ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY

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1 ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa, West Africa Abstract A ew weighted Weibull distributio has bee defied ad studied. Some mathematical properties of the distributio have bee studied ad the method of maximum likelihood was proposed for estimatig the parameters of the distributio. The usefuless of the ew distributio was demostrated by applyig it to a real lifetime dataset. Keywords: Weighted Weibull, Order statistic, Maximum likelihood estimatio, Azzalii Itroductio The Weibull distributio has received much attetio i literature because of the advatage it has over other distributios i modelig lifetime data. However, researchers cotiue to develop differet geeralizatios of the Weibull distributio to icrease its flexibility i modelig lifetime data. Merovci ad Elbatal (25) developed the Weibull-Rayleigh distributio ad demostrated its applicatio usig lifetime data. Almalki ad Yua (22) preseted the ew modified Weibull distributio by combiig the Weibull ad the modified Weibull distributio i a serial system. The hazard fuctio of the ewly proposed distributio is the sum of the Weibull hazard fuctio ad a modified Weibull hazard fuctio. Aother geeralizatio of the Weibull distributio is the expoetiated Weibull distributio of Mudholkar ad Srivastava (993). Mudholkar et al. (995) ad Mudholkar ad Husto (996) further studied the expoetiated Weibull distributio with some applicatio to bus-motor failure data ad flood. Pal et al. (26) gave a re-itroductio of the expoetiated Weibull distributio i more details. Al-Saleh ad Agarwal (26) proposed aother exteded versio of the Weibull distributio. They demostrated that the hazard fuctio ca exhibit uimodal ad bathtub shapes. Xie ad Lai (996) developed the additive Weibull distributio with bathtub shaped hazard fuctio obtaied as the sum of two hazard fuctios 34

2 of the Weibull distributio. Zhag ad Xie (27) employed the Marshall ad Olki (997) approach of addig ew parameter to a distributio to propose the exteded Weibull distributio. Lai et al. (23) proposed a modificatio of the Weibull distributio by multiplyig the Weibull cumulative hazard fuctio by e λx ad studied its properties. I this article, a ew geeralizatio of the Weibull distributio based o a modified weighted versio of Azzalii s (985) approach has bee proposed. If g (x) is a probability desity fuctio (pdf) ad G (x) is the correspodig survival fuctio such that the cumulative distributio fuctio (cdf),g (x), exist; The the ew weighted distributio is defied as: f(x; α,, λ) = Kg (x)g (λx) () where K is a ormalizig costat. Weighted Weibull Distributio I this sectio, the desity of the weighted Weibull distributio has bee derived based o the defiitio give i equatio (). Cosider a two parameter Weibull distributio with pdf give by: g (x) = αx e ( αx), x >, α >, > (2) The cdf is give by: G (x) = e ( αx ) (3) The survival fuctio is give by: G (x) = e ( αx ) (4) Usig equatios (), (2) ad (4) the pdf of the weighted weibull distributio is defied as: f(x; α,, λ) = ( + λ )αx e (αx +α(λx) ) (5) The correspodig cdf of the weighted Weibull distributio is give by: F(x; α,, λ) = e (αx +α(λx) ) (6) where α is a scale parameter ad, ad λ are shape parameters. Figure ad 2 illustrates possible shapes of the pdf ad the cdf of the weighted Weibull distributio for some selected values of the parameters α, ad λ. 35

3 Figure : Probability desity fuctio of weighted Weibull distributio Figure 2: Cumulative distributio fuctio of weighted Weibull distributio The survival fuctio is give by: F (x; α,, λ) = F(x; α,, λ) = e (αx +α(λx) ) (7) ad the hazard fuctio is: h(x; α,, λ) = ( + λ )αx (8) Figure 3 illustrates possible shapes of the hazard fuctio of the weighted Weibull distributio for some selected values of the parameters α, ad λ. 36

4 Figure 3: Hazard fuctio of weighted Weibull distributio Statistical Properties I this sectio, the statistical properties of the weighted Weibull distributio is studied. The quatile fuctio, skewess, kurtosis, mode, momet ad momet geeratig fuctio have bee derived. Quatile fuctio ad Simulatio Let Q(u), < u < deote the quatile fuctio for the weighted Weibull distributio. The Q(u) is give by: Q(u) = F (u) = [ l ( u ) α( + λ ) ] I particular, the distributio of the media is: l 2 Q(.5) = [ α( + λ ) ] To simulate from the weighted Weibull distributio is straight forward. Let u be a uiform variate o the uit iterval (,). Thus by meas of the iverse trasformatio method, we cosider the radom variable X give by: (9) 37

5 X = [ l ( u ) α( + λ ) ] () Mode Cosider the desity of the weighted Weibull distributio give i (5). The mode is obtaied by solvig at x = x is give by: ( ) x = [ α( + λ ) ] d l f(x) dx = for x. Therefore the mode () Skewess ad Kurtosis I this study, the quatile based measures of skewess ad kurtosis was employed due to o-existece of the classical measures i some cases. The Bowley s measure of skewess based o quartiles is give by: Q(3/4) 2Q(/2) + Q(/4) B = Q(3/4) Q(/4) ad the Moors kurtosis is o octiles ad is give by: M = Q(7/8) Q(5/8) Q(3/8) + Q(/8) Q(6/8) Q(2/8) where Q(. ) represets the quatile fuctio. Momet ad Momet Geeratig Fuctio I this sectio, the r th o cetral momet ad the momet geeratig fuctio have bee derived. Theorem. If a radom variable X has a weighted Weibull distributio, the the r th o cetral momet is give by the followig: r μ r = [ α( + λ ) ] r Γ ( + ) (2) Proof. This implies μ r = x r f(x) dx μ r = x r ( + λ )αx e (αx +α(λx) ) dx (3) Let y = αx + α(λx), dy = α( + λ )x dx ad x = [ y ] α(+λ ) 38

6 r μ y r = [ α( + λ ) ] e y dy = [ α( + λ ) ] r = [ α( + λ ) ] r Γ ( + ) This completes the proof. If r =, E(X) = [ r ] α(+λ ) y ( r +) e y dy Γ ( + If r = 2, E(X 2 2 ) = [ Γ ( + ) Therefore the variace is give by Var(X) = E(X 2 ) {E(X)} 2 α(+λ ) ] 2 ) (4) Theorem 2. Let X have a weighted Weibull distributio. The momet geeratig fuctio of X deoted by M X (t) is give by: M X (t) = ti [ i! α( + λ ) ] i= i Γ ( + i Proof. By defiitio M X (t) = E(e tx ) = e tx Usig Taylor series M X (t) = ) (5) f(x)dx ( + tx! + t2 x t x + ) f(x)dx 2!! = ti E(X i ) i! i= i = ti [ i! α( + λ ) ] i Γ ( + ) i= This completes the proof. Reyi Etropy If X be a radom variable havig a weighted Weibull distributio. A importat measure of the ucertaity of X is the Reyi etropy. The Reyi etropy is defied as: 39

7 I R (δ) = δ log[i(δ)] where I(δ) = [f(x)] δ dx, δ > ad δ. R Theorem 3. For a radom variable X havig a weighted Weibull distributio, the Reyi etropy is give by: I R (δ) = log [( δ + λ ) δ (α) δ ( )i (αδλ ) i (αδ) (δ+δ+i+) Γ(δ δ i! i= + i + )] (6) Where Γ( ) is the gamma fuctio. Reliability Reliability of a compoet plays a sigificat role i Stress-Stregth aalysis of a model. If X is the stregth ad Y is the stress, the compoet fails whe X Y. The the estimatio of the reliability of the compoet R is Pr (Y < X). R = f(x)f(x)dx = f(x)f (x)dx Theorem 4. If X is the stregth ad Y is the stress, the the reliability of the compoet R is give by: R = ( )h α h ( + λ ) h [α( + λ )] h h= h! Γ(h + ) (7) Order Statistics Let X () deote the smallest of {X, X 2,, X }, X (2) deote the secod smallest of {X, X 2,, X }, ad similarly X (k) deote the k th smallest of {X, X 2,, X }. The the radom variables X (), X (2),, X (), called the order statistics of the sample X, X 2,, X, has probability desity fuctio of the k th order statistic, X (k), as: g k: (x) =! (k )! ( k)! f(x)[f(x)]k [ F(x)] k for k =, 2, 3,,. The pdf of the k th order statistic is defied as: 4

8 μ r (k:) = g k: (x) =!! (k )! ( k)! ( + λ )αx e (αx +α(λx) ) [ e (αx +α(λx) ) ] k [e (αx +α(λx) ) ] k (8) The pdf of the largest order statistic X () is therefore: g : (x) = ( + λ )αx e (αx +α(λx) ) [ e (αx +α(λx) ) ] (9) ad the pdf of the smallest order statistic X () is give by: g : (x) = ( + λ )αx e (αx +α(λx) ) [e (αx +α(λx) ) ] (2) Theorem 5. The r th o cetral momet of the k th order statistics is give by: j ) ( )j+l+m (αj+αjλ l ) [α( k)+αλ ( k)] m m= l!m! (k )!( k)! l= j= (k [α( + λ )] (r+l+m ) r+l+m Γ ( + ) (2) Maximum Likelihood Estimatio I this sectio, the method of maximum likelihood was cosidered for the estimatio of the parameters of the weighted Weibull distributio. Cosider a radom sample of size, cosistig of value x, x 2,, x from the weighted Weibull desity f(x; α,, λ) = ( + λ )αx e (αx +α(λx) ) The likelihood fuctio of the above desity is give by: L(x; α, λ, ) = [( + λ )αx e (αx +α(λx) ) ] i= where x = [x, x 2,, x ]. The log-likelihood fuctio is give by: l L(x; α, λ, ) = l( + λ ) + l α + l + ( ) l x i α( + λ ) x i i= i= (22) Takig the partial derivatives of the log-likelihood fuctio i (22) with respect to the parameters α, λ ad yields: 4

9 l L(x; α, λ, ) = α α ( + λ ) x i l L(x; α, λ, ) λ l L(x; α, λ, ) i= = λ ( + λ ) αλ x i i= = λ l λ ( + λ ) + + l x i i= α [x i l x i + (λx i ) l(λx i )] i= (25) (23) (24) Settig equatios (23), (24) ad (25) to zero ad solvig them simultaeously yields the maximum likelihood estimates of the three parameters. By takig the secod partial derivatives of (23), (24) ad (25) the Fisher s iformatio matrix ca be obtaied by takig the egative expectatios of the secod partial derivatives. The iverse of the Fisher s iformatio matrix is the variace covariace matrix of the maximum likelihood estimators. Empirical Study I this sectio, a empirical study was carried out to ivestigate the effect of chage i the values of the ew parameter λ for α =.5 ad = 2.5. Table provides the mea, variace, Bowley s coefficiet of Skewess ad Moor s coefficiet of kurtosis. From Table, the mea ad the variace of the weighted Weibull distributio decreases for a icrease i the value of λ. However, the coefficiet of skewess ad kurtosis are ot affected by icrease i the value of λ. Table : Mea, Variace, Skewess ad Kurtosis of weighted Weibull distributio α =.5 = 2.5 λ Mea Variace Skewess Kurtosis

10 Applicatio I this sectio, the applicatio of the ew weighted Weibull distributio is demostrated usig the lifetime data of 2 electroic compoets (see Murthy et al., 24, pp. 83,). Teimouri ad Gupta (23) studied this data usig a three-parameter Weibull distributio. I this study, the weighted Weibull distributio is fitted to this data ad the results compared to that of Teimouri ad Gupta (23). The data is show i Table 2. From Table 3, the Aderso-Darlig (AD) statistics revealed that the weighted Weibull fits the data better tha the three-parameter Weibull distributio. Table 2: Lifetimes of 2 electroic compoets Table 3: Estimated Parameters of dataset Distributio Estimated Parameters AD Statistic Weighted Weibull α =.363 =.96 λ = Three-parameter Weibull α =.27 β = 2.72 μ = Coclusio A ew weighted Weibull distributio based o modified weighted versio of Azzalii s (985) approach has bee proposed. Some importat ad mathematical properties of the distributio have bee derived. A empirical study was carried out to determie the effect of the ew parameter o the mea, variace, skewess ad kurtosis of the distributio. The applicatio of the ew distributio has bee demostrated usig real life data. Future works iclude compariso of the ew distributio with other modified weibull distributios, applicatio of the distributio to cesored dataset ad compariso of differet techiques for estimatig the parameters of the distributio. Refereces: Almalki, S. J. ad Yua, J. (23). The ew modified Weibull distributio. Reliability Egieerig ad System Safety, : Al-Saleh, J. A. ad Agarwal, S. K. (26). Exteded Weibull type distributio ad fiite mixture of distributios. Statistical Methodology, 3: Azzalii, A. (985). A class of distributios which icludes the ormal oes. Scadiavia Joural of Statistics, 2: Lai, C. D., Xie, M. ad Murthy, D. N. P. (23). A modified Weibull distributio. IEEE Trasactios o Reliability, 52():

11 Marshall, A. W. ad Olki, I. (997). A ew method for addig a parameter to a family of distributios with applicatio to the expoetial ad Weibull families. Biometrika, 84(3): Merovci, F. ad Elbatal, I. (25). Weibull-Rayleigh distributio: Theory ad applicatios. Applied Mathematics ad Iformatio Scieces, 9(5): -. Mudholkar, G. S. ad Husto, A. D. (996). The expoetiated Weibull family: Some properties ad flood data applicatio. Commuicatios i Statistic-Theory ad Methods, 25(2): Mudholkar, G. S. ad Srivastava, D. K. (993). Expoetiated Weibull family for aalyzig bathtub failure-rate data. IEEE Trasactios o Reliability, 42(2): Mudholkar, G. S., Srivastava, D. K. ad Freimer, M. (995). The expoetiated Weibull family: A reaalysis of the bus-motor-failure data. Techometrics, 37(4): Murthy, D. N. P., Xie, M. ad Jiag, R. (24). Weibull models. Joh Wiley, New York. Pal, M., Ali, M. M. ad Woo, J. (26). Expoetiated Weibull distributio. Statistica, 66: Teimouri, M. ad Gupta, K. A. (23). O three-parameter Weibull distributio shape parameter estimatio. Joural of Data Sciece, : Xie, M. ad Lai, C. D. (996). Reliability aalysis usig a additive Weibull model with bathtub shaped failure rate fuctio. Reliability Egieerig ad System Safety, 52(): Zhag, T. ad Xie, M. (27). Failure data aalysis with exteded Weibull distributio. Commuicatios i Statistics-Simulatio ad Computatio, 36:

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