The New Probability Distribution: An Aspect to a Life Time Distribution

Transcription

1 Math. Sci. Lett. 6, No. 1, ) 35 Mathematical Sciece Letters A Iteratioal Joural The New Probability Distributio: A Aspect to a Life Time Distributio Diesh Kumar 1,, Umesh Sigh 1, Sajay Kumar Sigh 1 ad Souradip Mukherjee 1 Departmet of Statistics ad DST- CIMS, Baaras Hidu Uiversity, Uttar Pradesh- 1005, Idia Departmet of Statistics, The Uiversity of Burdwa, West Begal , Idia Received: 0 Aug. 015, Revised: 8 Jul. 016, Accepted: 10 Jul. 016 Published olie: 1 Ja. 017 Abstract: I the preset article, we have proposed a method to costruct a ew distributio o the basis of ay two baselie idepedet cotiuous distributios o the same spectrum ad studied some statistical properties of the proposed distributio. Further, for applicatio poit of view, we have derived it for Weibull distributio as a baselie distributio ad proved its applicatio i compariso to the other well kow distributios like gamma, Weibull ad expoetiated expoetial distributio i terms of fittig a real data through AIC, BIC test ad log likelihood LL) criterio of goodess of fit. Keywords: Life Time Distributio, Reliability Aalysis ad Maximum Likelihood Estimatio. 1 Itroductio I statistical literature, several life time distributios are available like Weibull, gamma, expoetiated expoetial ad may more to aalyze the data o medical field, egieerig ad fiace sectors etc. No doubt, modelig ad aalyzig the life time data are crucial. The quality of the outputs of the statistical aalysis depeds heavily o the assumed life time models or distributios see, Merovci [1]). Recetly applicatios from evirometal scieces, biomediacal scieces, fiace ad egieerig sectors ad may others shows that the available data sets followig the classical distributios are more ofte the exceptio rather tha the reality see, Aryal [3]). Therefore, it becomes very importat to fid more earer distributio i compariso to the classical distributios to get more accurate results. To solve such problems may developmet has take place. For example, Gupta et al. [1] proposed the cdf of the ew distributio as F 1 t) = [Ft)] α, for all α > 0, where Ft) is the cdf of ay baselie distributio. May works has bee published o the basis of it, usig differet classical baselie distributios for example, Gupta ad Kudu [10], Seeoi et al. [19], Elbatal ad Muhammed [18] etc.). Later o quadratic rak trasmuted map QRTM) has bee proposed by Shaw ad Buckley [8]. Accordig to the QRTM, the cdf F x) of the ew distributio correspodig to the base lie distributio havig cdf Fx), is give by, F x) = 1+)Fx) F x) for all 1. May researchers have used QRTM to develop ew life time distributio see, Kha ad Kig [5], Aryal ad Tsokos [6], Kha et al. [] etc.). I the preset article, we propose the cdf of ew distributio deotig it by Gx)) by the use of ay two may be the same) cdfs F 1 x) ad F x) of baselie cotiuous distributios) with commo spectrum; by the trasformatio, Gx)= F 1x)+F x) 1+F 1 x) 1) We will call trasformatio 1) as M trasformatio for its frequetly used purpose i our work or elsewhere. Theorem: Gx) possesses the property of a cdf. Proof: 1.F 1 x) ad F x) are the cdfs of ay two idepedet cotiuous radom variables with commo spectrum, which implies that F 1 x), F x) ad hece 1+F 1 x) Correspodig author diesh.ra77@gmail.com c 017 NSP

2 36 D. Kumar et al.: The New probability distributio are cotiuous fuctios see, Rohatgi ad Saleh [7]). Agai, 1+F 1 x) 0 x. It proves that Gx) is a cotiuous fuctio of x see, Mapa [11])..0 F 1 x) 1,0 F x) 1 x 0 Gx) 1 x R 3.G x)= f 1x)[1 F x)]+ f x)[1+f 1 x)] 0 x R [1+F 1 x)] 4.G )=0 & G )=1. Thus, Gx) satisfyig the sufficiet coditios for a fuctio to be a cdf ad hece so see, Rohatgi ad Saleh [7]) A particular case If we chose, F 1 x) = F x) = Fx), the particularly M trasformatio 1) reduces to the followig form, Gx)= Fx) 1+Fx) The pdf correspodig to cdf ) is give by, gx)= ) fx) 1+Fx)) 3) ad the correspodig hazard rate fuctio is give by, hx)= fx) 1 F x).1 Order Statistic from pdf 3) Let, X 1), X ),..., X ) be the order statistics of a radom sample X = X 1, X,..., X ) of size from M.)-distributio havig pdf 3), the the pdf of the distributio of first order statistic X 1) is give by see, Gu, Gupta ad Dasgupta [9]) 4) g 1 X 1) )= gx 1) ) [1 Gx 1) )] 1 [ ] 1 fx 1) ) 1 Fx1) ) = [1+Fx 1) )] 5) 1+Fx 1) ) Similarly, the pdf of the distributio of largest order statistic X) is give by, g X ) )= gx ) )) [Gx ) )] 1 [ ] 1 fx ) ) Fx) ) = [1+Fx ) )] 6) 1+Fx ) ) I geeral, the pdf of the distributio of r th order statistic x r) is give by, g X r)= =! r 1)! r)! gx r)) [ GX r) ) ] r 1[ 1 GXr) ) ] r! r 1)! r)! [ ] FXr) ) r 1 [ 1 FX ] r [ r)) 1+FX r) ) 1+FX r) ) fx r) ) 1+Fx r) )) 7) ] 3 M trasformatio of Weibull distributio usig ) The most flexible life distributio is Weibull distributio with its pdf give by, x 1 fx)= e x )k ; x> 0 8) ad the cdf of Weibull distributio havig pdf 8) is give by, Fx)=1 e x )k 9) where k> 0 ad > 0 are its shape parameter ad scale parameters respectively. For more applicatio related to the Weibull distributio, readers may refer to Mudholkar ad Srivastava [15], Mudholkar et al. [14] etc. Mudholkar ad Hutso [13] have modeled various failure time data sets with the proposed model with Weibull as the baselie distributio. Now, usig 9) i ) ad 8) i 3), we will get the cdf ad pdf of the M trasformatio of Weibull distributio with parameters k ad as 1 e x)k) Gx)= 10) e x )k ad gx)= k x 1 ) e ) x k e x) 11) respectively. We ame M trasformatio ) of Weibull distributio with the parameters k ad as M W k, ) distributio for frequetly use purpose i the preset article or elsewhere. Also, the hazard rate fuctio of M W k, ) distributio havig pdf 11) is obtaied as follows hx)= k x 1 ) e ) x k 1 1 e x )k) 1) The plots of gx) ad hx) are show i Figures 1 ad respectively, for differet combiatios of the parameters k, ). 3.1 Raw Momets ad Characteristic Fuctio: Raw Momets The r th momet about origi µ r raw momet) of M W k, ) distributio havig pdf 11) is obtaied as follows, µ r = k k 0 x k+r 1 e x )k e x )k) dx 13) c 017 NSP

3 Math. Sci. Lett. 6, No. 1, ) / 37 Above itegral is ot solvable i ice closed form, so we propose to use 19 pt. Gauss Lagurre quadrature formula GLQF) for its umerical solutio or some other techique, like Mote-Carlo simulatio etc. may be used. Characteristic Fuctio We have derived the characteristic fuctio of M W k, )- distributio havig pdf 11) ad the same is obtaied as follows, φ X t)= k k 0 x k 1 e itx x e x) where t Ris the dummy parameter. 3. Radom sample Geeratio: dx 14) Usig the method of iversio see, Merovci [1]) we ca geerate radom umbers from the M W k,). 1 e x )k) e = U x )k Solvig above equatio for x i terms of U, we get [ )] 1 U k x= l 15) U where U follow U0,1)- distributio. Oe ca use 15) to geerate radom samples from M W k,) whe parameters ad k are kow. ad hece the log- likelihood fuctio is give by: xi l L= C k l l e x i )k) 17) where C is a costat. It is obvious that the log-likelihood equatios for estimatig k ad are ot easily solvable simultaeously for k ad ; therefore we propose to use some umerical iteratio method for gettig their umerical solutios. 3.4 Order Statistics: Refer to sub-sectio 3.1, the pdf of the distributio of first order statistic X 1) of a radom sample of size from M W k,)- distributio is give by: g 1 X 1) )= k x k 1 1) k x1) e ) x1) ) +1 18) e Similarly, pdf of the distributio of the largest order statistics X) is give by, g X ) )= k x) 1 e ) 1 e x) x1) ) +1 19) k e 3.3 Maximum likelihood estimatio Let the lifes of idetical items put o a life testig experimet be X = X 1, X,..., X )where each X i follow M W k,) - distributio. The the likelihood fuctio for X is give by see, Potdar ad Shirke [4]), L= = gx i ) ) k xi 1 e x i )k e x i )k) k) [ x i ] k 1) e x i = e x) 16) k Fially, the pdf of the distributio of the r th order statistic X r) is give by, )! g r X r) )= r! r)! r k r x k 1 r) xr) r+1 xr) r 1 e 1 e xr) +1 k e 4 A particular case of M W k,) whe k= 1 I particular, if k=1, the M W k,)- distributio shall be treated as M E ); as the M trasformatio through) of Exp)- distributio. By the use of4), we get the hazard rate fuctio of M E )- distributio as follows, 0) hx)= ) 1) e x c 017 NSP

4 38 D. Kumar et al.: The New probability distributio It may be recalled that the hazard rate fuctio is costat for Exp )- distributio. Now, we wat to show the shape of hazard rate fuctio of M E )- distributio. Differetiatig 1) partially w.r.t. x, we get h e x x)= ) e x 0 x 0, 0 ) Table 1: LL, AIC ad BIC values of Gamma, Weibull, EED ad M W k,)- distributio for the data of the failure times of the air coditioig system of a air plae Distributios ˆ ˆk LL AIC BIC M W k,) EED Weibull Gamma Thus, we ca say that hazard rate fuctio of M E )- distributio is always decreasig. The pdfs ad hazard rate fuctios of M E )- distributio for differet values of are show i the Figures 3 ad 4 respectively. 5 Real Data Applicatio: Here, we have show the applicability of M W k,)- distributio o the real life data by showig that it is better fit i compariso to some well kow ad exploited existig distributios. For the purpose, we cosidered the followig data of the failure times of the air coditioig system of a air plae see, Lihart ad Zucchii [17]). The cosidered data set is used by may authors such as Gupta ad Kudu [10], ad Sigh et al. [1] etc. Fig. 1: Plots of pdf of M W k,)-distributio 3,61,87,7,10,14,6,47,5,71,46,1,4,0,5,1, 10,11,3,14,71,11,14,11,16,90,1,16,5,95. Gupta ad Kudu [10] cosidered above data set ad showed that Weibull distributio fits the best i terms of likelihood ad i terms of Chi- square as compared to EED ad gamma distributio ad they coclude that i certai circumstaces Weibull distributio might work better tha EED or gamma distributio. I our case, we have derived AIC Akaike iformatio criterio), BIC Bayesia iformatio criterio) ad LL criterio values of above data set for M W k,)- distributio, Weibull distributio, EED ad gamma distributio AIC= k m ad BIC = k l) m where k deotes the umber of ukow parameters i the model, is the sample size ad the maximized value of the log-likelihood fuctioll) uder the cosidered model is m. The results are show i Table 1. AIC, BIC ad LL criterio of fittig is used by several author authors, for example Gupta ad Kudu [10], ad Sigh et al. [0] etc. Fig. : Plots of hazard rate fuctio of M W k,)-distributio M W k,))- distributio might work better tha Webull distributio, gamma distributio ad EED. From Table 1, it is quite clear that M W k,)- distributio domiates Weibull distributio, gamma distributio ad EED i terms of AIC, BIC ad LL test values ad we may, therefore coclude that i certai circumstaces c 017 NSP

5 Math. Sci. Lett. 6, No. 1, ) / 39 Ackowledgemet The authors are grateful to the Editor ad the aoymous referees for careful checkig of the details ad for helpful commets that led to improvemet of the paper. We devote the preset article to the mother Mrs. Mithu Mukherjee of Souradip Mukherjee. Refereces Fig. 3: Plots of pdfs of M E )- distributio for differet values of Fig. 4: Plots of Hazard rate fuctio ofm E )- distributio for differet values of 6 Coclusio: I the preset article, we have proposed a ew distributio with the help of M trasformatio that uses ay two arbitrary cotiuous distributios, with commo spectrum, as baselie distributios). As a applicatio part, we have derived it by assumig Weibull distributio as a baselie distributio. The ew distributio, thus obtaied is much more flexible as its hazard rate fuctio covers differet shapes. By cosiderig a real data, we proved its applicability i compariso to other existig exploited lifetime distributios; like gamma, Weibull ad Expoetiated Expoetial distributios. Thus, we recommed the use of M trasformatio to get ew life time distributios. [1] Merovci, F. 014): Trasmuted Geeralized Rayleigh Distributio, Joural of Statistics Applicatio ad Probability, 3 1), pp [] Kha, et al. 014): Characterisatios of the trasmuted iverse Weibull distributio, ANZIAM J. 55 EMAC-013), C197 C17. [3] Aryal, G. R. 013): rasmuted Log-Logistic Distributio, Joural of Statistics Applicatio ad Probability., 1), pp [4] Potdar, K. G. ad Shirke, D. T. 013): Iferece for the parameters of geeralized iverted family of distributios. Prob. Stat. Forum, Vol. 6, pp [5] Kha, M. S. ad Kig, R. 013): Trasmuted modifed Weibull distributio: A eeralizatio of the modifed Weibull probability distributio, Europe. J. of Pure Appl. Math. 6 1), pp [6] Aryal, G. R. ad Tsokos, C. P. 011): Trasmuted Weibull Distributio: A Geeralizatio of the Weibull Probability Distributio, Europea joural of pure ad applied mathematics, 4 ), pp [7] Rohatgi, V. K. ad Saleh, A. K. Md. E. 010): A Itroductio to Probability ad Statistics, Secod editio, Joh Wiley ad Sos, Idia. [8] Shaw, W. T. ad Buckley, I. R. C. 009): The alchemy of probability distributios: beyod Gram-Charlier expasios, ad a skew-kurtotic-ormal distributio from a rak trasmutatio map, arxiv preprit arxiv: [9] Gu et al. 008): Fudametal of Statistics, Vol., World Press Pvt. Ltd. [10] Gupta, R. D. ad Kudu, D. 001): Expoetiated Expoetial Family: A Alterative to Gamma ad Weibull Distributios, Biometrical Joural, 43 1), pp [11] Mapa, S. K. 000): Itroductio to Real Aalysis, Secod Editio, Sarat Book House Pvt. Ltd. [1] Gupta et al. 1998): Modelig failure time data by Lehma alteratives, Commuicatios i Statistics - Theory ad Methods, 7 4), pp [13] Mudholkar, G. S. ad Hutso, A. D. 1996)): The expoetiated Weibull family: some properties ad a flood data applicatio, Commuicatios i Statistics - Theory ad Methods, 5 1), pp [14] Mudholkar et al. 1995): The expoetiated Weibull family: a reaalysis of the bus-motor-failure data, Techometrics, 37 4), pp [15] Mudholkar, G. S. ad Srivastava, D. K. 1993): Expoetiated Weibull family for aalyzig bath tub failure data, IEEE Trasactios o Reliability, 4 ), pp c 017 NSP

6 40 D. Kumar et al.: The New probability distributio [16] Glaser, R. E. 1980): Bathtub ad related failure rate characterizatios, Joural of the America Statistical Associatio, 75, pp [17] Lihart, H. ad Zucchii, W. 1986): Model selectio, Wiley. [18] Elbatal, I. ad Muhammed, H. Z. 014): Expoetiated geeralized iverse Weibull distributio, Applied Mathematical Scieces, 8 81), pp [19] Seeoi et al. 014): The legth-biased expoetiated iverted Weibull distributio, Iteratioal Joural of Pure ad Applied Mathematics, 9 ), pp [0] Sigh, S. K., Sigh, U. ad Kumar, M. 014): Bayesia iferece for expoetiated Pareto model with applicatio to bladder cacer remissio time, Statistics i Trasitio, 153), pp [1] Sigh, S. K., Sigh, U., Kumar, M. ad Vishwakarma, P. 014): Classical ad Bayesia Iferece for a Extesio of the Expoetial Distributio uder Progressive Type-II cesored data with biomial Removal, Joural of Statistics Applicatios ad Probability Letters, 1 3), pp stochastic model ad testig its suitability i demography is aother field of his iterest. Sajay Kumar Sigh is Professor of Statistics at Baaras Hidu Uiversity. He received the PhD degree i Statistics at Baaras Hidu Uiversity His mai area of iterest is Statistical Iferece. Presetly he is workig o Bayesia priciple i life testig ad reliability estimatio, aalyzig the demographic data ad makig projectios based o the techique. He also acts as reviewer i differet iteratioal jourals of repute. Diesh Kumar is Assistat Professor of Statistics at Baaras Hidu Uiversity. He received the Ph. D. degree i Statistics at Baaras Hidu Uiversity. He is workig o Bayesia Ifereces for lifetime models. He is tryig to establish some fruitful lifetime models that ca cover most of the realistic situatios. He also worked as reviewer i differet iteratioal jourals of repute Souradip Mukherjee is a very youg researcher. He recetly completed his M. Sc. Degree i Statistics from Burdwa Uiversity. Presetly he started research at Chemical Egieerig & Process Developmet CEPD), Chemical Laboratory, Idia ad tryig to model the disease data at populatio level. Umesh Sigh is Professor of Statistics ad coordiator of DST- Cetre for Iterdiscipliary Mathematical Sciece at Baaras Hidu Uiversity. He received the PhD degree i Statistics at Rajastha Uiversity. He is referee ad Editor of several iteratioal jourals i the frame of pure ad applied Statistics. He is the fouder Member of Idia Bayesia Group. He started research with dealig the problem of icompletely specified models. A umber of problems related to the desig of experimet, life testig ad reliability etc. were dealt. For some time he worked o the admissibility of prelimiary test procedures. After some time he was attracted to the Bayesia paradigm. At preset his mai field of iterest is Bayesia estimatio for life time models. Applicatios of Bayesia tools for developig c 017 NSP