. Zaindin. Modified Weibull distribution, Applied Sciences, Vol.11, 2009, pp. 123-136. 2009. [11] M. Shakil and M. Ahsanullah. Review on Order Statistics and Record Values from Fα Distributions. Pakistan Journal of Statistics and Operation Research. Vol. VIII No.1, pp. 101-120. 2012. REFERENCES 87 [12] W. Shaw and I. Buckley. The alchemy of probability distributions: beyond Gram-Charlier expansions and a skew-kurtotic-normal distribution from a rank transmutation map. Research report, 2007. [13] M. Zaindin and A. M. Sarhan. Parameters Estimation of the Modified Weibull Distribu- tion, Applied Mathematical Sciences, Vol. 3, no. 11, 541-550. 2009. Appendix Proof of Theorem 2 µk = ∫ ∞ 0 tk f (α,β ,η,λ, t)d t By substituting (2) into the above relation we have µk = ∫ ∞ 0 tk(α+ βηtβ−1)exp(−αt −ηtβ )(1−λ+ 2λexp(−αt −ηtβ ))d t (A1) Case A: In this case α,β ,η > 0 and |λ| ≥ 1. The exponent quantity is E x p(−ηtβ ) E x p(−ηtβ ) = ∞ ∑ i=0 (−1)iηi(t)iβ i! (A2) Here equation (A1) takes the following form µk = ∞ ∑ i=0 (−1)iηi i! � (1−λ) � Γ(iβ + k+ 1) αiβ+k + βηΓ(β(i+ 1) + k) αβ(i+1)+k �� + 2λ ∞ ∑ i=0 (−1)i(2η)i i! �� αΓ(iβ + k+ 1) 2αiβ+k+1 + βηΓ(β(i + 1)+ k) 2αβ(i+1)+k �� (A3) Case B: In this case α= 0, β ,η > 0 and |λ| ≥ 1. µk = ∫ ∞ 0 tk(βηtβ−1)exp(−ηtβ )(1−λ+ 2λexp(−ηtβ ))d t By substituting w = ηt(β) then we get µk = η −k β Γ � 1+ k β � � (1−λ) +λ2 −k β � (A4) Case C: In this case α > 0, β = 0, η = 0 and |λ| ≥ 1. µk = ∫ ∞ 0 tkαexp(−αt)(1−λ+ 2λexp(−αt))d t µk = α −kΓ(1+ k)((1−λ) +λ2−k) (A5) REFERENCES 88 Proof of Theorem 3 Mx (t) = ∫ ∞ 0 et x f (α,β ,η,λ, t)d t By substituting (2) into the above relation we have Mx (t) = ∫ ∞ 0 et x (α+ βηtβ−1)exp(−αt −ηtβ )(1−λ+ 2λexp(−αt −ηtβ ))d t (A6) Case A: In this case α,β ,η > 0 and |λ| ≥ 1. The exponent quantity is E x p(−ηtβ ). Here equation (A6) takes the following form Mx (t) = ∞ ∑ i=0 (−1)iηi i! � (1−λ) � αΓ(iβ + 1) (α− t)iβ+1 + βηΓ(β(i+ 1)) (α− t)β(i+1) �� + 2λ ∞ ∑ i=0 (−1)i(2η)i i! �� αΓ(iβ + 1) (2α− t)iβ+1 + βηΓ(β(i+ 1) + k) (2α− t)β(i+1) �� (A7) Case B: In this case α= 0, β ,η > 0 and |λ| ≥ 1. Mx (t) = ∫ ∞ 0 et x (βηtβ−1)exp(−ηtβ )(1−λ+ 2λexp(−ηtβ ))d t By substituting w = ηt(β) then we get Mx (t) = ∞ ∑ i=0 t i i! η −k β Γ � 1+ i β � � (1−λ)+λ2 −i β � (A8)