Weighted Half Exponential Power Distribution and Associated Inference

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1 Applied Mathematical Sciences, Vol. 0, 206, no. 2, 9-08 HIKARI Ltd, Weighted Half Exponential Power Distribution and Associated Inference M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad Department of Statistics and Operations Research Faculty of Science, Kuwait University, Kuwait Copyright c 205 M. E. Ghitany, S. M. Aboukhamseen and E. A. S. Mohammad. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The weighted exponential distribution has been generalized by many authors in recent years. Here, we introduce a new generalization, called the weighted half exponential power distribution, that can be used to model negative or positive skewed data. Various structural properties of the distribution are derived. The method of maximum likelihood is used to estimate the parameters of the distribution. Finally, a real data application is presented to compare the goodness-of-fit of the various generalized weighted exponential distributions. Mathematics Subject Classification: 60E05, 62F0 Keywords: Weighted distributions, probability density, hazard rate, mean residual life, moments, order statistics, maximum likelihood Introduction Azzalini [3] proposed a new method for introducing a skewness parameter to the normal distribution based on a weighted function and obtained the so called skew-normal distribution. Many authors used Azzalini s idea to generate new skew-symmetric distributions, for example, the skew-cauchy distribution due Arnold & Beaver []. A similar approach to Azzalini was introduced by Gupta & Kundu [8] to introduce a skewness parameter to the exponential

2 92 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad distribution and obtained the weighted exponential (WE) distribution with probability density function (PDF): f W E (x) = α + α βe βx ( e βx ), x > 0, α, β > 0. () They showed that the WE distribution can be used to model positively skewed data and data that arise from hidden truncation models, see for example, Arnold & Beaver []. However, the WE distribution has serious itations by iting its coefficient of variation, skewness and kurtosis to the range ( 2, ), ( 2, 2) and (6, 9), respectively. These itations motivated many authors to propose generalizations of the WE distribution by adding an extra shape parameter γ > 0. To the best of our knowledge, the following are the PDFs of all the three-parameter generalizations of the WE distributions presented in the literature. Each of these generalized distributions reduces to the WE distribution when the extra shape parameter γ =. (i) Beta exponential (BE) distribution, Nadarajah & Kotz [2]: f BE (x) = αβ B(/α, 2γ) e βx ( e αβx ) 2γ, x > 0, (2) where B(a, b) = 0 xa ( x) b dx, a, b > 0, is the beta function. The BE distribution includes the generalized weighted distribution of Kharazmi et al. [0] as a special case when γ = (m + )/2, m =, 2,.... (ii) Weighted Weibull (WW) distribution, Shahbaz et al. [4]: f W W (x) = (α + )βγ α x γ e βxγ ( e αβx γ ), x > 0. (3) (iii) Weighted gamma (WG) distribution, Jain et al. [9]: f W G (x) = β γ [ (α + ) γ ] Γ(γ) xγ e βx ( e αβx), x > 0, (4) where Γ(a) = 0 x a e x dx, a > 0, is the gamma function. (vi) Weighted exponentiated exponential (WEE) distribution, Mahdavi []: f W EE (x) = βγ γb(α +, γ) e βx ( e βx) γ ( e αβx ), x > 0. (5) In this paper, we introduce a new generalized WE distribution, called the weighted half exponential power (WHEP) distribution with PDF f(x) = C e βxγ ( e αβx γ ), x > 0, (6)

3 Weighted half exponential power distribution and associated inference 93 where C = [(α + )β] /γ [(α + ) /γ ] Γ(/γ + ). (7) The proposed distribution can be interpreted as follows. Interpretation. Let X be a random variable (RV) having the half exponential power (HEP) distribution, introduced by Gui [6], with PDF f X (x) = β /γ Γ(/γ + ) e βxγ, x > 0, β, γ > 0, and X 2 be a RV having the Weibull distribution with PDF f X2 (x) = βγ x γ e βxγ, x > 0, β, γ > 0. If X and X 2 are independent, then the RV X = X {α /γ X > X 2 }, α > 0, follows the WHEP distribution, since f X (x) = P (X = x, X 2 < α /γ x) P (α /γ X > X 2 ) = f X (x) F X2 (α /γ x) P (α /γ X > X 2 ) = C e βxγ ( e αβxγ). This formulation of the WHEP distribution extends the approach considered by Azzalini [3] to the case of independent non-normal RVs. Interpretation 2. If U and V are correlated RVs with joint PDF f U,V (u, v) = β/γ+ Γ(/γ + ) vγ e β(+u)vγ, u, v > 0, β, γ > 0, then the RV X = V {U α}, α > 0, follows the WHEP distribution, since f X (x) = P (U α, V = x) P (U α) = α 0 f U,V (u, x) du P (U α) = C e βxγ ( e αβx γ ). This formulation of the WHEP distribution is a hidden truncation model. In Sections 2-4, we study the shapes of the density, hazard rate and mean residual life functions of the proposed WHEP distribution. The moments and associated measures are derived in Section 5. The iting distributions of the order statistics are presented in Section 6. The method of maximum likelihood is used to estimate the parameters of the WHEP distribution in Section 7. Finally, in Section 8, we fit the various generalized WE distributions to a real data set and compare their performances.

4 94 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad 2 Probability density function The following theorem explores the shape of the PDF (6). Theorem. The PDF f(x) of the W HEP (α, β, γ) distribution is unimodal for all α, β, γ > 0 with f(0) = f( ) = 0. Proof. The its of f(x) at x = 0 and x = are clear. The derivative of f(x) can be written as f (x) = (α + e αβxγ ) βγx γ e αβxγ f(x). [ ] Therefore, f(x) has a unique maximum at the mode x m = ln(α + ) γ. αβ When γ =, Theorem states that the PDF f(x) of the WE distribution is unimodal for all α, β > 0 with f(0) = f( ) = 0. Moreover, in this case, the mode is x m = ln(α + ). αβ Figure shows the PDF of the WHEP distribution for selected values of the parameters. f x x Figure : PDF of the WHEP distribution with α = 2, β =, γ = 0.8 (solid), (dashed), 2(dotted).

5 Weighted half exponential power distribution and associated inference 95 3 Hazard rate function The survival function of the WHEP distribution is given by S(x) = x f(y) dy = (α + )/γ Γ(/γ, βx γ ) Γ(/γ, (α + )βx γ ), (8) [(α + ) /γ ] Γ(/γ) where Γ(a, z) = y a e y dy, a, z > 0, is the upper incomplete gamma z function. Now, using (6) and (8), the hazard rate function (HRF) of the WHEP distribution is given by h(x) = f(x) S(x) = γ[(α + )β] /γ e βxγ ( e αβxγ ) (α + ) /γ Γ(/γ, βx γ ) Γ(/γ, (α + )βx γ ). (9) When γ =, since Γ(, z) = e z, z > 0, we obtain the HRF of the WE distribution: h(x) = (α + )β e αβx, x > 0, α, β > 0. α + e αβx The following lemma provides simple sufficient conditions for the shape of the HRF of any PDF on (0, ). Lemma. (Glaser [5]) Let f(x), 0 < x <, be a twice differentiable PDF of a continuous random variable X. Define η(x) = (ln f(x)). (a) If η(x) is increasing, then h(x) is increasing. (b) If η(x) is unimodal and f(0) = 0, then h(x) is unimodal. The following theorem explores the shape of the HRF (9). Theorem 2. The HRF h(x) of the W HEP (α, β, γ) distribution is unimodal (increasing) if 0 < γ < (γ ) for all α, β > 0 with h(0) = 0 and 0, if 0 < γ <, h( ) = β, if γ =,, if γ >. Proof. For all α, β, γ > 0, we have h(0) = f(0) = 0, since S(0) =. Also, for all α, β, γ > 0, ) η(x) = (ln f(x)) = βγx ( γ α. e αβxγ

6 96 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad Now, using L Hospital rule, since f( ) = S( ) = 0, we have h( ) = f (x) x f(x) = η(x) = βγ x x xγ. Clearly, h( ) takes the values 0, β,, if 0 < γ <, γ =, γ >, respectively. The derivative of η(x) can be written as where η (x) = g(e αβxγ ) βγx γ 2 (e αβxγ ) 2, g(z) = (γ )z 2 α(γ )z + αγ(z + ) ln(z + ), z > 0. For γ, we have g(z) αγ[(z + ) ln(z + ) z] 0. Therefore, η (x) 0, i.e. is η(x) is increasing. Now, using part (a) of Lemma, h(x) is increasing. Now we consider the case 0 < γ <. Here, we have g (z) = 2(γ )z + α + αγ ln(z + ), and g (z) = 2(γ ) + αγ z +. The function g (z) is decreasing and negative (changes sign from positive to negative) when 0 < γ 2 < ( 2 < γ < ). α+2 α+2 Since g (0) = α and g ( ) =, it follows that g (z) changes sign from positive to negative. Finally, since g(0) = 0 and g( ) =, it follows that g(z) changes sign from positive to negative. This implies that η(z) is unimodal. Now, using part (b) of Lemma, h(x) is unimodal. When γ =, Theorem 2 states that the HRF h(x) of the WE distribution is increasing for all α, β > 0 with h(0) = 0 and h( ) = β. Figure 2 shows the HRF of the WHEP distribution for selected values of the parameters.

7 Weighted half exponential power distribution and associated inference 97 h x x Figure 2: HRF of the WHEP distribution with α = 2, β =, γ = 0.8 (solid), (dashed), 2(dotted). 4 Mean residual life function The mean residual life function (MRLF) of the WHEP distribution is given by µ(x) = = S(x) x yf(y) dy x (α + ) 2/γ Γ(2/γ, βx γ ) Γ(2/γ, (α + )βx γ ) [(α + )β] /γ [(α + ) /γ Γ(/γ, βx γ ) Γ(/γ, (α + )βx γ )] x. (0) When γ =, since Γ(, z) = e z and Γ(2, z) = ( + z)e z, z > 0, we obtain the MRLF of the WE distribution: µ(x) = (α + )2 ( + βx) [ + (α + )βx]e αβx (α + )β (α + e αβx ) x. The following lemmas provide simple sufficient conditions for the shape of a MRLF of any PDF on (0, ). Lemma 2. (Bryson & Siddique [4]) Let X be a non-negative continuous random variable with HRF h(x) and MRLF µ(x). If h(x) is increasing, then µ(x) is decreasing. Lemma 3. (Gupta & Akman [7]) Let X be a non-negative continuous random variable with HRF h(x) and MRLF µ(x). If h(x) is unimodal and h(0)µ(0) <, then µ(x) is anti-unimodal.

8 98 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad The following theorem explores the shape of the MRLF (0). Theorem 3. The MRLF µ(x) of the W HEP (α, β, γ) distribution is antiunimodal (decreasing) if 0 < γ < (γ ) for all α, β > 0 with µ(0) = [ (α + ) 2/γ ] Γ(2/γ) [(α + )β] /γ [(α + ) /γ ] Γ(/γ), and, if 0 < γ <, µ( ) = /β, if γ =, 0, if γ >. Proof. Using Γ(a, 0) = Γ(a), a > 0, in equation (0), the it µ(0) follows. Also, using L Hospital rule, the it µ(0) follows since µ( ) = x S(x) f(x) = h( ). For 0 < γ <, the HRF h(x) is unimodal and h(0)µ(0) = 0. Therefore, using Lemma 3, the MRLF µ(x) is anti-unimodal. For γ, the HRF h(x) is increasing. Therefore, using Lemma 2, the MRLF µ(x) is decreasing. Figure 3 shows the MRLF of the WHEP distribution for selected values of the parameters. 2 Μ x x Figure 3: MRLF of the WHEP distribution with α = 2, β =, γ = 0.8 (solid), (dashed), 2(dotted).

9 Weighted half exponential power distribution and associated inference 99 5 Moments and associated measures The rth raw moment of the WHEP distribution is given by µ r = = 0 x r f(x) dx [ (α + ) (r+)/γ ] Γ((r + )/γ). r =, 2,.... () [(α + )β] r/γ [(α + ) /γ ] Γ(/γ) When γ =, we obtain the rth raw moment of the WE distribution: µ r = [(α + )r+ ] r! α[(α + )β] r. r =, 2,.... Therefore, the mean and variance of the WHEP distribution, respectively, are µ = σ 2 = [ (α + ) 2/γ ] Γ(2/γ) [(α + )β] /γ [(α + ) /γ ] Γ(/γ), [ (α + ) 3/γ ] [ (α + ) /γ ] Γ(3/γ) Γ(/γ) [ (α + ) 2/γ ] 2 Γ 2 (2/γ) [(α + )β] 2/γ [(α + ) /γ ] 2. Γ 2 (/γ) The coefficient of variation, skewness and kurtosis measures of the WHEP distribution can be obtained from the expressions CV = σ µ, (2) CS = CK = E (X µ)3 σ 3 = µ 3 3µ 2µ + 2µ 3 σ 3, (3) E (X µ)4 σ 4 = µ 4 4µ 3µ + 6µ 2µ 2 3µ 4 σ 4, (4) upon substituting for the raw moments. Figure 4 shows the coefficient of variation, skewness and kurtosis of the WHEP distribution as a function of of γ for α = 0.5, 2, 0. This figure shows that each of these measures of the WHEP distribution can be smaller/larger than that of the WE distribution. Moreover, we observe that the skewness can be negative which is not possible for the WE distribution. Negative skewness of the WHEP distribution can be useful for modeling left skewed data.

10 00 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad CV Γ (a) coefficient of variation 3 2 CS Γ (b) coefficient of skewness 2 9 CK Γ (c) coefficient of kurtosis Figure 4: The coefficient of variation, skewness and kurtosis of the WHEP distribution, as a function of γ, with α = 0.5(solid), 2(dashed), 0(dotted).

11 Weighted half exponential power distribution and associated inference 0 6 Limiting distributions of order statistics The following theorem provides the iting distributions of the minimum and maximum of a random sample of size n from the WHEP distribution. Theorem 4. Let X :n and X n:n be the minimum and maximum of a random sample X, X 2,..., X n from W HEP (α, β, γ), respectively. Then: where (a) (b) n n { X:n a n P b n { Xn:n a n P b n } x } t = e xγ+, x > 0, = exp( e t ), < t <, a n = 0, b n = F (/n), a n = F ( /n), b n = with F (c) is the inverse function of F (c) = S(c), c > 0. nf(a n ), Proof. For the W HEP (α, β, γ), we have, by using L Hospital rule, ɛ 0+ F (ɛx) F (ɛ) = ɛ 0+ x f(ɛx) f(ɛ) = x ɛ 0+ e αβ(ɛx)γ e αβɛγ = x γ+. Therefore, by Theorem (ii) of Arnold et al. [2], the minimal domain of attraction of the W HEP (α, β, γ) distribution is the standard Weibull distribution with shape parameter γ +, proving part (a). Since ( ) d S(x) η(x) = + dx h(x) f(x) = + βγ C ) S(x) x ( γ α e αβxγ, e βxγ ( e αβxγ ) it follows that x d dx ( h(x) ) = + βγ C x S(x) x γ e βxγ = + βγ C y S(y /γ ) y /γ e βy. Now, by using L Hospital rule, we have x d ( ) dx h(x) = + βγ C y = + y ( e αβy ) = 0. f(y /γ ) γ y/γ [(/γ 2)y β]y /γ e βy

12 02 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad Therefore, by Theorem of of Arnold et al. [2], the maximal domain of attraction of the W HEP (α, β, γ) distribution is the standard Gumbel distribution, proving part (b). Now, we use Theorem 4 to find the iting distribution of any order statistic. Theorem 5. Let X :n X 2:n... X n:n be the order statistics of a random sample X, X 2,..., X n from W HEP (α, β, γ). Then, for any fixed i, (a) (b) n n { Xi:n a n P b n { Xn i+:n a n P b n } i x = } t = r=0 i r=0 e xγ+ x(γ+)r, x > 0, r! exp( e t ) e rt, < t <. r! Proof. The theorem follows from Equations (8.4.2) and (8.4.3) of Arnold et al. [2]. 7 Maximum likelihood estimation Given a random sample x, x 2,..., x n from the WHEP distribution with PDF (6), the log-likelihood function is given by { } l(θ) = n γ ln[(α + )β] ln[(α + )/γ ] ln Γ(/γ + ) n n ( ) β x γ i + ln e αβxγ i, (5) where θ (α, β, γ). The maximum likelihood estimate (MLE) θ of θ is the solution of the system of non-linear equations l α = n (α + )γ l β = n n βγ l γ = n γ 2 { x γ i + n } + (α + ) /γ { ln[(α + )β] + β αx γ i e αβxγ i = 0, n βx γ i e αβxγ i = 0, } ln(α + ) + Ψ(/γ + ) (α + ) /γ n n x γ i ln(x αβx γ i i) + ln(x i) e αβxγ i = 0,

13 Weighted half exponential power distribution and associated inference 03 where Ψ(z) = (ln Γ(z)) is the digamma function. The observed information matrix is given by J J 2 J 3 2 l J(θ) = J 2 J 22 J 23 = 2 l α β J 3 J 23 J 33 2 l α 2 α β 2 l β 2 2 l α γ 2 l β γ 2 l α γ 2 l β γ 2 l γ 2, where the diagonal elements are given by J = n[γ (γ + )(α + )/γ ] n (α + ) 2 γ 2 [(α + ) /γ ] + β 2 x 2γ i e αβxγ i 2 (e αβxγ i ), 2 J 22 = n n β 2 γ + α 2 x 2γ i e αβxγ i (e αβxγ i ), 2 J 33 = 2n γ 3 { } ln(α + ) ln[(α + )β] + + Ψ(/γ + ) (α + ) /γ + n { (α + ) /γ ln 2 } (α + ) Ψ (/γ + ) γ 4 [ (α + ) /γ ] 2 + n x γ i ln2 (x i ) + n and the off-diagonal elements are given by J 2 = n [ + (αβx γ i )eαβxγ i ] x γ i (e αβxγ i ) 2, J 3 = n{γ + [ln(α + ) γ](α + )/γ } (α + )γ 3 [(α + ) /γ ] 2 + J 23 = n n βγ + x γ 2 i ln(x i) + n αβ[ + (αβx γ i )eαβxγ i ] x γ i ln2 (x i ) (e αβxγ i ) 2, n β[ + (αβx γ i )eαβxγ i ] x γ i ln(x i) (e αβxγ i ) 2, α[ + (αβx γ i )eαβxγ i ] x γ i ln(x i) (e αβxγ i ) 2, where Ψ (z) is the trigamma function. Under conditions that are fulfilled for parameters in the interior of the parameter space, we have that ( θ θ) T D N 3 (0, J ( θ)), where T is the transpose of a vector, D means approximately distributed and N 3 (.,.) denotes the trivariate normal distribution. The asymptotic trivariate normal distribution can be used to construct approximate confidence intervals for the individual parameters.

14 04 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad 8 Application The following data represent fracture toughness of Alumina (Al 2 O 3 ) (in the units of MPa m /2 ), Nadarajah & Kotz [3]: 5.5, 5, 4.9, 6.4, 5., 5.2, 5.2, 5, 4.7, 4, 4.5, 4.2, 4., 4.56, 5.0, 4.7, 3.3, 3.2, 2.68, 2.77, 2.7, 2.36, 4.38, 5.73, 4.35, 6.8,.9, 2.66, 2.6,.68, 2.04, 2.08, 2.3, 3.8, 3.73, 3.7, 3.28, 3.9, 4, 3.8, 4., 3.9, 4.05, 4, 3.95, 4, 4.5, 4.5, 4.2, 4.55, 4.65, 4., 4.25, 4.3, 4.5, 4.7, 5.5, 4.3, 4.5, 4.9, 5, 5.35, 5.5, 5.25, 5.8, 5.85, 5.9, 5.75, 6.25, 6.05, 5.9, 3.6, 4., 4.5, 5.3, 4.85, 5.3, 5.45, 5., 5.3, 5.2, 5.3, 5.25, 4.75, 4.5, 4.2, 4, 4.5, 4.25, 4.3, 3.75, 3.95, 3.5, 4.3, 5.4, 5, 2., 4.6, 3.2, 2.5, 4., 3.5, 3.2, 3.3, 4.6, 4.3, 4.3, 4.5, 5.5, 4.6, 4.9, 4.3, 3, 3.4, 3.7, 4.4, 4.9, 4.9, 5. Figure 5 shows the empirical scaled total time on test (TTT)-transform for the considered data set where T (r/n) = r x i:n + (n r)x r:n n x, r =, 2,..., n. (6) i:n Inspection of Figure 5 shows concave behavior above the diagonal line, indicating that the considered data set is drawn from a distribution with an increasing hazard rate function T r n r n r,2,...,n Figure 5: Empirical scaled TTT plot for Alumina data. For the given data set, the sample coefficient of variation, skewness and kurtosis are , and , respectively. Each of these sample measures falls outside the range permitted by the WE distribution. This suggests that some available generalized WE distributions such as BE, WW, WG. WEE, and WHEP may be more suitable for such data.

15 Weighted half exponential power distribution and associated inference 05 For the various fitted models, Table shows the MLEs of the parameters, log-likelihood, Anderson-Darling goodness-of-fit test and sum of squares of the residuals: n [ ] 2 SSR = F (xi:n ) F n (x i:n ), where F (x) is the MLE of the distribution function F (x) = S(x) and F n (x) = n (number of x is x) is the empirical distribution function. Table : Summary of fitted models for Alumina data. α β γ Model (S.E.) (S.E.) (S.E.) l( α, β. γ) AD p-value SSR WE (0.252) (0.066) BE (0.094) (.350) (0.996) WW (0.65) (0.007) (0.23) WG (0.353) (2.34) (3.953) WEE (0.94) (0.23) (34.440) WHEP (0.86) (0.00) (0.353) Table shows the following points: (i) Both the WE and WEE models are not suitable for the Alumina fracture toughness data. This is also supported by inspection of the Probability- Probability (P P) plots in Figure 6(a) and Figure 6(e), respectively. (ii) Both the BE and WG models gave the same estimated log-likelihood function, test statistic, p value of the Anderson-Darling test, and SSR. The reason for this is that f BE (x) and f W G (x) are almost identical over the range of the data, i.e. x (.68, 6.8). (iii) The WHEP model, is the most suitable model for the Alumina fracture toughness data in terms of the highest value of the estimated log-likelihood function, the smallest test statistic, highest p value of the Anderson-Darling test, and smallest SSR. This is also supported by inspection of the P P plots in Figure 6.

16 06 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad P P plot P P plot Empirical probabilities Empirical probabilities Theoretical probabilities (a) WE distribution Theoretical probabilities (b) BE distribution P P plot P P plot Empirical probabilities Empirical probabilities Theoretical probabilities (c) WW distribution Theoretical probabilities (d) WG distribution P P plot P P plot Empirical probabilities Empirical probabilities Theoretical probabilities (e) WEE distribution Theoretical probabilities (f) WHEP distribution Figure 6: P-P plots of various distributions for Alumina data.

17 Weighted half exponential power distribution and associated inference 07 References [] B.C. Arnold, R.J. Beaver, The skew Cauchy distribution, Statistics & Probability Letteters, 49 (2000), [2] B.C. Arnold, N. Balakrishnan, H.N. Nagaraja, A First Course in Order Statistics, John Wiley & Sons, 992. [3] A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 2 (985), [4] M.C. Bryson, M.M. Siddique, Some criteria for aging, Journal of the American Statistical Association, 64 (969), [5] R.E. Glaser, Bathtub and related failure rate characterizations, Journal of the American Statistical Association, 75 (980), [6] W. Gui, Statistical inferences and applications of the half exponential power distribution, Journal of Quality and Reliability Engineering, 203 (203), Article ID 29473, 9. [7] Akman H. Olcay, Mean residual life function for certain types of nonmonotonic ageing, Communications in Statistics-Stochastic Models, (995), [8] R.D. Gupta, D. Kundu, A new class of weighted exponential distribution, Statistics, 43 (2009), [9] K. Jain, N. Singla, R.D. Gupta, A weighted version of gamma distribution, Discussiones Mathematicae, Probability and Statistics, 34 (204), [0] O. Kharazmi, A. Mahdavi, M. Fathizadeh, Generalized weighted exponential distribution, Communications in Statistics: Simulation and Computation, 44 (205), no. 6, [] A. Mahdavi, Two weighted distributions generated by exponential distribution, Journal of Mathematical Extension, 9 (205), no. 3, 2. [2] S. Nadarajah, S. Kotz, The beta exponential distribution, Reliability Engineering and System Safety, 9 (2006),

18 08 M. E. Ghitany, S. M. Aboukhamseen, E. A. S. Mohammad [3] S. Nadarajah, S. Kotz, Strength modeling using Weibull distributions, Journal of Mechanical Science and Technology, 22 (2008), [4] S. Shahbaz, M. Q. Shahbaz, N. S. Butt, A class of weighted Weibull distribution, Pakistan Journal of Statistics and Operations Research, 6 (200), no., Received: November 26, 205; Published: January 7, 206