Inverse Maxwell Distribution as a Survival Model, Genesis and Parameter Estimation

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1 Research Journal of Mahemaical and Saisical Sciences ISSN Inverse Maxwell Disribuion as a Survival Model, Genesis and Parameer Esimaion Absrac Kusum Laa Singh and R.S. Srivasava Deparmen of Mahemaics and Saisics, DDU Gorakhpur Universiy, Gorakhpur, INDIA Available online a: Received 31 s May 214, revised 7 h June 214, acceped 9 h June 214 If a random variable follows a paricular disribuion hen he disribuion of he inverse of ha random variable is called invered disribuion. Some auhors have discussed boh coninuous and discree invered disribuion and is applicaions o various disciplines eg. In social sciences o geological, engineering, environmenal and medical sciences ec. In his paper we have derived he probabiliy densiy funcion of Inverse Maxwell disribuion and sudied is properies and is suiabiliy as a survival model has been discussed by obaining is survival and hazard funcions, hese are ploed on a graph paper and heir properies have been discussed. The maximum likelihood esimaor, an momen equaion esimaor of he parameer have been obained Keywords: Maxwell disribuion, inverse disribuion, momens, maximum likelihood esimaion, momen equaion esimaion of parameer, survival and hazard funcions ec. Inroducion Le X be a random variable having he Maxwell disribuion, wih pdf given by: f (x;θ) = x >, θ > (1) Where θ is he scale parameer. The Maxwell disribuion, he raw momens are given by = Ґ( ) (2) The mean and variance are given as =()=2 ; (3) and =()= () If X is a random variable hen Y = 1/X, is described as invered random variable. There are many invered disribuions discussed in lieraure. Sephan 1 is one of he earlier auhors who discuss negaive momens of binomial and hyper geomeric disribuions. Grab and Savage 2 have consruced he ables for negaive momens of he binomial and Poisson disribuions. Mendenhall and Lehman 3, Govendarajulu 4,5, Tiku 6, Vijsokousku 7, Sancu 8, Skibusky 9, Kabe 1 Shahnbhag and Busawa 11, Chao and Srawderman 12, Gupa 13,14, Lepage 15, Kumar and Cousul 16, Cressie 17 e al, Cressie 18 e al, Ahmad and Sheikh 19,19,2,21 Jones 22, are some of he early auhors who have discussed various aspecs of invered disribuions or negaive momens. More recenly, Roohi 23, Jones and Zhiglijavsky 24, Rempala 25 and Ahmad and Roohi 26,27,28,29, have also discussed boh coninuous and discree invered disribuion. (4) In his paper we have obained he p.d.f. of inverse Maxwell disribuion and obained is mean, variance, harmonic mean and mode. The suiabiliy of Inverse Maxwell disribuion as a survival model has been discussed by obaining he hazard and survival funcions. Inverse Maxwell Disribuion If X has a Maxwell disribuion hen he random variable Y= is said o follow inverse Maxwell disribuion. The pdf of inverse Maxwell disribuion may be obained using he ransformaion echnique. We ge f (y; θ) = f X (1/y; θ) y (5) where f X (.;θ) is given by (1). Now (5) becomes f (y; θ) = Here we ge, f (y; θ).e y >, θ > (6) =1. Momens: The inverse Maxwell disribuion, he r h raw momens are given by E(Y r ) = = () = Ґ( ) (7) The mean, variance, harmonic mean and mode are obained as Mean = () = = () = (8) Inernaional Science Congress Associaion 23

2 Variance = = () Harmonic Mean = () = H = = () (9) (1) The mos likely value, ha is, yi ha has he highes probabiliy pi, or he y a which pdf is maximum, is called mode of y. Mode ()= M o = (11) I is clear from figure1 ha he densiy funcion of he Inverse Maxwell disribuion akes differen shapes for differen values of he parameer θ. The curve is posiively skewed and has a long righ ail. I is unimodal and is mode shifs in he righ side and he ail increases and he curve ends o be symmerical and flaer as θ decreases. Cumulaive disribuion funcion: The cumulaive disribuion funcion of a real-valued random variable Y is he funcion given by F Y (y) = P(Y y) where he righ-hand side represens he probabiliy ha he random variable Y akes on a value less han or equal o y. Now, he cumulaive disribuion funcion of Inverse Maxwell disribuion is defined as F () = f (y; θ) dy (12) =.e dy Pu =u ; = du Limis y = ; u = ; y = ; u = and we ge F() = du = 1 (, ) (12) The cdf of inverse Maxwell disribuion is lower incomplee gamma funcion depending on θ. I is S shaped. And is shape changes in accordance wih θ. Survival funcion: Le T be a coninuous random variable wih probabiliy densiy funcion (p.d.f.) f() and cumulaive disribuion funcion (c.d.f.) F() = Pr(T ), giving he probabiliy ha he even has occurred by duraion. hen he survival funcion is given as S () = Pr {T > } = 1 - F () (14) = 1- [1- (, )] =, (15) f() θ=.5 θ= Figure-1 pdf of Inverse Maxwell disribuion Inernaional Science Congress Associaion 24

3 F().6.4 θ= Figure-2 cdf of inverse Maxwell disribuion S().6.4 θ= Figure-3 Survival funcion of inverse Maxwell disribuion Inernaional Science Congress Associaion 25

4 For a survival funcion, he value on he graph sars a one and monoonically decreases o zero. The inverse Maxwell disribuion s survival funcion is shown in he figure below. Hazard funcion: he hazard funcion, or insananeous failure rae λ() funcion is defined by, λ () = lim { } The condiional probabiliy in he numeraor may be wrien as he raio of he join probabiliy ha T is in he inerval (; + d) and T > (which is, of course, he same as he probabiliy ha is in he inerval), o he probabiliy of he condiion T >. The former may be wrien as f()d for small d, while he laer is S() by definiion. Dividing by d and passing o he limi gives he useful resul.the hazard funcion of IMD is define as λ () = () (16) () where f() is probabiliy densiy funcion and S(), survival funcion of inverse Maxwell disribuion respecively. We have, λ () = =. (, ). (, ) ; >,> ;>,> (17) The hazard funcion of IMD increases iniially, hen decreases and evenually approaches zero. This means ha iems wih a inverse Maxwell disribuion have a higher chance of failing as hey age for some period of ime, bu afer survival o a specific age, he probabiliy of failure decreases as ime increases. The inverse Maxwell disribuion s hazard funcion is shown in he figure below. For θ =.5,1.,2..for smaller value of θ he hazard funcion are seeper. Esimaion of Parameers Momen equaion esimaion: In his mehod of momens replacing he populaion mean are equaed wih he mehod of momens esimaion he sample momens are equaed wih he corresponding populaion momens and solved for he parameer(s). Thus he corresponding sample mean and variance respecively, = (18) Puing he value of = Squaring boh he sides, we ge = in equaion (18),we have From he above equaion, we have he sample mean of is θ = π (19) λ() θ= Figure-4 hazard funcion of inverse Maxwell disribuion for differen value of θ. Inernaional Science Congress Associaion 26

5 Maximum Likelihood Esimaor (MLE) Case (i) when r<: Le n iems are pu o es for heir lifeimes, heir failure (life) imes are recorded and he experimen is erminaed as soon as r n ( preassigned) iems fail. Le he reordered lives are as y 1 < y 2 < < y r (r n). now likelihood funcion is L f (y; θ) [1 ( ;)] (2) L = k.e () (, ) () (21) Taking log on boh side, we have LogL= r [log 4 log ] + log + r[log 1- log] - +( ) + ( ) (, ) (22) Differeniaing equaion (22) w.r.. and seing he resuls equal o zero, we have = +( ) () + (, ) + = + = + +( ) - () +( ) - () (, (, ) (, ) = (23) I is a non-linear equaion in. which can no be solved direcly. Is soluion may be obained eiher by Newon- Raphson mehod or mehod of scoring. The maximum likelihood equaions involve he parameers in limi of inegral in he incomplee gamma funcion. So we canno ge simplified soluion of he above equaion. Case (ii) when r = n: The esimaions of he parameer of inverse Maxwell disribuion are obained by he mehod of MLE using equaion (6), he maximum log likelihood funcion of he IMD as follow: L = f (y; θ) (24) = =.e.e Now he log likelihood funcion for n observaion of Y is given by Log L = n [ log 4 log ] +log + n [log 1- log] - (25) Differeniaing equaion (25) w.r.. and seing he resuls equal o zero, we have = + now, leads o = = As he MLE of θ Conclusion (26) (27) In his paper we have obained he p.d.f. of inverse Maxwell disribuion by he help of ransformaion echnique an also ploed pdf in figure.1, and sudied heir properies. Is suiabiliy as a survival model has been discussed by obaining is survival and hazard funcions. These are ploed in a figures- 3 and 4 respecively, and heir properies have been discussed. The maximum likelihood esimaor of IMD when r <, a bi difficuly. Since he maximum likelihood equaion involve he parameer in limi of inegral, so we canno ge simplified soluion of he equaion. Now when r = n is, he MLE of θ is = and momen equaion esimaor of θ have been obained as = ; A suiabiliy as a survival model has been proposed by obaining he hazard funcion. he IMD may be a good survival model when he daa conforms wih is hazard funcion and survival funcion. Suiable example may be obained an lieraure. References 1. Sephan F.F., The expeced value and variance of he reciprocal and oher negaive powers of a posiive Bernoullian variae. Ann. Mah. Sais, 16, 5-61, (1945) 2. Grab E.L. and Savage I.R. Tables of he expeced value of 1/X for posiive Bernoulli and Poisson variables, J. Amer. Sais. Assoc., 49, (1954) 3. Mendenhall W. and Lehmann E.H., Jr. An approximaion o he negaive momens of he posiive binomial useful in life esing, Technomerics 2, (196) Inernaional Science Congress Associaion 27

6 4. Govindarajulu Z., The reciprocal of he decapiaed negaive binomial variable, J. Amer. Sais. Assoc., 57, (1962) 5. Govindarajulu Z., Recurrence relaions for he inverse momens of he posiive binomial variable, J. Amer. Sais. Assoc., 58, (1963) 6. Tiku M.L.A noe on he negaive momens of a runcaed Poisson variae, J. Amer. Sais. Assoc., 59, (1964) 7. Vijsokouska E.S., Reliabiliy of ools used in semiauomaic lahes, Russian Engineering Journal, XLVI(6), 46-5 (1966) 8. Sancu D.D., On he momens of negaive order of he posiive Bernoulli and Poisson variables. Sudia Universiais Babes Bolyai Series, Mahemaics and Physics, 1, (1968) 9. Skibusky M., A characerizaion of hyper-geomeric disribuions, J.Amer. Sais. Assoc., 65, (197) 1. Kabe D.G., Inverse momens of discree disribuions, The Canadian Journal of Saisics, 4(1), (1976) 11. Shahnbhag, D.N. and Busawa, T.V. On a characerizaion propery of he mulinomial disribuion, Ann. Mah. Sais, 42, 22 (1971) 12. Chao M.T. and Srawderman W.E., Negaive momens of posiive random variables, J. Amer. Sais. Assoc. 67, (1972) 13. Gupa R.C., Modified power series disribuion and some of is applicaions, Sankhya B., 36, (1974) 14. Gupa R.C., Esimaing he probabiliy of winning (losing) in a gambler s ruin problem wih applicaions, Journal of Saisical Planning and Inference, 9, (1984) 15. Lepage Y., Negaive facorial momens of posiive random variables, Indusrial Mahemaics, 28, 95-1 (1978) 16. Kumar A. and Consul P.C., Negaive momens of a modified power series disribuion and bias of he maximum likelihood esimaor, Communicaions in Saisics. A., 8, (1979) 17. Cressie N. and Borken M., The momen generaing funcion have is momens, Journal of Saisics Planning and Inference, 13, (1986) 18. Cressie N., Davis A.S. Folks J.L. and Policello G.E., The momen generaing funcion and negaive ineger momens, American Saisician., 35(3), (1981) 19. Ahmad M. and Sheikh A.K., Reliabiliy compuaion for Bernsein disribuion srengh and sress. Submied o he 1h Pak. Saisical Conference, Islamabad, Pakisan (1981) 2. Ahmad M. and Sheikh A.K., Some esimaion problem of a class of invered disribuions. 44h session of he Inernaional Saisical Insiue. Spain. 1-6, Ahmad: On he Theory of Inversion 51 (1983) 21. Ahmad M. and Sheikh A.K., Bernsein Reliabiliy Model: Derivaion and Esimaion of Parameers, Reliabiliy Engineering, 8, (1984) 22. Ahmad M. and Sheikh A.K., Some remarks on he hazard funcions of he invered disribuion, Reliabiliy Engineering, 11, 1-6. (1987) 23. Jones M.C., Inverse facorial momens, Saisics and Probabiliy Leers, 6, (1987) 24. Roohi A., Negaive Momens of Non-Negaive Random Variables. Un-published M. Phil disseraion, Universiy of Lahore, Lahore, 535 (22) 25. Jones M.C. and Zhigljavsky Anaoly A. Approximaing he negaive momens of he Poisson disribuion, Saisics and Probabiliy Leers, 66(2), (24) 26. Rempala G.A., Asympoic facorial powers expansion for binomial and negaive binomial reciprocals, Proc. Amer. Mah. Sociey, 132(1), (24) 27. Ahmad M. and Roohi A., Characerizaion of he Hyper- Poisson Probabiliy Funcion Using Firs Order Negaive Momens. Proceedings of he Iner- naional Conference on Disribuion Theory, Order Saisics and Inference in Honour of Barry C. Arnold, held a Universiy of Canabria, Sanander Spain June 16-18, (24a) 28. Ahmad M. and Roohi A., On sums of some hypergeomeric series funcions-i, Pak. J. Sais., 2(1), (24b) 29. Ahmad M. and Roohi A., On sum of some hypergeomeric series funcion-ii. Pak. J. Sais., 21(3), (24c) 3. Ahmad M. and Roohi A., Characerizaion of he Poisson Probabiliy Funcion, Pak. J. Sais., 2(2), (24d) Inernaional Science Congress Associaion 28