Estimation of Parameters of the Exponential Geometric Distribution with Presence of Outliers Generated from Uniform Distribution
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1 ustala Joual of Basc ad ppled Sceces, 6(: 98-6, ISSN Estmato of Paametes of the Epoetal Geometc Dstbuto wth Pesece of Outles Geeated fom Ufom Dstbuto Pavz Nas, l Shadoh ad Hassa Paza Depatmet of Statstcs, Uvesty of Payame Noo, Teha Bach, Teha, Ia bstact: two-paamete dstbuto wth deceasg falue s toduced wth pesece of outles geeated fom ufom dstbuto. Vaous popetes ae dscussed ad the estmato of paametes s studed by the methods of momet ad mamum lelhood. These estmatos ae compaed empcally whe all of the paametes ae uow. The bas ad mea squae eo (MSE ae vestgated wth help of umecal techques. Key wods: Compoudg, Epoetal Geometc dstbuto, Falue ate, Ufom dstbuto, Momet estmato, Mamum lelhood estmato, Newto-Raphso Method, Mote- Calo Smulato. INTRODUCTION The study of legth lfe of ogasms, devces, stuctues, mateals, etc., s of majo mpotace the bologcal ad egeeg sceces. substatal pat of such study s devoted to the mathematcal descpto of the legth of lfe by a falue dstbuto. Sometmes physcal cosdeatos of the falue mechasm may lead to a specfc dstbuto but moe ofte, the choce s made o the bass of how well the actual obsevatos of tmes to falue appea to be ftted by dstbuto. Stuatos whee the falue ate fucto deceases wth tme have bee epoted by seveal authos. Idcatve eamples ae the busess motalty Loma (954, falues the a-codtog equpmet of a fleet of Boeg 7 acafts of semcoductos fom vaous combed Poscha (963 ad the lfe of tegated ccut modules Saudes ad Myhe (983. I geeal, a populato s epected to ehbt deceasg falue ate (DFR whe ts behavo ove tme s chaactezed by wo hadeg ( egeeg tems o mmuty ( bologcal tems; sometmes the boade tem 'fat motalty' s used to deote the DFR pheomeo. Poscha (963 poved that the DFR popety s heet to mtues of dstbutos wth costat falue ate ad Glese (989 demostated the covese fo ay gamma dstbuto wth shape paamete less tha oe. ccodg to Dt, Mooe ad Baett (996, we assume that a set of adom vaables (X, X,..., X epeset the dstace of a fected sample plat fom a plot of plats oculated wth a vus. Some of the obsevatos ae deved fom the aboe dspesal of the spoes ad ae dstbuted accodg to the epoetal dstbuto. The othe obsevatos out of adom vaables (say ae peset because aphds whch ae ow to be caes of baley yellow mosac dwaf vus (BYMDV have passed the vus to the plats whe the aphds feed o the sap. These (ow aphds ae cosdeed to be epoetal dstbuted. Thus, we assume that the adom vaables (X, X,..., X ae such that of them ae dstbuted wth p.d.f g(,θ, g,, ad the emag ( adom vaables ae dstbuted wth p.d.f f(, θ, p, f,, p ( p e ( pe, (. (. I secto, we toduce the epoetal geometc dstbuto, ad secto 3, we have obtaed the jot Dstbuto of (X, X,..., X the pesece of outles geeated fom ufom dstbuto, also ths secto we peset the suvval falue ate fucto. I sectos 4 ad 5, we deal wth estmato Coespodg utho: Pavz Nas, Depatmet of Statstcs, Uvesty of Payame Noo, Teha Bach, Teha, Ia. 98
2 ust. J. Basc & ppl. Sc., 6(: 98-6, paametes by usg the methods of the momet ad mamum lelhood, espectvely. I secto 6, we have gve a umecal study ad cocluso. Epoetal Geometc Dstbuto: Suppose that Y, Y,..., Y z ae d wth desty f(, θ. y f( y, e, y, ad Z s a geometc vaable wth pobablty fucto PZ [ z] ; P Z z p p z p z ( (,,,..., (. (. If we cosde the adom vaable X m( X, X,..., X z the f p p e pe (,, ( (, (.3 Poof: We ow that! f z F z F z f z!( (, ( ( ( ze e ze ( z z the f( f(, z f( z, P( Z z z z z z ( p z p e z d ( pe z pe ( p pe z zdp ( pe pe,, p z The latte defes the dstbuto that we shall be efeg to the sequel as the epoetal geometc dstbuto (EG fo bevty. It ca be vefed by stadad techques that f X s a EG vaable, wth desty gve by (.3 the, the adom vaable P( P y p ( e, espectvely. z follows the Paeto dstbuto, wth shape ad scale paametes oe ad Jot Dstbuto of (X, X,..., X wth κ Outles: ccodg to Dt (989 model, we assume that the adom vaables (X, X,..., X ae such that of them ae (. ad emag (- adom vaables ae dstbuto as (.. The jot dstbuto of (X, X,..., X s gve as 99
3 ust. J. Basc & ppl. Sc., 6(: 98-6, Fg. : p.d.f. of the Epoetal Geometc ad Ufom Dstbuto fo dffeet values of p ad θ =.!( g ( f (,,...,, p f(, p f p! (, ( p e ( pe!(! I (, ( ( p e ( pe!( ( p e ( pe! e I (, ( ( pe (3. whee ad,,3,...,. Fo the moe detals see Dt et al. (996, Dt ad Nas ( ad Nas ad Paza (. Fo =, t s gve by damds ad Louas (998. Fom (3. the magal dstbuto of X s
4 ust. J. Basc & ppl. Sc., 6(: 98-6, f p e p e pe (,, ( (, (3. Fom (3. the suvval pobablty s gve by s (,, p F (,, p f( t,, pdt ( ( t t t e dt p e pe dt (3.3 The falue ate (also ow as hazad ate s h ( e ( p e ( pe f(,, p s (,, p ( e p ( p p ( pe Method of Momet: The aw momets of X may be detemed fom (3. be dect tegato. Fo (3.4, we fe that ( ( ( ( ( ( E X e p e pe d e d ( p e ( pe d p e pe d pe ( pe d dv Let u ad the p EX ( ( udv p p ( uv vdu p p p pe pe ( d p ( p pe d Sce pe
5 ust. J. Basc & ppl. Sc., 6(: 98-6, pe ( pe (4. Hece p E( X ( ( pe d p p ( ( pe d p p ( p (! p ( p ( ( p (4. Fo = ad, E( X s gve by EX ( ( ( p p (4.3 EX p p ( ( ( (4.4 m D m X j m j Let, whee. The fom (4.3 ad (4.4 we have D whee pˆ ( ( pˆ pˆ ( ( pˆ ˆp s estmate of p. Net, fom (4.3 we ca obta estmate of θ as followg ˆ ˆ ˆ m ( ( p p (4.5 (4.6
6 ust. J. Basc & ppl. Sc., 6(: 98-6, whee ˆp s gve by (4.5. Method of Mamum Lelhood: Oe sees fom (4. that momet estmates fo the paametes of the EG dstbuto wth desty fucto caot be obtaed closed foms ad theefoe that s lttle pot cosdeg the method ay futhe. Poceedg wth the method of mamum lelhood, the lelhood fucto fom a sample of obsevatos, (X, X,..., X s gve by!( L(, p ( p e ( pe! the e I (, ( ( pe (5. ( ˆ ˆ L(, p ( ( p e ( pe ˆ ˆ e ( pe (5. whee ˆ X ma( X, X,..., X ( (5.3 To estmate p, we cosde l L( ˆ, p ( ˆ l ( ˆ, ( l ( ( l ( ˆ L p p l ( pe as ˆ ˆ l e ( pe (5.4 Tag the devatve wth espect to p ad equatg to, we obta the omal equato as dl( p ( e dp ( p ( p e Sce dl( p dp, hece ˆ ( pe e ( pe (5.5 3
7 ust. J. Basc & ppl. Sc., 6(: 98-6, ˆ ˆ ( pe e ( ˆ ( p ˆ ˆ ( pe e ( pe (5.6 Hee, we eed to use ethe the scog algothm o the Newto-Raphso method to solve the o-lea equato. Hee, we solve (5.6 by Newto-Raphso method. Hece soluto of the equato s g( p p p,,,,... g( p (5.7 whee ˆ ˆ ( pe e ( ˆ ˆ ˆ g( p ( pe ( p e ( pe (5.8 ad ˆ ˆ e e ( ˆ ˆ ˆ g( p ( pe ( p e ( pe ˆ ( pe ˆ ˆ e ( pe Hee, the tal soluto p should be selected fom (4.5. (5.9 Numecal Study: I ths pape, we have addessed the poblem of estmatg paametes of Epoetal Geometc dstbuto pesece of outles. I ode to have some dea about Bas ad Mea Squae Eo (MSE of methods of momet ad MLE, we pefom samplg epemets usg a MTLB. The esults ae gve Tables to 3, fo = (5 4 ad 5, p =.5, θ = 3, ad =, ad 3. We epot the aveage estmates ad the MSEs based o eplcatos. Fom Tables to 3, we cojectue that the momet (MOM ad mamum lelhood (MLE estmatos of p ae udeestmato, but the momet ad mamum lelhood estmatos of θ ae oveestmato, ths s tue fo =, ad 3. ccodg to Table to 3, whe cease the the MSEs decease, ths s tue fo =, ad 3, but whe cease the MSEs cease. Tables ae show that the MSE of MLE estmatos of θ ae less tha the MSE of the MOM estmatos fo all values of ad. lso, the MSE of MLE estmatos of p ae less tha the MSE of the MOM estmatos fo all values of, but oly fo >. Ideed, fo = the MSE of MOM estmatos of p ae less tha the MSE of the MLE estmatos fo all values of. So the MLE estmatos of p ad θ ae moe effcet tha the MOM estmatos. We stogly feel MLE estmatos ae bette tha the MOM estmatos. Fom the pevous obsevatos, we suggest to use MLE method fo estmatg p ad θ Epoetal Geometc dstbuto wth pesece of outles. 4
8 Table : p =.5, θ = 3 ad =. ust. J. Basc & ppl. Sc., 6(: 98-6, Method Bas ˆp MSE ˆp Bas ˆ MSE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE Table : p =.5, θ = 3 ad =. Method Bas ˆp MSE ˆp Bas ˆ MSE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE Table 3: p =.5, θ = 3 ad =. Method Bas ˆp MSE ˆp Bas ˆ MSE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE MOM MLE REFERENCES damds, K. ad S. Louas, 998. lfetme dstbuto wth deceasg falue ate. Statstcs ad pobablty lette 39, ˆ ˆ ˆ 5
9 ust. J. Basc & ppl. Sc., 6(: 98-6, Dt, U.J., 989. Estmato of paametes of the gamma dstbuto the pesece of outles. Commucatos statstcs theoy ad methods, 8: Dt, U.J., Moo, K.L. ad V. Baett, 996. O the estmato of the powe of the scale paamete of the epoetal dstbuto the pesece of outles geeated fom ufom dstbuto, Meto, 54: -. Dt, U.J. ad P.F. Nas,. Estmato of paametes of the epoetal dstbuto the pesece of outles geeated fom ufom dstbuto, Meto, 49(3-4: Glesse, L.J., 989. The gamma dstbuto as a mtue of epoetal dstbutos. me. Statst., 43: 5-7. Loma, K.S., 954. Busess falues: aothe eample of the aalyss of falue data. J. me. Statst. ssoc., 49: Nas, P.F. ad H. Paza,. Bayesa ad o-bayesa estmatos o the Geealzed Epoetal dstbuto the pesece outles, Joual of Statstcal Theoy ad Pactce, 4(3: Poscha, F., 963. Theoetcal eplaato of obseved deceasg falue ate. Techometcs, 5: Saudes, S.C. ad J.M. Myhe, 983. Mamum lelhood estmato fo two-paamete deceasg hazad ate dstbuto usg cesoed data. J. me. Statst. ssoc., 78:
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