Volume 7, Issue 3, Marh 2017 ISSN: 2277 128X Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig Researh Paper Available olie at: wwwijarsseom O Size-Biased Weighted Trasmuted Weibull Distributio Moa Abdelghafour Mobarak, Zohdy Nofal, Mervat Mahdy Departmet of Statistis, Mathematis ad Isurae, College of Commere, Beha Uiversity, Egypt Abstrat- This paper offers a ew weighted distributio alled size biased weighted trasmuted weibull distributio, deoted by (SBWTWD) Various useful statistial properties of this distributio are derived i this paper suh as, the umulative distributio futio, Reliability futio, hazard rate, reversed hazard rate ad the rth momet Plots for the probability desity futio at differet values of shape parameters are provided The maximum likelihood estimators of the ukow parameters of the proposed distributio are obtaied Oe data set has bee aalyzed for illustrative purposes Keywords- Weighted distributios, Trasmuted distributio, Weibull distributio, Maximum likelihood method I INTRODUCTION Addig a extra parameter to a existig family of distributio futios is very ommo i the statistial distributio theory Ofte itroduig a extra parameter brigs more flexibility to a lass of distributio futios ad it a be very useful for data aalysis purposes Espeially the weibull distributio ad its geeralizatios i the literature attrat the most of the researhers due to its wide rage appliatios The Weibull distributio iludes the expoetial ad the Rayleigh distributios as sub models, the usefuless ad appliatios of parametri distributios iludig Weibull, Rayleigh are see i various areas iludig reliability, reewal theory, ad brahig proesses whih a be see i papers by may authors suh as i {[16, [17, [25} Differet geeralizatios of the Weibull distributio are ommo i the literature as i {[4, [5, [21, [22, [28, [38} ad aother geeralizatio of the weibull distributio usig the oept of weighted distributios is available as i {[6, [8, [19, [24, [30, [34, [36, [37} The use ad appliatio of weighted distributios i researh related to reliability, bio-mediie, eology ad several other areas are of tremedous pratial importae i mathematis, probability ad statistis These distributios arise aturally as a result of observatios geerated from a stohasti proess ad reorded with some weight futio The oept of these distributios has bee employed i wide variety appliatios i may fields of real life suh as mediie, reliability, ad survival aalysis, aalysis of family data, eology ad forestry It a be traed to the work of Fisher [14 i oetio with his studies o how method of asertaimet a ifluee the form of distributio of reorded observatios Azzalii [1 was first to itrodue a shape parameter to a ormal distributio depedig o a weight futio whih is alled the skew-ormal distributio Differet works o itroduig shape parameters for other symmetri distributios are available i the literature, several properties ad their iferee proedures are disussed by several authors see for example i {[2, [3} O the other side, Reetly several authors itrodued shape parameters for o-symmetri distributios as be show i {[7, [9, [10, [12, [13, [15,[18, [23, [26, [29, [32, [33, [35} I this paper we ostrut the size biased weighted trasmuted weibull distributio ad the sub-models whih are the speial ases of our proposed distributio Various useful statistial properties of this model are derived i the ext setios We also preset a umerial example of the proposed distributio osiderig the real life data-set for illustrative purposes This paper is orgaized as follows Setio 2 defies some basi materials ad i Setio 3, we provide the derivatio of PDF of the proposed model ad some partiular ases are obtaied i Setio 4 Setio 5 disusses the differet statistial properties of this model Estimatio of the ukow parameters of the proposed model by maximum likelihood method is arried out i Setio 6 The real data-set has bee aalyzed i Setios 7 ad setio 8 gives some brief olusio II MATERIALS AND METHODS Weighted distributios oept a be traed from the study of Fisher ad Rao Let X be a o-egative radom variable with its probability desity futio (pdf), f(x), the the pdf of the weighted radom variable X w is give by: f w w(x) f(x ) (x) =, 0 < E[w(X) <, x > 0 (1) E[w(X) where, f(x ) is the pdf of the base distributio ad the weight futio w(x) is a o- egative futio, that may deped o the parameter Whe the weight futio depeds o the legth of uits of iterest, w(x) = x, the resultig distributio is alled legth-biased whih fids various appliatios i biomedial areas suh as early detetio of a disease Rao [27 also used this distributio i the study of huma families ad wild-life populatios I this ase the pdf of a legth-biased radom variable is defied as: 2017, IJARCSSE All Rights Reserved Page 317
Mobarak et al, Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig 7(3), Marh- 2017, pp 317-325 f LB x f(x) (x) =, x > 0, 0 < E[X < E[X More geerally, whe the samplig mehaism selets uits with probability proportioal to some measure of the uit size, whe w(x) = x, > 0, the the resultig distributio is alled size-biased ad the pdf of a size-biased radom variable is defied as: f SB (x) = x f(x) E[X, 0 < E[X <, > 0 This type of samplig is a geeralizatio of legth- biased samplig I this paper we use this weight futio,w(x) = x, osiderig the trasmuted weibull distributio as baselie distributio to get a ew weighted distributio Aordig to the Quadrati Rak Trasmutatio Map (QRTM) approah proposed by Shaw ad Bukley [31 a radom variable X is said to have trasmuted probability distributio if its df, F T (x) ad pdf, f T (x) are give by: F T (x) = (1 + )F(x) F(x) 2, 1, ad, f T (x) = f(x)[(1 + ) 2F(x), where, F(x), f(x), are the df, pdf of the base distributio, respetively ad is the trasmuted, shape parameter The, the df ad the pdf of the trasmuted weibull distributio (TWD) are give as follow: ad, F T (x) = [1 e λx (1 + e λx ), f T (x) = λx 1 e λx [1 + 2e λx, where, λ > 0, > 0 are the sale, shape parameters respetively, the pdf, f(x), ad the df, F(x), of the weibull distributio take the forms as follow: f(x) = λx 1 e λx, λ > 0, > 0, x > 0, ad F(x) = [1 e λx The distributio i equatio (2) iludes espeially the trasmuted expoetial ad trasmuted Rayleigh distributios as speial ases where = 1 ad = 2, respetively III DERIVATION OF THE SIZE BIASED WEIGHTED TRANSMUTED WEIBULL DISTRIBUTION I this setio, we derive the probability desity futio of size biased weighted trasmuted weibull distributio The plot of pdf of this distributio at various hoies of shape parameters values a also be show i this setio We a get the pdf of size biased weighted trasmuted weibull distributio as follows: Whe, Substitutig (3) ad (2) i(1) the we get: Hee, E(X ) = w(x) = x Γ ( + 1) [1 + f SBWTWD (x, λ,,, ) = λ +1 x + 1 e λx [1 + 2e λx, x > 0 λ > 0, > 0, > 0, 1 (4) Γ ( + 1) [1 + The desity futio (4) a be kow as size biased weighted trasmuted weibull distributio, deoted by SBWTWD Figures a, b ad () represet the possible shapes of probability desity futio of the SBWTWD at differet values of shape parameters, ad, respetively whe the sale parameter, λ = 1 λ (2) (3) 2017, IJARCSSE All Rights Reserved Page 318
Mobarak et al, Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig 7(3), Marh- 2017, pp 317-325 IV SOME PARTICULAR CASES OF SBWTWD This setio presets some sub-models that dedued from Equatio (4) are: Case1 Puttig = 1, the resultig distributio is legth biased weighted trasmuted weibull distributio (LBWTWD)give as: 1 f(x; λ,, ) = λ +1 2 x e λx [1 + 2e λx, x > 0, λ > 0, > 0, 1 Γ ( 1 ) [1 + 1 Case2 Puttig = 1, = 1, the resultig distributio is legth biased weighted trasmuted expoetial distributio (LBWTED)give as: 2 f(x; λ, ) = (2 ) λ2 xe λx [1 + 2e λx, x > 0, λ > 0, 1 Case3 Puttig = 0, the resultig distributio is sized biased weighted weibull distributio (SBWWD)give as: f(x; λ,, ) = λ +1 x + 1 e λx Γ ( + 1), x > 0, λ > 0, > 0, > 0 Case4 Puttig = 1, the resultig distributio is sized biased weighted weibull distributio (SBWWD)give as: f(x; λ,, ) = (2λ) +1 x + 1 e 2λx Γ ( + 1), x > 0, λ > 0, > 0, > 0 Case5 Puttig = 1, = 1, = 1, the resultig distributio is legth biased weighted expoetial distributio (LBWED)give as: f(x; λ) = (2λ) 2 xe 2λx, x > 0, λ > 0 Case6 Puttig = 1, = 2, = 1, the resultig distributio is legth biased weighted Rayleigh distributio (LBWRD)give as: f(x; λ) = 25 2(λ) 3 2x 2 e 2λx2 Γ ( 3 2 ), x > 0, λ > 0 Case7 Puttig = 1, = 1, the resultig distributio is legth biased weighted weibull distributio (LBWWD)give as: f(x; λ, ) = (2λ)1 +1 x e 2λx Γ ( 1 + 1), x > 0, λ > 0, > 0 Case8 Puttig = 0, = 1, = 1, the resultig distributio is legth biased weighted expoetial distributio (LBWWD)give as: f(x; λ) = λ 2 xe λx, x > 0, λ > 0 Case9 Puttig = 0, the resultig distributio is trasmuted weibull distributio (TWD)give as: f(x; λ,, ) = λx 1 e λx [1 + 2e λx, x > 0, λ > 0, > 0, 1 Case10 Puttig = 0, = 0, the resultig distributio is weibull distributio (WD)give as: f(x; λ, ) = λx 1 e λx, x > 0, λ > 0, > 0 Case11 Puttig = 0, = 1, the resultig distributio is trasmuted expoetial distributio (TED)give as: f(x; λ, ) = λe λx [1 + 2e λx, x > 0, λ > 0, 1 Case12 Puttig = 1, = 2, the resultig distributio is legth biased weighted trasmuted Rayleigh distributio (LBWTRD)give as: 2017, IJARCSSE All Rights Reserved Page 319
Mobarak et al, Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig 7(3), Marh- 2017, pp 317-325 f(x; λ, ) = 2λ3 2x 2 e λx2 [1 + 2e λx2, x > 0, λ > 0, 1 Γ ( 3 ) [1 + 2 (2) 1 2 Case13 Puttig = 0, = 2, the resultig distributio is trasmuted Rayleigh distributio (TRD)give as: f(x; λ, ) = 2λxe λx2 [1 + 2e λx2, x > 0, λ > 0, 1 Case14 Puttig = 0, = 2, = 0, the resultig distributio is Rayleigh distributio (RD)give as: f(x; λ) = 2λxe λx2, x > 0, λ > 0 Case15 Puttig = 0, = 0, = 2 ad multiplyig by ( 1), this model gives the iverse Rayleigh distributio (IRD)give as: f(x; λ) = 2λx 3 e λx 2, x > 0, λ > 0 Case16 Puttig = 1, = 0, = 2 ad multiplyig by ( 1), this model gives the iverse Rayleigh distributio (IRD)give as: f(x; λ) = 2(2λ)x 3 e 2λx 2, x > 0, λ > 0 Case17 Puttig = 0, = 1, the resultig distributio is legth biased weighted weibull distributio (LBWWD)give as: 1 f(x; λ, ) = λ +1 x e λx Γ ( 1 + 1), x > 0, λ > 0, > 0 Case18 Puttig = 1, = 2, = 0, the resultig distributio is legth biased weighted Rayleigh distributio (LBWRD)give as: f(x; λ) = 2λ3 2x 2 e λx2 Γ ( 3 2 ), x > 0, λ > 0 Case19 Puttig = 0, = 1, the resultig distributio is weibull distributio (WD)give as: f(x; λ, ) = 2λx 1 e 2λx, x > 0, λ > 0, > 0 Case20 Puttig = 0, = 1, = 1, the resultig distributio is expoetial distributio (ED)give as: f(x; λ) = 2λe 2λx, x > 0, λ > 0 Case21 Puttig = 0, = 2, = 1, the resultig distributio is Rayleigh distributio (RD)give as: f(x; λ) = 2(2λ)xe 2λx2, x > 0, λ > 0 Case22 Puttig = 0, = 1, = 0, the resultig distributio is expoetial distributio (ED)give as: f(x; λ) = λe λx, x > 0, λ > 0 Case23 Puttig = 1, the resultig distributio is size biased weighted trasmuted expoetial distributio (SBWTED) give as: f(x; λ,, ) = λ+1 x e λx [1 + 2e λx Γ( + 1) [1 +, x > 0, λ > 0, > 0, 1 (2) Case24 Puttig = 2, the resultig distributio is size biased weighted trasmuted Rayleigh distributio (SBWTRD) give as: f(x; λ,, ) = 2λ 2 +1 x +1 e λx2 [1 + 2e λx2, x > 0, λ > 0, > 0, 1 Γ ( + 1) [1 + 2 (2) 2 V THE STATISTICAL PROPERTIES OF SBWTWD I this setio, we preset some basi statistial properties of SBWTWD iludig, the umulative distributio futio (CDF), reliability futio, hazard futio ad the reverse hazard futio, rth momet, the mea, variae ad order statistis as follow: i The CDF of SBWTWD is defied as: Therefore, The CDF of SBWTWD is give as: F SBWTWD (x) = f SBWTWD (t)dt F SBWTWD (x, λ,,, ) = where, γ(s, x) is the lower iomplete gamma futio defied as: 0 x 2017, IJARCSSE All Rights Reserved Page 320 x γ(s, x) = t s 1 e t dt 0 γ ( + 1, λx ) Γ ( + 1),
Mobarak et al, Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig 7(3), Marh- 2017, pp 317-325 ii The Reliability (Survival) futio of SBWTWD is defied as: R SBWTWD (x, λ,,, ) = 1 F SBWTWD (x, λ,,, ), γ ( + 1, λx ) = 1 Γ ( + 1) Table (1) otais the values of survival futio of SBWTWD Lookig at this table we a see that the survival probability of the distributio ireases with irease i the value of for a holdig x, λ ad at a fixed level Further, from the table we a see that; for fixed, λ ad ; the survival probability dereases with irease i x iii iv Table 1: Survival futio of SBWTWD The hazard rate futio of the radom variable X w follows SBWTWD is defied by: H SBWTWD (x, λ,,, ) = f SBWTWD (x) 1 F SBWTWD (x) = λ +1 x + 1 e λx [1 + 2e λx [1 + [Γ ( + 1) γ ( + 1, λx ) The reversed hazard rate futio of the radom variable X w follows SBWTWD is give as: H SBWTWD (x, λ,,, ) = f SBWTWD (x) F SBWTWD (x) = λ +1 x + 1 e λx [1 + 2e λx γ ( + 1, λx ) [1 + v The rth momet of the radom variable X w follows SBWTWD is give as: SBWTWD M r = Or M r SBWTWD a be writte as: Γ ( r+ + 1) Γ ( + 1) [1 + where,γ r = Γ ( r+ + 1), k r = [1 + For the ase, r = 1,2,3,4 we have, vi vii where, λ = 1, = 1 x 1 12 14 16 18 01 0995 0998 0999 0999 1 02 0982 099 0994 0996 0998 03 0963 0976 0985 0991 0994 04 0938 0958 0972 0981 0988 05 091 0936 0955 0969 0979 06 0878 091 0935 0953 0967 07 0844 0882 0912 0935 0953 08 0809 0852 0887 0914 0936 09 0772 082 0859 0891 0917 1 0736 0787 083 0866 0896 11 0699 0753 08 084 0873 r+ (λ) r [1 + SBWTWD Γ r k r M r = Γ k a r, (2) r+, r = 1,2,3,, x > 0, x > 0, Γ = Γ ( + 1), k = [1 + ad a = (λ) 1 μ 1 = Γ 1k 1 Γk a, μ 2 = Γ 2k 2 Γk a 2, μ 3 = Γ 3k 3 Γk a 3, μ 4 = Γ 4k 4 Γk a 4 The variae of the radom variable X w follows SBWTWD is give as:: σ 2SBWTWD = μ 2 μ 1 2 = ΓkΓ 2k 2 [Γ 1 k 1 2 Γ 2 k 2 a 2 The first etral momets of SBWTWD are give by: μ 1 = 0, μ 2 = σ 2 = μ 2 μ 1 2, μ 3 = μ 3 3μ 1μ 2 + 2μ 1 3, μ 4 = μ 4 4μ 1μ 3 + 6μ 1 2 μ 2 3μ 1 4, μ 3 = Γ 3k 3 Γ k a 3 3 Γ 1k 1 Γ 2 k 2 (Γ k) 2 a 3 + 2 [ Γ 3 1k 1 Γ k a = (Γk)2 Γ 3 k 3 3ΓkΓ 1 k 1 Γ 2 k 2 + 2[Γ 1 k 1 3 [Γka 3, 2017, IJARCSSE All Rights Reserved Page 321
Mobarak et al, Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig 7(3), Marh- 2017, pp 317-325 μ 4 = Γ 4k 4 Γk a 4 4 Γ 1k 1 Γ 3 k 3 (Γk) 2 a 4 + 6[Γ 1k 1 2 Γ 2 k 2 (Γk) 3 a 4 3 [ Γ 4 1k 1 Γ k a viii ix = [Γk3 Γ 4 k 4 4(Γk) 2 Γ 1 k 1 Γ 3 k 3 + 6Γk[Γ 1 k 1 2 Γ 2 k 2 3[Γ 1 k 1 4 [Γka 4 The oeffiiet of variatio is give as: CV = σ ΓkΓ 2k 2 [Γ 1 k 1 2 Γ = [Γk a 2 k 2 a 2 = ΓkΓ 2k 2 [Γ 1 k 1 2 μ 1 Γ 1 k 1 Γ 1 k 1 Coeffiiet of Skewess (SK) is give by: SK = μ 3 σ 3 = (Γk)2 Γ 3 k 3 3ΓkΓ 1 k 1 Γ 2 k 2 + 2[Γ 1 k 1 3 3 = (Γk)2 Γ 3 k 3 3ΓkΓ 1 k 1 Γ 2 k 2 + 2[Γ 1 k 1 3 2 [Γka 3 [ΓkΓ 2 k 2 [Γ 1 k 1 2 3 2 [ ΓkΓ 2k 2 [Γ 1 k 1 2 Γ 2 k 2 a 2 x Coeffiiet of Kurtosis (ku) is give by: Ku = μ 4 σ 4 3 = [Γk3 Γ 4 k 4 4(Γk) 2 Γ 1 k 1 Γ 3 k 3 + 6Γk[Γ 1 k 1 2 Γ 2 k 2 3[Γ 1 k 1 4 [ΓkΓ 2 k 2 [Γ 1 k 1 2 2 3 xi The mode is the value of the radom variable x whih makes the pdf is a maximum Takig logarithm of the pdf of SBWTWD as: log e f SBWTWD (x, λ,,, ) = ( + 1) log e λ + log e + ( + 1) log e x λx + log e [1 + 2e λx log e Γ ( + 1) log e [1 + (2) log e f SBWTWD (x, λ,,, ) ( + 1) = λx 1 2λx 1 e λx x x [1 + 2e λx (5) The mode of the SBWTWD is obtaied by solvig the equatio (5)with respet tox as follow: ( + 1) λx 1 2λx 1 e λx = 0 (6) x [1 + 2e λx By solvig the oliear equatio (6), a be alulated the mode of the SBWTWD xii The order statistis have great importae i life testig ad reliability aalysis Let X 1, X 2,, X be radom variables ad its ordered values is deoted as x 1, x 2,, x The pdf of order statistis is obtaied usig the below futio:! f s:, (x) = (s 1)! ( s)! f(x)[f(x)s 1 [1 F(x) s (7) To obtai the smallest value i radom sample of size put s = 1 i (7), the the pdf of smallest order statistis is give by f 1:, (x) = f(x)[1 F(x) 1 Therefore, the pdf of smallest order statistis for the SBWTWD is: f 1:, (x) = λ +1 x + 1 e λx [1 + 2e λx Γ ( + 1) [1 + γ ( + 1, λx ) [1 Γ ( + 1) 1, λ, > 0, x > 0 To obtai the largest value i radom sample of size put s = i (7), the the pdf of order statistis is give by: f :, (x) = f(x)[f(x) 1 Therefore, the pdf of largest order statistis for the SBWTWD is: f :, (x) = λ +1 x + 1 e λx [1 + 2e λx [γ ( 1 + 1, λx ), x > 0 [Γ ( + 1) [1 + VI MAXIMUM LIKELIHOOD ESTIMATION OF THE SBWTWD Let x 1, x 2,, x be a idepedet radom sample from the SBWTWD, the the likelihood futio, L(x; λ,,, ), of SBWTWD is give by: Substitutig from (4)ito (8), we have, L(x; λ,,, ) = f SBWTWD (x, λ,,, ) (8) 2017, IJARCSSE All Rights Reserved Page 322
Mobarak et al, Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig 7(3), Marh- 2017, pp 317-325 L(x; λ,,, ) = [1 + λ ( +1) x i [Γ ( + 1) So, logarithm likelihood futio log e L(x; λ,,, ), is give as: log e L(x; λ,,, ) = log e λ + log e λ + 1 e λ x i [1 + 2e λx i + log e log e [1 + (2) log e Γ ( + 1) + ( + 1) log e x i λ x i + log e [1 + 2e λx i (9) Differetiatig (9) with respet to λ,,,ad, respetively, as follows: log e L(x; λ,,, ) = λ λ + λ x i (2x i )e λx i [1 + 2e λx i, (10) log e L(x; λ,,, ) = 2 log l 2 e λ + 2 [1 + 2 ψ ( + 1) + loge x i λ (x i l x i ) 2λe λx i (x i l x i ), (11) [1 + 2e λx i where, ψ ( + 1) is the digamma futio [1 1 log e L(x; λ,,, ) = (1 2e λx i ) [1 + [1 + 2e λx i, (12) log e L(x; λ,,, ) = log l 2 e λ + 2 [1 + ψ ( + 1) + log e x i (13) Settig the equatios(10), (11), (12)ad (13)equal to zero, we have the followig equatios: λ + λ x i 2 log e λ [1 1 [1 + (2x i )e λx i = 0, (14) [1 + 2e λx i l 2 (2) 2 [1 + + 2 ψ ( + 1) + log e x i λ (x i l x i ) 2λe λx i (x i l x i ) = 0, (15) [1 + 2e λx i (1 2e λx i ) = 0, (16) [1 + 2e λx i log l 2 e λ + 2 [1 + ψ ( + 1) + log e x i = 0 (17) We a get MLEs of the ukow parameters by solvig the equatios(14),(15), (16)ad (17)to estimate the parameters λ,, ad usig umerial tehique methods suh as ewto Raphso method beause it is ot possible to solve these equatios aalytially By takig the seod partial derivatives of (10), (11), (12)ad (13) the Fisher s iformatio matrix a be obtaied by takig the egative expetatios of the seod partial derivatives The iverse of the Fisher s iformatio matrix is the variae ovariae matrix of the maximum likelihood estimators 2017, IJARCSSE All Rights Reserved Page 323
Mobarak et al, Iteratioal Joural of Advaed Researh i Computer Siee ad Software Egieerig 7(3), Marh- 2017, pp 317-325 VII APPLICATION I this setio, we provide a appliatio of the proposed distributio to show the importae of the ew model The data set (gauge legths of 10 mm) from Kudu ad Raqab [20 This data set osists of, 63 observatios as the followig: 1901, 2132, 2203, 2228, 2257, 2350, 2361, 2396, 2397, 2445, 2454, 2474, 2518, 2522, 2525, 2532, 2575, 2614, 2616, 2618, 2624, 2659, 2675, 2738, 2740, 2856, 2917, 2928, 2937, 2937, 2977, 2996, 3030, 3125, 3139, 3145, 3220, 3223, 3235, 3243, 3264, 3272, 3294, 3332, 3346, 3377, 3408, 3435, 3493, 3501, 3537, 3554, 3562, 3628, 3852, 3871, 3886, 3971, 4024, 4027, 4225, 4395, 5020 This data set is previously studied by Afify et al [11 to fit the trasmuted weibull lomax distributio We fit both trasmuted weibull (TW) ad size biased weighted trasmuted weibull (SBWTW) distributios to the subjet data We also estimate the parameters λ,, ad usig Newto-Raphso method by takig the iitial estimates λ 0 = 25321, 0 = 11577, 0 = 09, 0 = 0216 ad the estimated values of the parameters a be show i table 2 To see whih oe of these models is more appropriate to fit the data set, we alulate Akaike Iformatio Criterio (AIC), the Cosistet Akaike Iformatio Criterio (CAIC) ad Bayesia Iformatio Criterio (BIC) The best distributio orrespods to lower for ( 2)log-likelihood, AIC, BIC, ad CAIC statistis values, where, AIC = 2l + 2k, CAIC = 2l + 2k ( k 1), ad BIC = 2l + k(l ) where l deotes the log-likelihood futio evaluated at the maximum likelihood estimates, k is the umber of parameters ad is the sample size These umerial results are obtaied usig the MATH- CAD PROGRAM Table 2 The Estimated Values of the Parameters Parameters estimates values Model λ TW 00041 4718-02542 - SBWTW 02564 25007 06819 96564 Table (2) otais the estimated values of the parameters for the (TWD) ad SBWTWD Table 3 The Statistis ( 2l), AIC, BIC ad CAIC for Gauge Legths of 10 MM Data Set Models 2l AIC BIC CAIC TW 124414 130414 136843 13082 SBWTW 116861 124861 133433 125551 Table (3) otais the values of ( 2l), AIC, BIC ad CAICstatistis We ote that the SBWTW model gives the lowest values for ( 2l), AIC, BIC ad CAIC statistis so that SBWTWD leads to a better fit to these data tha TWD VIII CONCLUSION I this paper we propose a ew four-parameter model, alled size biased weighted trasmuted weibull distributio whih is a geeralizatio of trasmuted weibull distributio We preset some of its statistial properties The ew distributio is very flexible model that approahes to differet life time distributios whe its parameters are haged We disuss maximum likelihood estimatio We osider Akaike Iformatio Criterio (AIC), the Cosistet Akaike Iformatio Criterio (CAIC) ad Bayesia Iformatio Criterio (BIC) statistis to ompare the model with trasmuted weibull model A appliatio of the size biased weighted trasmuted weibull distributio to real data shows that the proposed distributio a be used quite effetively to provide better fits tha the trasmuted-weibull distributio REFERENCES [1 Azzalii, A (1985) A lass of distributios whih iludes the ormal oes, Sadiavia Joural of Statistis,12, 171-185 [2 Azzalii, A ad Dalla Valle, A (1996) The multivariate skew-ormal distributio Biometrika, 83, 715 26 [3 Arold, B C ad Bever, R J (2000) The skew Cauhy distributio Statistis & Probability Letters, 49, 285-290 [4 Al-Saleh, J A ad Agarwal, S K (2006) Exteded Weibull type distributio ad fiite mixture of distributios Statistial Methodology, 3, 224-233 [5 Aryal, G R ad Toskos, C P (2011) Trasmuted Weibull Distributio: A Geeralizatio of the Weibull Europea Joural of Pure ad Applied Mathematis, 4, 89-102 [6 Aleem, M, Sufya, M ad Kha, N S (2013) A lass of modified weighted Weibull distributio ad its properties Ameria Review of Mathematis ad Statistis, 1, 29 37 [7 Al-Kadim, K ad Hatoosh, A F (2013) Double weighted distributio & double weighted expoetial distributio Mathematial Theory ad Modelig, 3, 2224 5804 [8 Al-Kadim, K A ad Hatoosh, A F (2014) Double weighted iverse Weibull Distributio Pakista Publishig Group, 978-969-9347-16-0 2017, IJARCSSE All Rights Reserved Page 324
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