A Class of Weighted Weibull Distributions and Its Properties. Mervat Mahdy Ramadan [a],*

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1 Studies in Mathematical Sciences Vol. 6, No. 1, 2013, pp. [35 45] DOI: /j.sms ISSN [Print] ISSN [Online] A Class of Weighted Weibull Distributions Its Properties Mervat Mahdy Ramadan [a],* [a] Department of Statistics, Mathematics Insurance, College of Commerce, Benha University, Egypt. * Corresponding author. Address: Department of Statistics, Mathematics Insurance College of Commerce, Benha University, Egypt; drmervat.mahdy@fcom.bu.edu.eg Received: December 18, 2012/ Accepted: February 13, 2013/ Published: February 28, 2013 Abstract: The Weibull distribution is a well known common distribution. In this article, a skewness parameter to a Weibull distribution is introduced using an idea of Azzalini, which creates a new class of weighted Weibull distributions. This new distribution has a probability density function with skewness representing a general case of weighted probability density function of the Extreme value distribution, the Rayleigh distribution Exponential distribution. Different properties of this new distribution are discussed the inference of the old parameters the skewness parameter is studied. Key words: Hazard function; Moment generating function; Truncated distribution; Weighted gamma distribution; Selection model; Constrained normal mean vector; Probabilistic representation Mahdy, M. 2013). A Class of Weighted Weibull Distributions Its Properties. S- tudies in Mathematical Sciences, 6 1), Available from index.php/sms/article/view/j.sms DOI: /j.sms INTRODUCTION The usefulness applications of parametric distributions including the Weibull distribution, the Raleigh distribution the extreme value distribution in various areas including reliability, renewal theory, branching processes can be seen in recent papers by several authors including Oluyede [19] in references therein. 35

2 A Class of Weighted Weibull Distributions Its Properties Weighted distributions are used to adjust the probabilities of the events as observed recorded. Patil Rao [20] discussed how, for example, truncated distributions damaged observations can give rise to weighted distributions. Azzalini [4] was the first to introduce the skew-normal distribution to incorporate a shape/skewness parameter to a normal distribution depending on a weighted function denoted by F αx) where α is a skewness parameter. Since then extensive work has been done to introduce a skewness parameter to some symmetric distributions, for instance skew-t, skew-cauchy, skew-laplace, skew-logistic. In general, skew-symmetric distributions have been defined several of their properties inference procedures have been discussed, see for example, Arnold Beaver [3], Gupta Kundu [14] the recent monograph by Genton [11]. Arnold Beaver [2] provided a nice interpretation of Azzalini s skew-normal distribution as a hidden truncation model, although the same interpretation may not be true for other skewed distributions. Actually, Azzalini s method has been used extensively for several symmetric distributions non-symmetric distributions. In this article, it will be observed that if we apply Azzalini s method to the weibull distribution, then it produces a new class of weighted weibull W W λ,, α) distributions with an additional parameter called sensitive skewness parameter. From now we denote a member of this new class of weighted distributions as W W λ,, α) distribution. The sensitive skewness parameter governs essentially the shape of the probability density function of the W W λ,, α) distribution. The main aim of this article is to introduce this distribution study its properties. It will be observed that although the distribution has been obtained as a W W λ,, α) distribution, it has several other special cases can be considered as a hidden truncation model. It can be seen that the W W λ,, α) distribution function has a compact form all its moments can be computed explicitly, for instance mean, variance, skewness, kurtosis, coefficient of variation, hazard function HF), mean residual lifetime. The rest of the article is organized as follows. In Section 2, we provide the definition of W W λ,, α) distribution. Different properties are discussed in Section 3, also we introduce special cases by defining a weighted the Extreme value distribution, a weighted Rayleigh distribution a weighted Exponential distribution in Section 4. The inference of old parameters sensitive skewness parameter is studied in Section DEFINITION OF WEIGHTED WEIBULL DISTRIBUTION Let X be a non-negative rom variable with an absolutely continuous distribution function F X probability density function f X. Let l X = inf{x R : F X x) > 0}, u X = sup{x R : F X x) < 1}, S X = {x : l X x u X }. Let w be a non-negative function defined on the real line. Suppose that the realization x of X will be recorded with probability proportional to wtx)). Then 36

3 Mahdy, M./Studies in Mathematical Sciences, 6 1), 2013 the recorded tx) is not an observation on X but it is an observation on the so-called weighted rom variable X {w} with density function given by: f X {w} x) = wtx))f Xx) E[wtX))] for < x <, where 0 < E [wtx))] <. The rom variable X {w} is called the weighted version of X, its distribution relative to X is called the weighted distribution of X with weight function w. Let l X 0 wtx)) = x, x S X, for some positive integer. Then the corresponding weighted distribution is called a size biased distribution. If this distribution is of order one, then it is simply called a length-biased distribution see Blumenthal [6], Gupta Kirmani [12], Mahfoud Patil [16]). Now, the new class of weighted distributions is defined by the corresponding density function: with f X {α} x) = F Xαx)f X x), for x > 0, 1) E [F αx)] wtx)) = F αx). According to 1) the new class of weighted Weibull distributions can be derived as follows. Let X be distributed according to a Weibull distribution with parameters λ, density function as follow f X {λ,} = λx 1 e λx, 2) where λ is a scale parameter, is a shape parameter. The distribution function of X {, λ} is F X {λ,} x) = 1 e λx. Hence, we get E [ F X {λ,} αx) ] = Inserting into 1) yields F X {λ,} αx) = 1 e λαx), x F X {λ,} αx)f X {λ,} x)dx = α, ) λ )x 1 e λx 1 e λxα) f X {λ,,α} x) = α, for x > 0, 3) f X {λ,,α} x) = 0 otherwise. The density function 3) is referred to as W W λ,, α). Suppose X 1 X 2 are two i.i.d. rom variables with probability density function f Y distribution function F Y. Then for any α > 0, = 2, consider the new rom variable X with X = X 1 given that αx 1 > X 2. Azzalini [4] obtained the weighted skew-normal distribution from two i.i.d. normal distributions Mahdy [15] obtained the weighted gamma distribution from two i.i.d. gamma distributions as follows f X {λ,2,α} x) = 1 P X1,X 2 {x 1, x 2 ) : αx 1 > x 2 }) f Y x)f Y αx). 4) 37

4 A Class of Weighted Weibull Distributions Its Properties Now, 1) can be obtained explicitly from 4) by inserting P X1,X 2 {x 1, x 2 ) : αx 1 > x 2 }) = f X {λ,2} x) = 2λxe λx2, F X {λ,2} x) = 1 e λx2, αx f X1,X 2 x 1, x 2 )dx 2 dx 1 = α SOME PROPERTIES OF THE W W λ,, α) MODEL In this section we study the W W λ,, α) distribution. Without loss of generality, the density function of the W W λ,, α) distribution is provided by 3). The graphs of the density function of W W for different values of α are displayed in Figure 1. Figure 1 Weighted Weibull Densities It is easy to see that as α increases, the skewness of the distribution increases. If X is a rom variable with probability density function W W λ,, α), then 38

5 Mahdy, M./Studies in Mathematical Sciences, 6 1), 2013 the kth moment of X {λ,, α} is given by E W W λ,,α) X k ) = λ k/ ) ) k + α Γ 1 ) ) k+)/. 5) From 5),the first moment of W W λ,, α) is obtained: E W W λ,,α) X) = λ 1/ ) ) 1 + α Γ 1 ) ) 1+)/. where, Similarly, the variance is obtained as i).a 1 = E W W λ,,α) X 2 ) = λ 2/ α V W W λ,,α) x) = a 1 a 2, ) Γ ) ) ) 2+)/, [ λ ii).a 2 = EW 2 1/ W λ,,α) X) = ) ) 1 + α Γ 1 ) )] 2 1+)/. Moreover, we get E W W λ,,α) X 3 ) = λ 3/ α ) Γ ) ) ) 3+)/. Some other measures, like the coefficient of variation CV W W λ,,α) of W W λ,, α) the skewness coefficient τ W W λ,,α) can also be easily obtained in explicit forms: CV W W λ,,α) x) = ) λ 1/ a 1 a 2 / ) ) 1 + α Γ 1 ) ) 1/, τ W W λ,,α) = s 1 s 2 a 1 a 2 ) 3/2, where s 1 = λ 3/ ) ) 3 + α Γ 1 ) ) 3+)/ 3a 1 a 2 ) λ 1/ ) ) 1 + α Γ 1 ) ) 1+)/ ; λ 1/ s 2 = ) ) 1 + α Γ 1 ) )) 3 1+)/. Next, we provide the distribution function, survival function, hazard function, reversed hazard function, mean residual life function for the W W λ,, α) distribution: 39

6 A Class of Weighted Weibull Distributions Its Properties i). The distribution function F W W λ,,α) of W W λ,, α) is given by F W W λ,,α) x) = 1 [ 1 + α ) ) α 1 e λx + e λx 1+α ) 1]. ii). The survival function F W W λ,,α) of W W λ,, α) is given by F W W λ,,α) x) = 1 1 [ 1 + α ) ) ] α 1 e λx + e λx 1+α ) 1. 6) iii). The hazard function h W W λ,,α) of W W λ,, α) is given by ) λx 1 exp λx ) [ 1 exp λxα) )] h W W λ,,α) = [ ) e λx exp λx )) a ]. iv). The reversed hazard function r W W λ,,α) of W W λ,, α) is given by λ ) ) x 1 e λx 1 e λxα) r W W λ,,α) = ) 1 e ) λx + e λx 1+α ) 1. v). The mean reversed residual function m W W λ,,α) of W W λ,, α) is given by x 0 m W W λ,,α) x) = F W W λ,,α)u)du, F W W λ,,α) x) which equals to λx 1 xα + ) e λx e λx 1+α ) ) m W W λ,,α) x) = λx 1 ) 1 e ) λx λx 1 1 e ). λx 1+α ) It can be shown that m W W λ,,α) is an increasing function of x. Table 1 contains the values of survival function 6). Looking at this table we can see that the survival probability of the distribution decreasing with increase in the value of α for a holding x λ at a fixed level. Further, from the table we can see that; for fixed α, λ ; the survival probability decreases with increase in x. 4. SPECIAL CASES 4.1. The Weighted Extreme Value Distribution In many fields of modern science, engineering insurance, extreme value distribution is well established see e. g. Embrechts et al. [10], Reiss Thomas [21]). Recently, more more research has been undertaken to analyze the extreme variations that financial markets are subject to, mostly because of currency crises, stock market crashes large credit defaults. The tail behavior of financial series has, among others, been discussed in McNeil Frey [18], Coles [7], Beirlant et al. [5], Mira [17], Cooley et al. [8] Andjelic et al. [1]. An interesting discussion about the potential of extreme value theory in risk management is given in Diebold et al. [9]. 40

7 Mahdy, M./Studies in Mathematical Sciences, 6 1), 2013 Table 1 Survival Function of Weighted Weibull Distribution for λ = 1; = 1 x α = 1 α = 1.2 α = 1.4 α = 1.6 α = 1.8 α = 2 α = Let rom variable X has a Weibull distribution in 2), then the weighted probability density function of Y = log Xλ 1/) is given by ) f Y {,α} y) = e y e e y 1 e α e y), α which is weighted extreme value distribution The Weighted Rayleigh Distribution The Rayleigh distribution is an important distribution in statistics operations research. It is applied in several areas such as health, agriculture, biology, other sciences. Let rom variable Y has the weighted Rayleigh distribution W Rσ, α), then Y has the weighted Weibull distribution W W 1/σ 2) 2, 2, α ). Therefore, if w tx)) = F αx), the weighted Rayleigh distribution has an probability density function distribution function are given by f Y { 1/σ 2) 2,2,α } y) = 2) / σ 2 α 2)) y exp 0.5y/σ) 2) 1 exp 0.5yα/σ) 2)), F W R 1/σ 2) ) 2,2,α y) = 2) 1 exp 0.5y/σ) 2)) ) + exp 0.5y/σ) 2 2 ) 1. 41

8 A Class of Weighted Weibull Distributions Its Properties 4.3. The Weighted Exponential Distribution The second distribution can be obtained following Gupta Kundu [13] who introduced a new weighted exponential distribution but not fixed λ. Suppose = 1 in 2), then we get the density function of the weighted exponential distribution of the rom variable Y {λ, α} as f Y {λ,α} y) = λ ) exp λy) 1 exp λyα)) /α, for y > 0, f Y {λ,α} otherwise also, the distribution function can be written as F W Eλ,α) y) = 1 [) 1 exp λy)) + exp λy )) 1]. α 5. PARAMETER ESTIMATION This section is devoted to the estimation of the unknown parameter α in case = 2. For this case, the maximum likelihood the moment estimator is derived their asymptotic distributions are given Maximum Likelihood Estimates The weighted Weibull distribution is obtained by means of 3) in case = 2. We can write it as follows: f X {λ,α} x) = 2λ 2) x exp λx 2) 1 exp λxα) 2)) /α 2, for x > 0. The likelihood function based on the observed sample {x 1, x 2,..., x n } is L x 1, x 2,..., x n λ, α) = 2 n λ n 2) n Π n x i exp λ x i 2 ) Π n 1 exp λx i α) 2)) /α 2n. From 7) we obtain the log-likelihood function l x 1, x 2,..., x n λ, α): 7) l x 1, x 2,..., x n λ, α) = n ln 2 + n ln λ + n ln 2) + + lnx i λ [ ln 1 e λxiα)2] 2n ln α. x 2 i 8) For determining the maximum likelihood estimators MLE) of λ α, the expression 8) must be maximized with respect to λ α. Thus, firstly, the MLE of λ is obtained as a solution of the following fixed-point type equation: ) g ˆλ = ˆλ, 9) 42

9 Mahdy, M./Studies in Mathematical Sciences, 6 1), 2013 where ) g ˆλ = C 1 + ˆλ C 2, C 3 C 1 = n, C 2 = x 2 i x i α) 2 exp λx i α) 2), C 3 = x 2 i [ ln 1 exp λx i α) 2)]. The solution of 9) can be obtained by a simple iterative procedure. Suppose we start with an initial guess ˆλ 0, then the next iteration ˆλ 1 can be obtained as ˆλ 1 = gˆλ 0 ), similarly, ˆλ 2 = gˆλ 1 ), so on. Finally the iterative procedure should be stopped when ˆλ i ˆλ i+1 < ɛ, where ɛ is a preassigned tolerance value. Secondly, the MLE of α is obtained as a solution of the following fixed-point type equation: g ˆα) = ˆα, 10) where g ˆα) = S 1 S 2, ) 1 + ˆα 2 S 1 = ) 1 + ˆα 2 2λˆαx i ) 2 exp λx i ˆα) 2) S 2 = 2n n ln 1 exp λx i ˆα) 2)). The solution of 10) can be obtained by a simple iterative procedure. Suppose we start with an initial guess ˆα 0, then the next iteration ˆα 1 can be obtained as ˆα 1 = gˆα 0 ), similarly, ˆα 2 = gˆα 1 ), so on. Finally the iterative procedure should be stopped when ˆα i ˆα i+1 < ɛ, where ɛ is a preassigned tolerance value The Moment Estimators Next, we discuss the moment estimators of λ α when = 2. If m 2 denotes the second non-central moment, then by equating the second moment, we obtain the moment estimator of α as 0 = λ 1 α 2 2 ) 1 2) 2 ) m 2. Thus, the moment estimators of α can be obtained as a solution of the following fixed-point type equation: g ˆα) = ˆα, 43 ˆα,

10 A Class of Weighted Weibull Distributions Its Properties where λ 1 g ˆα) = 1 + ˆα 2 ) m ˆα 2) ) ) 1/2 2, This solution can be obtained by a simple iterative procedure, analogously as in the case of the solution of 10). Also, we can be obtained the moment estimator of λ as ˆλ = 1 α 2 m 2 2 ) 1 2) 2 ). REFERENCES [1] Andjelic, G., Milosev, I., & Djakovic, V. 2010). Extreme value theory in emerging markets. Economic Annals, LV 185), [2] Arnold, B. C., & Beaver, R. J. 2000a). Hidden truncation models. Sankhya, A62), [3] Arnold, B. C., & Beaver, R. J. 2000b). The skew Cauchy distribution. S- tatistics & Probability Letters, 49, [4] Azzalini, A. 1985). A class of distributions which includes the normal ones. Scinavian Journal of Statistics, 12, [5] Beirlant, J., Teugels, J., Goegebeur, Y., & Segers, J. 2004). Statistics of extremes: theory applications. New York: Wiley. [6] Blumenthal, S. 1967). Proportional sampling in life-length studies. Technometrics, 9, [7] Coles, S. G. 2001). An introduction to statistical modeling of extreme values. New York: Springer-Verlag. [8] Cooley, D., Nychka, D., & Naveau, P. 2007). Bayesian spatial modeling of extreme precipitation return levels. Journal of the American Statistical Association, 102, [9] Diebold, F. X., Schuermann, T., & Stroughair, J. D. 1998). Pitfalls opportunities in the use of extreme value theory in risk management. In Refenes, A. P., Burgess, A., & Moody, J. Eds.), Decision technologies for computational finance pp. 3-12). Kluwer Academic Publishers. [10] Embrechts, P., Kläuppelberg, C., & Mikosch, T. 1999). Modelling extremal events for insurance finance: applications of mathematics. New York: Springer. [11] Genton, M. G. 2004). Skew-elliptical distributions their applications: a journey beyond normality 1st ed.). New York, Chapman & Hall/CRC. [12] Gupta, R. C., & Kirmani, S. N. U. A. 1990). The role of weighted distributions in stochastic modeling. Communications in Statistics, 19, [13] Gupta, R. C., & Kundu, D. 2007). Skew-logistic distribution. Technical Report, Indian Institute of Technology Kanpur, India. [14] Gupta, R. C., & Kundu, D. 2009). A new class of weighted exponential distributions. Statistics, 43, [15] Mahdy, M. 2011). A weighted gamma distribution its properties. Journal of Economic Quality Control, 26 2), [16] Mahfoud, M., & Patil, G. P. 1982). On weighted distributions, in statistics 44

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