The Inverse Weibull-Geometric Distribution

Transcription

1 Article Iteratioal Joural of Moder Mathematical Scieces, 016, 14(: Iteratioal Joural of Moder Mathematical Scieces Joural homepage: The Iverse Weibull-Geometric Distributio Adil H. Kha *, T.R. Ja ISSN: X Florida, USA Departmet of Statistics, Uiversity of Kashmir, Sriagar, Idia, * Author to whom correspodece should be addressed; khaadil_19@yahoo.com Article history: Received 6 November 015; Received i revised form 1 Jauary 016; Accepted 15 April 016; Published 3 April 016. Abstract: Iverse Weibull-Geometric Distributio which geeralizes the Iverse Expoetial-Geometric distributio, Iverse Weibull distributio, Iverse Expoetial distributio ad Iverse Rayleigh distributio has bee itroduced i this paper. The model ca be cosidered as aother useful 3-parameter distributio. Model characterizatio is studied, we derive the cumulative distributio ad hazard fuctios, the desity of the order statistics ad calculate expressios for its momets ad for the momets of the order statistics. Mixture model of two Iverse Weibull-Geometric distributios is ivestigated. Estimates of parameters usig method of maximum likelihood have bee obtaied through simulatios. Two real life example are provided oe for complete data aother for cesored data to show the flexibility ad potetiality of the proposed distributio ad compariso with Iverse Weibull distributio, Iverse Expoetial Geometric distributio, Iverse Expoetial distributio ad Iverse Rayleigh distributio is also discussed. The proposed model compares well with other competig models to fit the data. Keywords: Iverse Weibull distributio, Iverse Rayleigh distributio Hazard rate, Mixture distributio, Cesored data. Mathematics Subject Classificatio Code (010: 68M15, 6F15 1. Itroductio The iverse Weibull distributio is oe of the most commoly used lifetime distributio which ca be used i the reliability egieerig disciplie. The Iverse Weibull distributio is used to model a

2 It. J. Moder Math. Sci. 016, 14(: variety of failure characteristics such as ifat mortality, useful life ad wear-out periods ad ca be used to determie the cost effectiveess ad maiteace periods of reliability cetered maiteace activities. I literature various other distributios have bee proposed to model lifetime model. Adamidis ad Loukas [] proposed the two-parameter expoetial-geometric distributio with decreasig failure rate. Kus [13] itroduced the expoetial-poisso distributio with decreasig failure rate ad discussed several of its properties. Adamidis, Dimitrakopoulou ad Loukas [1] proposed the exteded expoetialgeometric distributio which geeralizes the expoetial-geometric distributio. The hazard fuctio of the exteded expoetial-geometric distributio ca be mootoe decreasig, icreasig or costat ad they discussed several reliability features ad its properties. Wager, Alice ad Gauss [4] proposed the Weibull-Geometric distributio which geeralizes the expoetial-geometric distributio proposed by Adamidis, Dimitrakopoulou ad Loukas. The hazard fuctio of Weibull-Geometric distributio takes more geeral forms. The Weibull-Geometric distributio is useful for modelig uimodal failure rates. Wag ad Elbatal [0] proposed a modified Weibull geometric distributio which have mootoically icreasig, decreasig, bathtub-shaped, ad upside-dow bathtub-shaped hazard rate fuctios. Saboor, Kamal ad Ahmad [18] proposed a trasmuted expoetial Weibull distributio which have a bathtubshaped ad upside-dow bathtub-shaped hazard rate fuctios. Kacha, Neetu ad Suresh Kumar [10] proposed Geeralized Iverse Geeralized Weibull (GIGW distributios ad discussed some mathematical properties of the distributio with a real life example. Iverse distributios, amely, Iverse Gamma, Iverse Geeralized Gamma, Iverse Weibull [17], ad Iverse Rayleigh [19], have also bee studied i literature [6, 7, 9]. Kha ad Ja [11, 1] worked o mixture models ad discussed the stress-stregth problem of the systems, where the stregth follows fiite mixture of two parameter Lidley distributio ad stress follows expoetial, Lidley distributio ad mixture of two parameter Lidley distributio ad obtaied geeral expressios for the reliabilities of a system. I this paper, we itroduced a three parameter cotiuous distributio model, the Iverse Weibull Geometric Distributio (IWGD. It is the distributio of reciprocal of a variable distributed accordig to the geeralized Weibull distributio. A comprehesive descriptio of some mathematical properties of the IWGD are discussed with the hope that it will attract wider applicatios i reliability, egieerig ad other areas of research.. The Iverse Weibull Geometric Distributio (IWGD The Weibull geometric distributio with parameters p (0,1, > 0 ad > 0 is defied by its probability desity fuctio (pdf f(x; p,, = (1 px 1 e (x {1 pe (x } ; x > 0 (1

3 It. J. Moder Math. Sci. 016, 14(: The cumulative distributio fuctio (cdf of this distributio is F(x = 1 e (x 1 pe (x ; x > 0 ( The iverse of Weibull geometric distributio called Iverse Weibull Geometric (IWGD with parameters p (0,1, > 0 ad > 0 is defied by f IWGD (x; p,, = (1 px (+1 e ( x x,, > 0 ad pε(0,1 {1 pe ( x } The cumulative distributio fuctio of the distributio is give by F IWGD (x; p,, = (1 pe ( x The Survival fuctio ad Hazard rate fuctio of IWGD are = 1 e ( x (3 1 pe ( x ; x,, > 0 ad pε(0,1 (4 S IWG (x 1 pe ( x (5 h IWGD (x = (1 px (+1 e ( x (1 e ( x (1 pe ( x (6 Special Cases 1 Whe For = 1, the ew distributio is obtaied called Iverse Expoetial Geometric distributio (IEGD with pdf ad cdf respectively give as f IEGD (x; p, = (1 px e x {1 pe x} ; x, > 0 ad pε(0,1 F IEGD (x; p, = (1 pe x 1 pe x ; x, > 0 ad pε(0,1 Whe p approaches to zero, IWGD approaches to Iverse Weibull Distributio (IWD with parameter ad.

4 It. J. Moder Math. Sci. 016, 14(: Whe p approaches to zero ad =, we get Iverse Rayleigh Distributio (IRD with parameter. 4 Whe p approaches to zero ad = 1, we get Iverse Expoetial Distributio (IED with parameter. Fig. 1: Plots for IWGD fuctio Fig. : Plots for Hazard Rate of IWGD 3. Momets of IWGD The k th order momet aroud zero for X~IWG(,, p is writte as

5 It. J. Moder Math. Sci. 016, 14(: Substitutig ( x = y, we get E(X k = x k (1 px (+1 e ( x 0 {1 pe ( x } E(X k = (1 p k y k e y (1 pe y dy 0 E(X k = (1 p k Γ (1 k pj (j + 1 k For = 1, we get k th order momet of Iverse Expoetial distributio. The momet geeratig fuctio of IWGD of X ca be writte as M(t = (1 p [ tk k! k Γ (1 k pj (j + 1 ] k, for t < 1 k=0 The cumulative geeratig fuctio of IWGD of X is give by K(t = logm(t = log {(1 p [ tk k! k Γ (1 k pj (j + 1 ] k }, for t < 1 The mea ad variace of IGWD are k=0 E(X = (1 pγ (1 1 pj (j V(X = (1 p Γ (1 pj (j + 1 [(1 pγ (1 1 pj (j + 1 ] 1 V(X = (1 p {Γ (1 pj (j + 1 (1 p [Γ (1 1 pj (j + 1 ] 1 } dx 3.1. Mixture of Two IWGD ad Properties Various mixture of distributios have bee discussed i past by umber of authors. Maclachla ad Krisha [14], Everitt ad Had [8], Al-Hussaii ad Sulta [3], Maclachla ad Peel [15]. They also discussed the properties of mixture distributio ad have foud them useful i may complex problems. The desity fuctio of mixture of two IWG distributios is called Mixed Iverse Weibull Geometric (MIWG distributio ad is give by f MIWG (x; θ = λ 1 f 1 (x, θ 1 + λ f (x, θ

6 It. J. Moder Math. Sci. 016, 14(: where, f MIWG (x; θ = λ r f r (x, θ r θ 1 = (p 1, 1, 1 T ad θ = (p,, T, f r (x, θ r correspods to r th compoet of the mixture ad is give by k=1 r r f r (x, θ r = r r (1 p r x (r+1 e ( r x {1 p r e ( r x } ; x, r, r > 0 ad p r ε(0,1 ad λ 1 + λ = 1. Properties of MIGWD The Survival fuctio ad Hazard rate fuctio of MIWG distributio are S MIWG (x, θ = λ r ( 1 e ( r=1 r r x 1 p r e ( r x r h MIWG (x = (λ r r r (1 p r x ( r+1 e ( r x k=1 ( λ r ( 1 e ( r=1 r r x 1 p r e ( r x r The k th order momet aroud zero for MIGWD ca be writte as E(X k = x k λ r r r (1 p r x ( r+1 e ( r x r=1 0 1 r r r {1 p r e ( r x } r {1 p r e ( r x } dx E(X k = λ r (1 p r k r Γ (1 k j p r (j + 1 k r r r=1 3.. Estimatio of Parameters Let x 1, x,, x be a radom sample from IGWD with ukow parameter vector φ = (p,, T. The log likelihood for l = l(φ; x for φ is l = [log + log + log(1 p] ( + 1 logx i log (1 pe ( x i ( x i (7

7 It. J. Moder Math. Sci. 016, 14(: The score fuctio U(φ = ( l p, l, l T has compoets l p = (1 p 1 + e ( x i l = 1 + log logx i [1 pe ( 1 x i ] ( x i log ( {1 + pe ( x i [1 pe ( 1 x i ] } x i l = 1 1 x i {1 + pe ( x i [1 pe ( 1 x i ] } The maximum likelihood estimate (MLE φ of φ ca be obtaied by solvig o-liear equatios U(φ = 0. These equatios caot be solved aalytically but statistical software ca be used to solve them umerically, for example, through the R-laguage or ay iterative methods such as the NR (Newto-Raphso,BFGS (Broyde-Fletcher-Goldfarb-Shao, BHHH (Berdt-Hall-Hall-Hausma, NM (Nelder-Mead, L-BFGS-B (Limited-Memory Quasi-Newto code for Boud-Costraied Optimizatio ad SANN (Simulated-Aealig. Cesored Case: Cesorig is the coditio i which value of the observed value of some variables is ukow. I reliability studies particularly i survival aalysis cesorig occurs whe the iformatio about the survivals time of the observatios uder study is icomplete ad therefore we are geerally ecoutered with cesored data. Let the idepedet radom variables Z i ad C i respectively deote the lifetime of the i th idividual ad the cesorig time ad let t i = mi(z i, C i for i = 1,,3,,. The distributio of C i does ot deped o the ukow parameters of Z i, where each Z i follows the IGWD with parameters φ = (p,, T. For cesored case, while writig the likelihood fuctio the data set splits ito two parts oe correspods to cesored data ad aother correspods to those observatios that are ot cesored. Let the sets of cesored ad ucesored observatio respectively be deoted by C ad F, ad r the o of failures. The likelihood fuctio for cesored case ca be writte as l = f(t i S(t i where, f(t i ad S(t i are the desity fuctio ad survival fuctio of the IGWD, respectively. Hece from (3 ad (4 the log likelihood for the IGWD ca be writte as i C

8 It. J. Moder Math. Sci. 016, 14(: l = [log + log + log(1 p] ( + 1 logt i + log (1 e ( t i i C ( t i log (1 pe ( t i The score fuctio U(φ = ( l p, l, l T has compoets l p = (1 p 1 + e ( t i l = 1 + log logt i i C i C [1 pe ( 1 x i ] log (1 pe ( t i 1 + e ( t i [1 pe ( t i ] i C log ( 1 {1 + pe ( t i [1 pe ( t i ] } ( t i t i (1 p ( t e ( t i log ( i t i (1 e ( t i (1 pe ( t i l = 1 1 {1 + pe ( t i 1 [1 pe ( t i ] } t i t i t i (1 1 pe( i C (1 e ( t i (1 pe ( t i The maximum likelihood estimate (MLE φ of φ ca be obtaied by solvig o-liear equatios U(φ = 0 usig umerical methods. 4. Simulatios ad Applicatios Estimatio Based o simulatios: For checkig the theoretical results, we simulate data by geeratig observatios from IGWD for differet sample sizes with umber of repetitios 10,000 ad values of parameters are chose arbitrary. The values of the parameters are estimated usig quasi- Newto method i R. the estimates of parameters with correspodig sample sizes are give i Table 1.

9 It. J. Moder Math. Sci. 016, 14(: Table 1: Estimates of parameters for IGWD p p

10 It. J. Moder Math. Sci. 016, 14(: Real Data Illustratio I this sectio we compare the results of fittig the IWGD, IWD, IEGD, IED ad IRD to the data set studied by Meeker ad Escobar [16], which gives the times of failure ad ruig times for a sample of devices from a eld-trackig study of a larger system. At a certai poit i time, 30 uits were istalled i ormal service coditios. Two causes of failure were observed for each uit that failed: the failure caused by ormal product wear ad failure caused by a accumulatio of radomly occurrig damage from power-lie voltage spikes durig electric storms. The times are:.75, 0.13, 1.47, 0.3, 1.81, 0.30, 0.65, 0.10, 3.00, 1.73, 1.06, 3.00, 3.00,.1, 3.00, 3.00, 3.00, 0.0,.61,.93, 0.88,.47, 0.8, 1.43, 3.00, 0.3, 3.00, 0.80,.45,.66. I order to compare the distributio models, we cosider criteria like log(l, AIC (Akaike Iformatio Criterio, BIC (Bayesia Iformatio Criterio, CAIC (Cosistat Akaike Iformatio Criterio ad HQIC (Haa-Qui Iformatio Criterio for the data set. The better distributio correspods to smaller l, AIC, CAIC ad HQIC. The MLE for IWGD, IWD, IEGD, IED ad IRD are give table 1 alog with Akaike Iformatio Criterio, Bayesia Iformatio Criterio, Haa-Qui Iformatio Criterio ad Cosistet Akaike Iformatio Criterio for the data set. Table shows parameter MLEs to each oe of the two fitted distributios for data set, values of log(l, AIC, CAIC ad HQIC. The values i Table 1 idicate that the IWGD model performs sigificatly better tha its sub-models used here for fittig data set. Table : The ML estimates, stadard error, AIC, BIC ad CAIC of the models based o data set Model -LL Estimates St. Error AIC BIC CAIC HQIC = IWGD = p = IWD = = IEGD = p = IED = IRD =

11 It. J. Moder Math. Sci. 016, 14(: I the secod example we cosidered the cesored provided by David W. Hosmer ad Staley Lemeshow [5]. The survival times i moths of 100 HIV patiets are give below ad * idicates that the data is cesored. 5, 6*, 8, 3,, 1*, 7, 9, 3, 1, *, 1, 1, 15, 34, 1, 4, 19*, 3*,, *, 6, 60*, 7*, 60*,11, *, 5, 4*, 1*, 13, 3*, *, 1*, 30, 7*, 4*, 8*, 5*, 10, *, 9*, 36, 3*, 9*, 3*, 35, 8*, 1*, 5*, 11, 56*, *, 3*, 15, 1*, 10, 1*, 7*, 3*, 3*, *, 3, 3*, 10*, 11, 3*, 7*, 5*, 31, 5*, 58, 1*, *, 1, 3*, 43, 1*, 6*,53, 14, 4*, 54, 1*, 1*, 8*, 5*, 1*, 1*, *, 7*, 1*, 10, 4*, 7*, 1*, 4*, 57, 1*, 1*. The MLE for IWGD, IWD, IEGD, IED ad IRD are give table 3 alog with Akaike Iformatio Criterio, Bayesia Iformatio Criterio, Haa-Qui Iformatio Criterio ad Cosistet Akaike Iformatio Criterio for the cesored data. Table 3: The ML estimates, stadard error, AIC, BIC ad CAIC of the models based o cesored data Model -LL Estimates St. Error AIC BIC CAIC HQIC IWGD = = p = IWD = = IEGD = p = IED = IRD = Table 3 shows that values of log(l, AIC, CAIC ad HQIC are lowest for IWGD. So, we ca coclude that IWGD model performs sigificatly better tha its sub-models used here for fittig cesored data set. 5. Coclusio Here, we propose a ew model, the so-called the Iverse Weibull Geometric Distributio which exteds the Iverse Weibull distributio, Iverse Expoetial Geometric Distributio, Iverse Expoetial Distributio ad Iverse Rayleigh Distributio i the aalysis of data with real support. A proouced reaso for geeralizig a stadard distributio is because the geeralized form exted larger

12 It. J. Moder Math. Sci. 016, 14(: flexibility i modellig real data. We derive expressios for the momets ad for the momet geeratig fuctio. The estimatio of parameters is approached by the method of maximum likelihood. We cosider the likelihood ratio statistic to compare the model with its baselie model. A applicatio to real data show that the ew distributio ca be used adequately to provide better fits tha the Iverse Weibull distributio, Iverse Expoetial Geometric distributio, Iverse Expoetial distributio ad Iverse Rayleigh distributio. Refereces [1] Adamidis K, Dimitrakopoulou, T, Loukas S. O a geeralizatio of the expoetial-geometric distributio. Statist. ad Probab. Lett., 73(005: [] Adamidis K, Loukas S. A lifetime distributio with decreasig failure rate. Statist. ad Probab. Lett. 39(1998: [3] AL-Hussaii E. K., Sulta K. S. Reliability ad hazard based o fiite mixture models. I Hadbook of Statistics, N. Balakrisha ad C. R. Rao, Eds., Elsevier, Amsterdam, The Netherlads, 0(001: [4] Barreto-S.W., De Morais A.L., Cordeiro G.M. The Weibull-Geometric Distributio. Joural of Statistical Computatios ad Simulatios, 81(5 (011: [5] David W. Hosmer, Staley Lemeshow. Applied Survival Aalysis, Wiley Itersciece, (1999. [6] Drapella A. ComplemetaryWeibull distributio: ukow or just forgotte. Quality ad Reliability Egieerig Iteratioal, 9(1993: [7] Erto P., Rapoe M. No-iformative ad practical Bayesia cofidece bouds for reliable life i the Weibull model. Reliability Egieerig, 7(3 (1984: [8] Everitt B. S., D. J. Had. Fiite Mixture Distributios. Chapma & Hall, Lodo, UK, (1981. [9] Johso R., Kotz S., ad Balakrisha N.Cotiuous Uivariate Distributios.Wiley-Itersciece, New York, NY, USA, d editio, (1995. [10] Kacha Jai, Neetu Sigla ad Suresh Kumar Sharma. The Geeralized Iverse Geeralized Weibull Distributios ad its properties. Joural of Probability, Article ID : (014. [11] Kha Adil H., Ja T. R. Estimatio of Stress-Stregth Reliability Usig Fiite Mixture of Expoetial ad Gamma Distributios. Iteratioal Coferece oadvaces i Computers, Commuicatio ad Electroic Egieerig, ISBN: , 1(015: [1] Kha Adil H., Ja T. R. Estimatio of Stress-Stregth Reliability Model Usig Fiite Mixture of Two Parameter Lidley Distributios. Joural of Statistics Applicatios & Probability, 4(1 (015: [13] Kus C. A ew lifetime distributio. Computat. Statist. Data Aalysis, 5(1(007:

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