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On Wrapped Binomial Model Characteristics Mathematics and Statistics 2(7): 231-234, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020701 On Wrapped Binomial Model Characteristics S.V.S.Girija1,*, A.V. Dattatreya Rao2, G.V.L.N. Srihari 3 1Mathematics, Hindu College, Guntur, India 2Acharya Nagarjuna University, Guntur, India 3Mathematics, Aurora Engineering College, Bhongir *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In this paper, a new discrete circular model, the latter case, the probability density function f (θ ) the Wrapped Binomial model is constructed by applying the method of wrapping a discrete linear model. The exists and has the following basic properties. characteristic function of the Wrapped Binomial Distribution f (θ ) ≥ 0 is also derived and the population characteristics are studied. 2π Keywords Characteristic Function, Circular Models, ∫ fd(θθ) =1 Trigonometric Moments, Wrapped Models 0 ff(θ) =( θπ + 2 k) , for any integer k, That is , f (θ ) is periodic with period 2.π 1. Introduction 2.1. Wrapped Discrete Circular Random Variables Circular or directional data arise in many diverse fields, such as Biology, Geology, Physics, Meteorology, and If X is a discrete random variable on the set of integers, + psychology, Medicine, Image Processing, Political Science, then reduction modulo 2π mm( ∈ ) wraps the integers Economics and Astronomy (Mardia and Jupp (2000). on to the group of mth roots of unity which is a sub group of Wrapped circular models (Girija (2010)), stereographic unit circle. circular models (Phani (2013)) and Offset circular models i.e. Φ=2ππxm( mod 2 ) (Radhika (2014)) provide a rich and very useful class of models for circular as well as l-axial data. Jacob and More precision Φ is a mapping from a set of integers G Jayakumar (2013) initiated the construction of Wrapped which is a group with respect to ‘+’ to the set of mth roots Geometric distribution. Motivated by the above of unity which is a group with respect to ‘.’ is defined as contributions, in this paper we construct a new discrete G ' 2πix 2πix circular model, named as the Wrapped Binomial model and where xG∈ , then Φ is its characteristic function is derived and also the population Φ=()xem eGm ∈ ' characteristics are studied. called wrapped discrete circular random variable. Clearly Φ is a homomorphism 2. Circular Distributions 2(πix+ y ) i.e.(1) Φ+ = m A circular distribution is a probability distribution whose ()xy e total probability is concentrated on the unit circle. Since each 2πix 2πiy point on the unit circle represents a direction, it is a way of = m m assigning probabilities to different directions or defining a e . e directional distribution. The range of a circular random variable θ measured in radians, may be taken to be = ΦΦ( xy). ( ) (0, 2π ] or [−ππ,.] 2πi (0) (2) Circular distributions are of two types, they may be Φ=(0) e m discrete-assigning probability masses only to a countable number of directions, or may be absolutely continuous. In = e0 = 1 where 0 ∈ G, 1 ∈ G ' 232 On Wrapped Binomial Model Characteristics + Since Φ contains a finite number of elements they are Where n≥ m −∋1, mn ∈ Z 2π r denoted by Φ=rm =0,1,2..... − 1 which is lattice on m Here p1 is the probability of success and q1 is the the unit circle. probability of failure. Where np,1 and mare parameters. 3. Probability Mass Function 6. Distribution Function of Wrapped Binomial Distribution Suppose if θ is a wrapped discrete circular random variable then probability mass function of θ is denoted by The probability mass function of the wrapped binomial 2πr distribution is prθ = which is defined as m nr− 2πr ∞ 2π r m prθ = = p(r + km) pr()θ = = n pr+ km q n −+() r km ∑ ∑ cr+ km 11 m k =−∞ m k =0 (1) Where rm= 0,1,2,3,...,− 1 and mZ∈ + But to exist the probability mass function it satisfies the nr− following properties. Let = b 2πr m 1. prθ = ≥0 ++ m Where m,n∈Z ∋≥ nm −1& bZ ∈ m−1 b 2πr r+ km n −+() r km 2. prθ = =1 = np q ∑ ∑ cr+ km 11 r =0 m k =0 θ= θπ + Now the distribution function of the wrapped binomial 3. pr( ) pr( 2 k ) for any integer k i.e. pr distribution is defined as is a periodic function. r nr− rm+ n −+() rm k npqcc11 + n p1 q 1 F ()θ = r rm+ w ∑ r+ bm n −+() r bm r=0 ++............. np q 4. Distribution Function cr+ bm 11 Suppose if θ is a wrapped discrete circular random where k=0,1,2,……b variable then distribution function of θ is denoted by Fw (θ ) and it is defined as k ∞ 7. Probability Generating Function of the Wrapped Binomial Distribution Fw ()θ = ∑∑P ( r+ km ) rk=0 =−∞ The probability generating function of the wrapped binomial distribution is given by m−1 5. Wrapped Binomial Distribution r 2π r G() s=∑ s pr θ = Suppose if x follows binomial distribution then the r=0 m probability mass function of the wrapped binomial mb−1 distribution is defined by using (1) = sr npr+ km q n −+() r km n−r ∑∑cr+ km 11 rk=00 = 2πr m prθ = = ∑ p(r + km) m k =0 r nr− rm+ n −+() rm m−1 npq+ n p q n−r r ccr 11 rm+ 1 1 = s m ∑ r+ bm n −+() r bm r+km n−r−km r=0 ++............. np q = n p q cr+ bm 11 ∑ cr+km 1 1 k =0 And also we have G(1)=1 Mathematics and Statistics 2(7): 231-234, 2014 233 8. Characteristic Function of Wrapped 222 −1 y Then xyr+=and θ = tan . Binomial Distribution x nn Since X follows binomial distribution then the Now ( x+= iy) ( rcosθθ + ir sin ) characteristic function of the binomial distribution is defined = (rnncos nθθ+ ir sin n ) as nn Then αp = rcos n θβ andp = rn sin θ itx φ = 2 2 X (t) Ee( ) Where r = x + y 2 n 2πp 2πp itx = q + p cos + p2 sin 2 = e p(X = x) 1 1 m 1 m ∑ x=0 2πp 2πp 2πp = q2 + p2 cos2 + 2q p cos + p2 sin 2 n 1 1 m 1 1 m 1 m = eitxn p xq n−x ∑ cx 1 1 2 2 2πp x=0 r = q1 + p1 + 2q1 p1 cos m n x − −1 y = n (p eit ) q n x and θ = tan ∑ cx 1 1 x x=0 2π p p1 sin n −1 = (q + p eit ) = Tan m 1 1 2π p qp11+ cos n m it where tR∈ ∴φX (t) = (q1 + p1e ) Then π But at an integer p the characteristic function of n 2 p p1 sin 2π p 2 − unwrapped distribution is equal to characteristic function of α =q22 ++ p2 q p cos cos n Tan 1m p 1 1 11 2π p wrapped distribution (Jammalamadaka and Sengupta (2001). m qp+ cos 11 m i.e. ϕφpX= ( p) And π n π 2 ip n 2 p 2 p1 sin m 22 2π p −1 m ϕ p = q1 + p1e β =q ++ p2 q p cos sin n Tan p 1 1 11 2π p m qp+ cos 11 m ϕ is also called as pth trigonometric moment of θ . p α β th Clearly, Here p , p are called p trigonometric moments. iµ 22 ipθ p Clearly ρ= αβ + ϕ = ρ e p pp p Ee( ) = p n 2πp =αβ + i ρ = q2 + p2 + 2q p cos pp p 1 1 1 1 m 2πip n β And µ = Tan -1 p = + m p q1 p1e α p 2πp n p1 sin 2πp 2πp −1 m = q + p cos + ip sin ∴µ p = nTan 1 1 1 2πp m m q + p cos 1 1 m n =( x + iy) Now the circular mean direction is defined as 2π 2πp 2πp p sin where x = q + p cos and y = p sin 1 1 1 1 −1 m m m µ1 = nTan 2π Now converting the Cartesian complex numbers in terms q + p cos 1 1 m of polar components by putting xr= cosθ , yr= sinθ . 234 On Wrapped Binomial Model Characteristics µ ∗ If 1 is denoted by µ then where α= αcos µβpp + sin µ and pp( p ) 2π β∗ = β µα − µ p1 sin pp( cospp p sin ) µ = nTan−1 m 2π Now the circular skewness for Wrapped Binomial + q1 p1 cos β m distribution is defined as 2 * and circular kurtosis for γ1 = 3 2 Now ρ1 represents the concentration towards the mean V0 direction which is defined as γ Wrapped Binomial distribution is denoted by 2 and it is n α *−− (1V ) 4 2 2 2π defined as γ = 20 ρ1 = q1 + p1 + 2q1 p1 cos 2 2 V0 m ρ If ρ1 denoted by then n 2π ρ = q2 + p2 + 2q p cos 1 1 1 1 m REFERENCES Now the circular variance is denoted by VO and it is [1] Abramowitz, M. k Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover, New York. defined as V0 =1 − ρ [2] A.W.Kemp, “Characterization of a Discrete Normal n Distribution,” Journal of Statistical Planning and Inference, 22 2π V0=−1 q1 ++ p 12 qp 11 cos Vol.63, No.2, 1997, pp.223-229. m [3] A.Pewsey, “A Wrapped Skew-Normal Distribution on the circle,” Communication in Statistics: Theory and Methods, The circular standard deviation is denoted by σ 0 and it is Vol.29, No.11, 2000, pp.2459-2472.
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