Wrapped Geometric Distribution: A new Probability Model for Circular Data

Transcription

1 Joural of Statistical Theory ad Applicatios, Vol., No. 4 Deceber 03), Wrapped Geoetric Distributio: A ew Probability Model for Circular Data Sophy Jacob ad K. Jayakuar MES Asabi College, P. Veballur, Kerala, Idia sophyjacob@rediffail.co Uiversity of Calicut, Kerala, Idia jkuar9@rediffail.co Received August 0 Accepted August 03 We propose a ew faily of circular distributios, obtaied by wrappig geoetric distributio o Z+ 0,,..., aroud a uit circle. The properties of this ew faily of distributios are studied. Keywords: Circular distributios, trigooetric oets, wrapped geoetric distributio.. Itroductio Circular data arise i various ways. Two of the ost coo correspod to two circular easurig istruets, the copass ad the clock. Data easured by copass usually iclude wid directios, ocea curret directios, the directio ad orietatios of birds ad aials, ad orietatio of geological pheoea like rock cores ad fractures. Data easured by clock icludes ties of arrival of patiets at a hospital eergecy roo, icideces of a disease throughout the year, ad the uber of tourists daily or othly) i a city withi a year, where the caledar is regarded as a oeyear clock. Circular or directioal data also arise i ay scietific fields, such as Biology, Geology, Meteorology, Physics, Psychology, Medicie ad Astrooy see Mardia ad Jupp 000)). Study o directioal data ca be dated back to the 8th cetury. I 734 Daiel Beroulli proposed to use the resultat legth of oral vectors to test for uifority of uit vectors o the sphere see Mardia 97)). I 98 vo Mises itroduced a distributio o the circle by usig characterizatio aalogous to the Gauss characterizatio of the oral distributio o a lie see Mardia 97)). Later, iterest was reewed i spherical ad circular data by Fisher 953), Greewood ad Durad 955), ad Watso ad Willia 956). Wrapped distributios provide a rich ad useful class of odels for circular data. The special cases of the wrapped oral, wrapped Cauchy, wrapped Poisso are widely discussed i Mardia 97). The wrapped Expoetial ad Laplace distributios have bee itroduced ad studied by Jaalaadaka ad Kozubowski 004). I Sectio, we itroduce ad study wrapped geoetric 348

2 Sophy Jacob ad K. Jayakuar distributio. Properties of the wrapped geoetric distributio are studied i Sectio 3. Sectio 4 deals with estiatio of paraeters.. Wrapped Geoetric Distributio Oe of the coo ethods to aalyze circular data is kow as wrappig approach. I this ethod, give a kow distributio o the real lie, we wrap it aroud the circuferece of a circle with uit radius. Techically this eas that if X is a rado variable o the real lie with distributio fuctio Fx), the rado variable X w of the the wrapped distributio is give by X w X od π).) ad the distributio fuctio F w θ) is give by F w θ) k {Fθ + πk) Fπk)}..) More precisely, if X is a rado variable o the itegers, the X w, defied by X w πx od π) is a rado variable o the lattice { πr, r0,,..., } o the circle. Now, cosider the geoetric distributio o Z + with paraeter. Its probability ass fuctio is give by px;) ) x, x0,,... ; > 0. The probability fuctio of X w is give by P X w πr ) k pr+k;), r0,,...,. That is, P X w πr ) k0 ) r+k )r ).3) where N ad r0,,...,. Agai r0 ) r ). Therefore, P ) represets a probability ass fuctio. 349

3 Wrapped Geoetric Distributio: A ew Probability Model for Circular Data If Φt) is the characteristic fuctio of a liear rado variable X, the the characteristic fuctio of X w is Φp). For the wrapped geoetric distributio, we have ) Φp) r e iπ rp r0 ), p0,±,±,... )e i x iy, ) ) where x )cos ad y )si, where p 0 od ). O expadig Φp), we have )) Φp){ )cos + )si where ρ p e iµ p, )) ρ p { )cos + )si { } )) arcta ei )) } } )si π p ) )cos π p ).4) ad )si µ p arcta )cos ) ). Thus the p th trigooetric oet of the wrapped geoetric distributio is give by where Φ p α p + iβ p, α p Ecos pθ)ρ p cos µ p ad The cetral trigooetric oets are give by β p Esi pθ)ρ p si µ p. α p ρ p cosµ p pµ ), β p ρ p siµ p pµ ). 350

4 Sophy Jacob ad K. Jayakuar The legth of the resultat vector, ρ ρ is ρ α + β + ) )cos π ) ) + ) cos π )).5) ad the ea directio, )si ) π µ arcta )cos ). π The circular variace, V 0 is give by V 0 ρ + ) )cos π ). The circular stadard deviatio, σ 0 lρ l ) + ) cos π ) ). The easure of skewess is deteried by the forula The easure of kurtosis, γ 0 β V 3 0 ρ siµ µ ) ) + ) cos π )) γ 0 ρ cosµ µ ) ρ 4 V0. Propositio.. Let Θ,...,Θ be idepedet ad idetically distributed wrapped geoetric rado variables with paraeters ad. The ΘΘ + +Θ od π) follows wrapped egative bioial distributio with paraeters p,, where p, N. ) 3 35

5 Wrapped Geoetric Distributio: A ew Probability Model for Circular Data Proof. Θ i Wrapped geoetric distributio with paraeters ad. Therefore, Φ Θi p) )e i Let Φ Θ p) deotes the characteristic fuctio of Θ. The Φ Θ p)φ Θ +Θ + +Θ Φ Θ j p) j ) Q Pe i. od π)p) where Q, P, Q P, > 0, which is the characteristic fuctio of wrapped egative bioial distributio. 3. Divisibility Properties 3.. Ifiite Divisibility Defiitio 3.. A agular rado variable Θ is said to be ifiitely divisible if for ay iteger there exists idepedetly ad idetically distributed i.i.d.) agular rado variables Θ,Θ,...,Θ such that Θ + Θ + + Θ od π) d Θ. Sice a circular rado variable obtaied by wrappig a ifiitely divisible rado variable is ifiitely divisible see Mardia ad Jupp, 000), the wrapped geoetric rado variable is ifiitely divisible. We have, the characteristic fuctio of wrapped geoetric rado variable Θ, for every, where Q 3.. Geoetric Ifiite Divisibility Φ Θ p) )e i ) Q Pe i ),, P, Q P, > 0. Defiitio 3.. A agular rado variable Θ is said to be geoetrically ifiitely divisible if for ay q 0,) there exists i.i.d. agular rado variables Θ, Θ,... such that where ν q has the geoetric distributio Θ + Θ + + Θ νq od π) d Θ, Pν q k) q) k q, k,,... 35

6 Sophy Jacob ad K. Jayakuar Let Θ, Θ,... be i.i.d. wrapped geoetric rado variables. Coditioig o the distributio of ν q, we have [ ] E e ipθ +Θ + +Θ νq ) od π) k k [ ] E e ipθ +Θ + +Θ k ) q) k q Φ k p q) k q q Φ p q) Φ p q ) Q Pe i q) A Be i ) where A Q +q q, B P q ifiitely divisible. ad A B. Hece the wrapped geoetric distributio is geoetrically 3.3. A Geeralizatio of Wrapped Geoetric Distributio ad its Divisibility Properties The probability ass fuctio of the wrapped geoetric rado variable Θ with paraeters ad is give by P Θ πr ) )r ; 0< <, N, r0,,... ) The probability geeratig fuctio p.g.f.) of Θ is give by Ps) r0 ) r s r ) ) ) s )) r r0 ) s) ) ) ) )s). We have P). The series Ps) coverges for s. Whe se i, we get the characteristic fuctio of wrapped geoetric distributio. As a geeralizatio of geoetric distributio, we ow cosider the discrete Mittag-Leffler distributio itroduced by Pillai ad Jayakuar 995). The p.g.f. of the discrete Mittag-Leffler distributio with paraeter α is give by Ps) Therefore, its characteristic fuctio, Φ t ) s) α ). ) e it ) α ). 353

7 Wrapped Geoetric Distributio: A ew Probability Model for Circular Data Hece the wrapped discrete Mittag-Leffler distributio has characteristic fuctio, Φ p ) e i ) α ) + ) e i +c e ipπ where c, 0<α <, p0,±,±,..., N, p 0, od ). Whe α, we get the characteristic fuctio of wrapped geoetric distributio. We have the characteristic fuctio of wrapped discrete Mittag-Leffler distributio, ) α, Φ p +c e i +c Φ p, ) α e i ) α ) α ) ), for every ad Φ p, is the characteristic fuctio of wrapped discrete Liik distributio. Agai, we have [ E e ipθ +Θ + +Θ νq ) od π) ] k k [ ] E e ipθ +Θ + +Θ k ) ) k Φ p ) k ) k Φ p ) Φ p + c e i +σ e i ) α ) α where c, σ c ifiitely divisible. > 0. So wrapped discrete Mittag-Leffler distributio is geoetrically 4. Estiatio 4.. Method of Moets Let θ θ,...,θ ) be a rado saple of size take fro the wrapped geoetric distributio with paraeters ad. I this ethod, estiates of the paraeters ca be obtaied by equatig 354

8 Sophy Jacob ad K. Jayakuar saple oets to the correspodig populatio oets. We have, the p th saple trigooetric oet about the zero directio, p a p+ ib p, where a p jcospθ j ) ad b p j sipθ j ). Equatig a p to α p ad b p to β p, we get )si a p ) + ) cos π )) cos ta )cos ) ) 4.) ad )si b p ) + ) cos π )) si ta )cos ) ) 4.) where p 0 od ). Solvig 4.) ad 4.) for a fixed value of ad p, we get R cos ) π ± R R cos ) π R )cos ) π R where R a + b, the saple ea legth of the resultat. Refereces [] R.A. Fisher, Dispersio o a Sphere, Proceedigs of the Royal Society of Lodo, A, 7, ). [] J.A. Greewood ad D. Durad, The distributio of legth ad copoets of the su of rado uit vectors, Aals of Matheatical Statistics, 6, ). [3] Jaalaadaka, S. Rao ad T.J. Kozubowski, New failies of wrapped distributios for odelig skew circular data, Couicatios i Statistics: Theory ad Methods, 33, ). [4] Jaalaadaka, S. Rao ad A Segupta, Topics i Circular Statistics, World Scietific, Sigapore,00). [5] K.V. Mardia, Statistics of Directioal Data, Acadeic Press, Lodo 97). [6] K.V. Mardia ad P.E. Jupp, Directioal Statistics, d Editio, Wiley, New York 000). [7] R.N. Pillai ad K. Jayakuar, Discrete Mittag-Leffler distributios, Statistics ad Probability Letters, 3, ). [8] G.S. Watso ad E.J. Willia, O the costructio of sigificace tests o the circle ad the sphere, Bioetrika, 43, ). 355