Iteratioal Joural of Probability Statistics 5, 4(): 65-75 DOI:.59/j.ijps.54. Akash Distributio Its Applicatios Rama Shaker Departmet of Statistics, Eritrea Istitute of Techology, Asmara, Eritrea Abstract A ew oe parameter lifetime distributio amed Akash distributio for modelig lifetime data has bee itroduced. Some importat mathematical properties of the proposed distributio icludig its shape, momets, skewess, kurtosis, hazard rate fuctio, mea residual life fuctio, stochastic orderig, mea deviatios, order statistics, Boferroi Lorez curves, Reyi etropy measure, stress-stregth reliability have bee discussed. The coditio uder which Akash distributio is over-dispersed, eui-dispersed, uder-dispersed are preseted alog with the coditios uder which epoetial Lidley distributios are over-dispersed, eui-dispersed uder-dispersed. The estimatio of its parameter has bee discussed usig maimum likelihood estimatio method of momets. The usefuless the applicability of the proposed distributio have bee discussed illustrated with two real lifetime data sets from medical sciece egieerig. Keywords Lifetime distributio, omets, Hazard rate fuctio, ea residual life fuctio, ea deviatios, Order statistics, Estimatio of parameter, Goodess of fit. Itroductio The modelig aalyzig lifetime data are crucial i may applied scieces icludig medicie, egieerig, isurace fiace, amogst others. There are a umber of cotiuous distributios for modelig lifetime data such as epoetial, Lidley, gamma, logormal, Weibull their geeralizatios. The epoetial, Lidley the Weibull distributios are more popular tha the gamma the logormal distributios because the survival fuctios of the gamma the logormal distributios caot be epressed i closed forms both reuire umerical itegratio. Though each of epoetial Lidley distributios has oe parameter, the Lidley distributio has oe advatage over the epoetial distributio that the epoetial distributio has costat hazard rate whereas the Lidley distributio has mootoically decreasig hazard rate. The probability desity fuctio (p.d.f.) the cumulative distributio fuctio (c.d.f.) of Lidley (958) distributio are give by f ( ; ) ( + e ) ; >, > + F( ; ) + e ; >, > + * Correspodig author: shakerrama9@gmail.com (Rama Shaker) Published olie at http://joural.sapub.org/ijps Copyright 5 Scietific & Academic Publishig. All Rights Reserved (.) (.) The desity (.) is a two-compoet miture of a epoetial distributio havig scale parameter a gamma distributio havig shape parameter scale parameter with their miig proportios + respectively. A detailed study about its various + mathematical properties, estimatio of parameter applicatio showig the superiority of Lidley distributio over epoetial distributio for the waitig times before service of the bak customers has bee doe by Ghitay et al (8). The Lidley distributio has bee geeralized, eteded, modified its detailed applicatios i reliability other fields of kowledge by differet researchers icludig Hussai (6), Zakerzadeh Dolati (9), Nadarajah et al (), Deiz Ojeda (), Bakouch et al (), Shaker ishra ( a, b), Shaker et al (), Elbatal et al (), Ghitay et al (), erovci (), Liyaage Pararai (4), Ashour Eltehiwy (4), Oluyede Yag (4), Sigh et al (4), Sharma et al (5), Shaker et al (5), Alkari (5), Pararai et al (5), Abouammoh et al (5) are some amog others. Although the Lidley distributio has bee used to model lifetime data by may researchers Hussai (6) has show that the Lidley distributio is importat for studyig stress-stregth reliability modelig, there are may situatios i the modelig of real lifetime data where the Lidley distributio may ot be suitable from a theoretical or applied poit of view. Therefore, to obtai a ew distributio which is fleible tha the Lidley distributio for modelig lifetime data i reliability i terms of its hazard rate shapes, we itroduced a ew distributio by cosiderig a two-
66 Rama Shaker: Akash Distributio Its Applicatios cosiderig a two- compoet miture of a epoetial distributio havig scale parameter a gamma distributio havig shape parameter scale parameter with their miig proportios + + respectively. The probability desity fuctio (p.d.f.) of a ew oe parameter lifetime distributio ca be itroduced as f ; + e ; >, > + (.) We would call this distributio, Akash distributio. This distributio ca be easily epressed as a miture of, with their miig epoetial ( ) gamma proportios + respectively. We have + where (, ) + ( ) f pg p g e p, g e, g + The correspodig cumulative distributio fuctio (c.d.f.) of (.) is give by F ( ) + + + e ; >, > (.4) The graphs of the p.d.f. the c.d.f. of Lidley Akash distributios for differet values of are show i figures.. Figure. Graphs of the pdf of Lidley Akash distributios for differet values of parameter. Left h side graphs are for Lidley distributio right h sides graph are for Akash distributio
Iteratioal Joural of Probability Statistics 5, 4(): 65-75 67 Figure. Graphs of the cdf of Lidley Akash distributio for differet values of parameter. Left h side graphs are for Lidley distributio right h side graphs are for Akash distributio. omets Related easures The r the momet about origi of Akash distributio (.) has bee obtaied as r! + r+ r+ r ; r,,,... r ( + ) so the first four momets about origi as + 6 ( + ) ( + ), ( + ) 6, ( + ) ( + ) ( + ) ( + ) 4 4 4 Thus the momets about mea of Akash distributio are obtaied as 4 + 6 + ( + ), 6 4 ( + + + ) ( + ) 8 6 4 ( + + + + ) 4 ( + ) 6 4 8 48 576 4 4 4 The coefficiet of variatio ( CV. ) β, coefficiet of kurtosis skewess ( ), coefficiet of β ide of dispersio ( ) of Akash distributio are thus obtaied as 4 σ + 6 + CV. + 6 6 4 ( + + 6 + 4) 4 ( + 6 + ) 8 6 4 ( + + + + ) 4 ( + 6 + ) β / / 8 48 576 4 4 β
68 Rama Shaker: Akash Distributio Its Applicatios 4 σ + 6 + + + ( 6) It ca be easily show that Akash distributio is over-dispersed ( < σ ), eui-dispersed ( σ ) uder-dispers ( σ ) > for.5546 < >. It would be recalled that Lidley distributio is over-dispersed ( < σ ), eui-dispersed ( σ ) uder-dispers ( > σ ) for < ( ) >.786487 while epoetial distributio is over-dispersed ( < σ ), eui-dispersed ( σ ) uder-dispersed ( > σ ) for < >.. omet Geeratig Fuctio The momet geeratig fuctio of Akash distributio (.) are obtaied as ( t ) X t e ( + ) d + + + t t ( ) k k t k + t + k k k + k + k + k + t + k 4. Hazard Rate Fuctio ea Residual Life Fuctio Let X be a cotiuous rom variable with p.d.f. f ( ) c.d.f. F( ). The hazard rate fuctio (also kow as the failure rate fuctio) the mea residual life fuctio of X are respectively defied as h ( < + > ) F P X X f lim (4.) m E X X > F ( t) dt F (4.) The hazard rate fuctio, h( ) the mea residual life fuctio, m( ) of Lidley distributio are give by Ad h m ( + ) ( + ) + + + ( + + ) (4.) (4.4) The correspodig hazard rate fuctio, h( ) the mea residual life fuctio, m( ) of the Akash distributio are give by h ( + ) ( + ) + ( + ) (4.5) ( + ) + ( + ) + t t t m e dt ( ) ( ) + + + e + It ca be easily verified that h( ) f ( ) + m( ) ( + ) h( ) m( ) that h( ) is a icreasig fuctio of,, whereas + 4 + ( + 6) ( + ) + ( + ) (4.6) + 6. It is also obvious from the graphs of m is a decreasig fuctio of,. The hazard rate fuctio the mea residual life fuctio of the Akash distributio show its fleibility over Lidley distributio epoetial distributio. The graphs of the hazard rate fuctio mea residual life fuctio of Lidley Akash distributios are show i figures 4.
Iteratioal Joural of Probability Statistics 5, 4(): 65-75 69 Figure. Graphs of hazard rate fuctio of Lidley Akash distributios for differet values of parameter. Left h side graphs are for Lidley distributio right h side graphs are for Akash distributio Figure 4. Graphs of mea residual life fuctio of Lidley Akash distributios for differet values of parameter. Left h side graphs are for Lidley distributio right h side graphs are for Akash distributio
7 Rama Shaker: Akash Distributio Its Applicatios 5. Stochastic Orderigs Stochastic orderig of positive cotiuous rom variables is a importat tool for judgig their comparative behavior. A rom variable X is said to be smaller tha a rom variable Y i the (i) stochastic order ( X st Y) if FX FY for all (ii) hazard rate order ( X hr Y) if hx hy for all (iii) mea residual life order ( X mrl Y) if mx my for all (iv) likelihood ratio order fx ( X lr Y) if fy decreases i. The followig results due to Shaked Shathikumar (994) are well kow for establishig stochastic orderig of distributios X lr Y X hr Y X mrl Y (5.) X st Y The Akash distributio is ordered with respect to the strogest likelihood ratio orderig as show i the followig theorem: Y Theorem: Let X Akash distributo. If, the X lr Y hece X hr Y, X mrl Y X st Y. Proof: We have Akash distributio X Now log ( + ) ( + ) fx ( ) e fy ( + ) ( + ) ; > fx log ( f ) Y d f This gives log X ( f ) Y d d fx Thus for, log <. This meas that fy ( d ) lr Y hece X hr Y, X mrl Y X st Y. 6. ea Deviatios The amout of scatter i a populatio is measured to some etet by the totality of deviatios usually from mea media. These are kow as the mea deviatio about the mea the mea deviatio about the media defied by δ ( X ) f d δ X f d, respectively, where E( X) edia ( X) The measures δ δ X usig the relatioships X ( ) + ( ). ca be calculated δ X f d f d δ F f d F + f d F + f d F f d (6.) ( ) + ( ) X f d f d F f d F + f d + f d f d (6.) Usig p.d.f. (.) epressio for the mea of Akash distributio, we get f f d { ( + ) + ( + ) + 6( + ) ( + ) d { ( + ) + ( + ) + 6( + ) ( + ) e e (6.) (6.4) Usig epressios from (6.), (6.), (6.), (6.4), the mea deviatio about mea, δ ( X ) the mea deviatio about media, δ ( X ) of Akash distributio are obtaied as δ ( X ) { ( + ) + ( + ) ( + ) e (6.5)
Iteratioal Joural of Probability Statistics 5, 4(): 65-75 7 ( X ) { ( + ) + ( + ) + ( + ) ( + ) 6 e δ (6.6) 7. Order Statistics Let X, X,..., X be a rom sample of size from Akash distributio (.). Let X < X( ) <... < X( ) deote the correspodig order statistics. The p.d.f. the c.d.f. of the k th order statistic, say Y X( k ) are give by respectively, for k,,,...,.! k fy ( y) F y F y f y!! ( k ) ( k) k! k l ( k ) ( k)!! l FY y F y F y j k j j { j j j kl j l k { l k+ l F ( y) f ( y) j l j+ l F ( y), Thus, the p.d.f. the c.d.f of k th order statistics are give by! + e k k l + + + fy ( y) e ( + )( k! ) ( k)! l l + 8. Boferroi Lorez Curves ( + ) + ( + ) j j l FY ( y) ( ) e j kl j l + The Boferroi Lorez curves (Boferroi, 9) Boferroi Gii idices have applicatios ot oly i ecoomics to study icome poverty, but also i other fields like reliability, demography, isurace medicie. The Boferroi Lorez curves are defied as B( p) f d f d f d f d p p p j+ l k+ l (8.) L( p) f d f d f d f d respectively or euivaletly (8.)
7 Rama Shaker: Akash Distributio Its Applicatios respectively, where E( X) F ( p). The Boferroi Gii idices are thus defied as respectively. Usig p.d.f. (.), we get Now usig euatio (8.7) i (8.) (8.), we get p B p F d p (8.) p L p F d (8.4) B B p dp (8.5) G L p dp (8.6) { ( ) 6( ) ( + ) + + + + + e f d (8.7) B p L( p) { ( + ) + ( + ) + 6( + ) e p + 6 { ( + ) + ( + ) + 6( + ) e + 6 Now usig euatios (8.8) (8.9) i (8.5) (8.6), the Boferroi Gii idices are obtaied as B { ( + ) + ( + ) + 6( + ) e + 6 { ( + ) + ( + ) + ( + ) 6 e G + + 6 (8.8) (8.9) (8.) (8.) 9. Reyi Etropy A etropy of a rom variable X is a measure of variatio of ucertaity. A popular etropy measure is Reyi etropy (96). If X is a cotiuous rom variable havig probability desity fuctio f (.), the Reyi etropy is defied as where >. ( ) TR f d log { Thus, the Reyi etropy for the Akash distributio (.) is obtaied as log ( ) TR + e d + log j e d ( ) j j + log j e d ( ) j j + log + j j ( ) j+ e d
Iteratioal Joural of Probability Statistics 5, 4(): 65-75 7 log j + ( ) j log j + j ( ) j. Stress-stregth Reliability ( j ) j+ ( ) Γ + ( j ) j+ ( ) Γ + The stress- stregth reliability describes the life of a compoet which has rom stregth X that is subjected to a rom stress Y. Whe the stress applied to it eceeds the stregth, the compoet fails istatly the compoet will fuctio satisfactorily till X > Y. R PY< X is a measure of compoet Therefore, reliability i statistical literature it is kow as stress-stregth parameter. It has wide applicatios i almost all areas of kowledge especially i egieerig such as structures, deterioratio of rocket motors, static fatigue of ceramic compoets, agig of cocrete pressure vessels etc. Let X Y be idepedet stregth stress rom variables havig Akash distributio (.) with parameter respectively. The the stress-stregth reliability R is obtaied as R PY< X PY< X X f d ( ; ) ( ; ) f F d X + 4 + ( + 4) ( ) ( ) 5 ( + )( + )( + ) 6 5 4 4 + + + + + 4 + + + +.. Estimatio of Parameter.. aimum Likelihood Estimatio,,,..., be a rom sample from Akash distributio (.). The likelihood fuctio, L of (.) is give by Let L ( + i ) e + i so its log likelihood fuctio is thus obtaied as Now log log log L + ( + i ) + i dlog L d + where is the sample mea. The maimum likelihood estimate, d solutio of the euatio log L so it ca be d obtaied by solvig the followig o-liear euatio ˆ of is the + 6 (..).. ethod of omet (o) Estimatio Euatig the populatio mea of the Akash distributio to the correspodig sample mea, the method of momet (O) estimate,, of is the same as give by euatio (..).. Applicatios of Akash Distributio The Akash distributio has bee fitted to a umber of data sets from medical sciece egieerig. I this sectio, we preset the fittig of Akash distributio to two real data sets compare its goodess of fit with the oe parameter epoetial Lidley distributios. Data set : The first data set represets the lifetime s data relatig to relief times (i miutes) of patiets receivig a aalgesic reported by Gross Clark (975, P. 5). The data are as follows:.,.4,.,.7,.9,.8,.6,.,.7,.7, 4.,.8,.5,.,.4,.,.7,.,.6,. Data set : The secod data set is the stregth data of glass of the aircraft widow reported by Fuller et al (994) 8.8,.8,.657,.,., 4.5, 4., 5.5, 5.5, 5.8, 6.69, 6.77, 6.78, 7.5, 7.67, 9.9,.,.,.7,.76,.89, 4.76, 5.75, 5.9, 6.98, 7.8, 7.9, 9.58, 44.45, 45.9, 45.8 I order to compare distributios, l L, AIC (Akaike Iformatio Criterio), AICC (Akaike Iformatio Criterio Corrected), BIC (Bayesia Iformatio Criterio), K-S Statistics (Kolmogorov-Smirov Statistics) for two real data sets have bee computed. The formulae for computig AIC, AICC, BIC, K-S Statistics are as follows: k( k + ) AIC l L + k, AICC AIC +, ( k ) BIC l L + k l
74 Rama Shaker: Akash Distributio Its Applicatios Table. LE s, - l L, AIC, AICC, BIC, K-S Statistics of the fitted distributios of data sets odel Parameter estimate l L AIC AICC BIC K-S statistic Akash.569 59.5 6.7 6.7 6.5.5 Data Lidley.86 6.5 6.5 6.7 6.5.4 Epoetial.56 65.7 67.7 67.9 68.7.895 Akash.976 4.7 4.7 4.8 44..664 Data Lidley.699 54. 56. 56. 57.4. Epoetial.46 74.5 76.7 76.7 77.9.464 D Sup F F, where k the umber of parameters, the sample size F is the empirical distributio fuctio. The best distributio correspods to lower l L, AIC, AICC, BIC, K-S statistics. It ca be easily see from above table that the Akash distributio is better tha the Lidley epoetial distributios for modelig life time data thus Akash distributio should be preferred to epoetial distributio Lidley distributios for modelig lifetime data-sets.. Coclusios A oe parameter lifetime distributio amed, Akash distributio has bee proposed. Its mathematical properties icludig shape, momets, skewess, kurtosis, hazard rate fuctio, mea residual life fuctio, stochastic orderig, mea deviatios, order statistics have bee discussed. The coditio uder which Akash distributio is over-dispersed, eui-dispersed, uder-dispersed are preseted alog with the coditios uder which epoetial Lidley distributios are over-dispersed, eui-dispersed uder-dispersed. Further, epressios for Boferroi Lorez curves, Reyi etropy measure stress-stregth reliability of the proposed distributio have bee derived. The method of momets the method of maimum likelihood estimatio have also bee discussed for estimatig its parameter. Fially, the goodess of fit test usig K-S Statistics (Kolmogorov-Smirov Statistics) for two real lifetime data- sets have bee preseted to illustrate its applicability superiority over epoetial Lidley distributios. ACKNOWLEDGEENTS The author is thakful to the aoymous reviewer for some useful commets. REFERENCES [] Abouammoh, A.., Alshagiti, A.. Ragab, I.E. (5): A ew geeralized Lidley distributio, Joural of Statistical Computatio Simulatio, preprithttp://d.doi.org/. 8/ 949655.4.995. [] Alkari, S. (5): Eteded Power Lidley distributio-a ew Statistical model for o- mootoe survival data, Europea joural of statistics probability, (), 9 4. [] Ashour, S. Eltehiwy,. (4): Epoetiated Power Lidley distributio, Joural of Advaced Research, preprit http://d.doi.org/.6/ j.jare. 4.8.5. [4] Bakouch, H.S., Al-Zaharai, B. Al-Shomrai, A., archi, V. Louzad, F. (): A eteded Lidley distributio, Joural of the Korea Statistical Society, 4, 75 85. [5] Boferroi, C.E. (9): Elemeti di Statistca geerale, Seeber, Fireze. [6] Deiz, E. Ojeda, E. (): The discrete Lidley distributio-properties Applicatios, Joural of Statistical Computatio Simulatio, 8, 45 46. [7] Elbatal, I., erovi, F. Elgarhy,. (): A ew geeralized Lidley distributio, athematical theory odelig, (), -47. [8] Fuller, E.J., Friema, S., Qui, J., Qui, G., Carter, W. (994): Fracture mechaics approach to the desig of glass aircraft widows: A case study, SPIE Proc 86, 49-4. [9] Ghitay,.E., Atieh, B. Nadarajah, S. (8): Lidley distributio its Applicatio, athematics Computig Simulatio, 78, 49 56. [] Ghitay,., Al-utairi, D., Balakrisha, N. Al-Eezi, I. (): Power Lidley distributio associated iferece, Computatioal Statistics Data Aalysis, 64,. [] Gross, A.J. Clark, V.A. (975): Survival Distributios: Reliability Applicatios i the Biometrical Scieces, Joh Wiley, New York. [] Hussai, E. (6): The o-liear fuctios of Order Statistics Their Properties i selected probability models, Ph.D thesis, Departmet of Statistics, Uiversity of Karachi, Pakista. [] Lidley, D.V. (958): Fiducial distributios Bayes theorem, Joural of the Royal Statistical Society, Series B,, - 7.
Iteratioal Joural of Probability Statistics 5, 4(): 65-75 75 [4] Liyaage, G.W. Pararai.. (4): A geeralized Power Lidley distributio with applicatios, Asia joural of athematics Applicatios,. [5] erovci, F. (): Trasmuted Lidley distributio, Iteratioal Joural of Ope Problems i Computer Sciece athematics, 6, 6 7. [6] Nadarajah, S., Bakouch, H.S. Tahmasbi, R. (): A geeralized Lidley distributio, Sakhya Series B, 7, 59. [7] Oluyede, B.O. Yag, T. (4): A ew class of geeralized Lidley distributio with applicatios, Joural of Statistical Computatio Simulatio, 85 (), 7. [8] Pararai,., Liyaage, G.W. Oluyede, B.O. (5): A ew class of geeralized Power Lidley distributio with applicatios to lifetime data, Theoretical athematics &Applicatios, 5(), 5 96. [9] Reyi, A. (96): O measures of etropy iformatio, i proceedigs of the 4 th berkeley symposium o athematical Statistics Probability,, 547 56, Berkeley, uiversity of Califoria press. [] Shaked,. Shathikumar, J.G. (994): Stochastic Orders Their Applicatios, Academic Press, New York. [] Shaker, R. ishra, A. ( a): A two-parameter Lidley distributio, Statistics i Trasitio-ew series, 4 (), 45-56. [] Shaker, R. ishra, A. ( b): A uasi Lidley distributio, Africa Joural of athematics Computer Sciece Research, 6(4), 64 7. [] Shaker, R., Sharma, S. Shaker, R. (): A two-parameter Lidley distributio for modelig waitig survival times data, Applied athematics, 4, 6 68. [4] Shaker, R., Hagos, F, Sujatha, S. (5): O modelig of Lifetimes data usig epoetial Lidley distributios, Biometrics & Biostatistics Iteratioal Joural, (5), -9. [5] Sharma, V., Sigh, S., Sigh, U. Agiwal, V. (5): The iverse Lidley distributio- A stress-stregth reliability model with applicatios to head eck cacer data, Joural of Idustrial &Productio Egieerig, (), 6 7. [6] Sigh, S.K., Sigh, U. Sharma, V.K. (4): The Trucated Lidley distributio- iferece Applicatio, Joural of Statistics Applicatios& Probability, (), 9 8. [7] Zakerzadeh, H. Dolati, A. (9): Geeralized Lidley distributio, Joural of athematical etesio, (), 5.