200026570_LSTA33_09_R2_090704

COMMUNICATIONS IN Theory and Methods 1 Vol. 33, No. 9, pp. 2059–2074, 2004 2 3 4 5 6 7 8 New Families of Wrapped Distributions for Modeling 9 10 Skew Circular Data 11 12 1, 13 Sreenivasa Rao Jammalamadaka * and 2 14 Tomasz J. Kozubowski 15 1 16 Department of Statistics and Applied , University of California, Santa Barbara, Santa Barbara, California, USA 17 2 18 Department of Mathematics and Statistics, University of Nevada-Reno, Reno, Nevada, USA 19 20 21 22 ABSTRACT 23 24 We discuss circular distributions obtained by wrapping the classical 25 exponential and Laplace distributions on the real line around the cir- 26 cle. We present explicit forms for their densities and distribution 27 functions, as well as their trigonometric moments and related param- 28 eters, and discuss main properties of these laws. Both distributions 29 are very promising as models for asymmetric directional data. 30 31 32 33 34 35 *Correspondence: Sreenivasa Rao Jammalamadaka, Department of Statistics 36 and Applied Probability, University of California, Santa Barbara, Santa Barbara, 37 CA 93106-3110, USA; Fax: (805) 893-2334; E-mail: [email protected]. 38 39 2059 40 41 DOI: 10.1081/STA-200026570 0361-0926 (Print); 1532-415X (Online) 42 Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com ORDER REPRINTS

200026570_LSTA33_09_R2_090804

2060 Jammalamadaka and Kozubowski

1 Key Words: Angular data; Asymmetric ; Bird 2 orientations; Circular data; Double ; 3 . 4 5 Mathematics Subject Classification: Primary 62H11; Secondary 62E10. 6 7 8 1. INTRODUCTION 9 10 Let X be a real (r.v.) with probability density 11 function (p.d.f.) f and characteristic function (ch.f.) f. Then the corre- 12 sponding wrapped r.v. 13 14 X ¼ Xðmod 2pÞð1:1Þ 15 w 16 has the density 17 18 X1 19 fwðyÞ¼ fðy þ 2kpÞ; y 2½0; 2pÞ; ð1:2Þ 20 k¼1 21 22 and the characteristic function (the discrete Fourier transform) 23 ¼ ipXw ¼ ð Þ; ¼ ; ; ; ...; ð : Þ 24 fp Ee f p p 0 1 2 1 3 25 26 see, e.g., Fisher (1993), Jammalamadaka and SenGupta (2001) and 27 Mardia and Jupp (2000). 28 The cases of wrapped Cauchy, normal, and stable distributions 29 have been studied extensively (see, e.g., Gatto and Jammalamadaka, 30 2003; Levy, 1939). In this work, we present basic theory of wrapped 31 exponential and double-exponential (Laplace) distributions, which were 32 introduced recently in Jammalamadaka and Kozubowski (2001, 2003). 33 The exponential distribution, with its important memoryless 34 property, provides a standard model in diverse fields such as reliability, 35 queueing theory, and others (see, e.g., Barlow and Proschen, 1996). The 36 classical Laplace distribution and its skew generalizations are compe- 37 titors to normal and other symmetric distributions in stochastic model- 38 ing, particularly in financial applications (see, e.g., Kotz et al., 2001 and 39 references therein). We believe that their circular analogs can find inter- 40 esting applications in directional data, when they resemble the characteri- 41 stic shape of the Laplace distribution (with its possible skewness and 42 sharp peak at the mode). Such data frequently result from orientation ORDER REPRINTS

200026570_LSTA33_09_R2_090704

Wrapped Distributions 2061

1 experiments in biology (see, e.g., Batschelet, 1981; Bruderer, 1975; Jander, 2 1957; Matthews, 1961; Schmidt-Koenig, 1964). As an example of a typi- 3 cally skew distribution, Batschelet (1981) shows a histogram of directions 4 chosen by migrating birds (see Fig. 2.6.1. in Batschelet, 1981). Commen- 5 ting on the ‘‘asymmetry in the choice of directions’’, he remarked that it 6 ‘‘may be caused by a mixture of two or three distributions’’. The wrapped 7 Laplace distribution can in fact be represented as such a mixture. 8 Mardia (1972) presents several examples of skew empirical distribu- 9 tions, and states that ‘‘symmetrical distributions on the circle are 10 comparatively rare’’ (see Mardia, 1972, p. 10). One of his examples is 11 studied in detail by Pewsey (2002), as an illustration of his test of symme- 12 try for circular data. Pewsey (2002) considered grouped data on the 13 frequencies of thunder during various times of the day recorded at 14 Kew (England) during the summers of 1910–1935 (see Table 1.3 in 15 Mardia, 1972). Mardia (1972) classified these data as unimodal, ‘‘slightly 16 asymmetrical’’ and ‘‘positively skew’’ (see Mardia, 1972, p. 10). When 17 testing these data, Pewsey (2002) obtained the p-value of zero, rejecting 18 the symmetry. Incidentally, when assessing the performance of his test 19 of symmetry, Pewsey (2002) used wrapped (symmetric) Laplace distri- 20 bution (as one of four symmetric distributions) and wrapped exponen- 21 tial distribution (as one of four skew alternatives) in his simulation 22 study. 23 Below we sketch a theory of wrapped exponential and Laplace distri- 24 butions. First, we define them in Sec. 2, where we also present their basic 25 properties. Then, we summarize their important properties in Sec. 3. 26 More detailed information with estimation, applications, and proofs 27 can be found in Jammalamadaka and Kozubowski (2001, 2003). 28 29 30 2. DEFINITIONS AND BASIC PROPERTIES 31 32 When we apply (1.2) and (1.3) to the exponential distribution with 33 p.d.f. fðxÞ¼lelx, x > 0, we obtain a wrapped exponential distri- 34 bution, denoted by WEðlÞ with l > 0. The following Table 1 provides 35 36 37 Table 1. The wrapped exponential distribution WEðlÞ. 38 ch.f. f ¼ 1 ; p ¼ 0; 1; 2; ... 39 p 1ip=l ð Þ¼ lely ; 2½ ; Þ 40 p.d.f. fw y 1e2pl y 0 2p 41 1ely c.d.f. F ðyÞ¼ ; y 2½0; 2pÞ 42 w 1e 2pl ORDER REPRINTS

200026570_LSTA33_09_R2_090704

2062 Jammalamadaka and Kozubowski

1 the characteristic function (ch.f.), density function (p.d.f.), and cumula- 2 tive distribution function (c.d.f.) of the wrapped exponential distribution 3 WEðlÞ, l 2 R. 4 The p.d.f. should be extended in a periodic fashion for the values of y 5 outside of the interval ½0; 2pÞ. The case l ¼ 0 is defined by a continuous 6 extension and corresponds to the circular uniform distribution (when ! þ y < 7 l 0 then the c.d.f. in Table 1 converges to 2p). The case l 0 results 8 from wrapping the ‘‘negative’’ exponential distribution with parameter 9 jlj > 0, whose p.d.f. is fðxÞ¼jljejljx, x < 0. Moreover, we have the relation 10 11 Y WEðlÞ if and only if 2p Y WEðlÞ: ð2:1Þ 12 13 Remark 2.1. We note a very interesting and curious property: The 14 restriction of the linear exponential r.v. X to the interval ½0; 2pÞ (that is 15 XjX < 2p) has the same distribution as the wrapped r.v. Xw given by (1.1). 16 Consider now an asymmetric Laplace r.v. X with the density function 17 ( 18 lkjxj; ; e for x 0 19 fðxÞ¼lð1=k þ kÞ 1 ð2:2Þ ð = Þj j 20 e l k x ; for x < 0; 21 22 and the ch.f. 23 24 1 fðtÞ¼ 25 1 þ t2=l2 itð1=k kÞ=l 26 1 1 27 ¼ ; t 2 R; ð2:3Þ 1 it=ðlkÞ 1 þ it=ðl=kÞ 28 29 30 where l; k>0; see Kotz et al. (2001) for theory and applications of these 31 distributions and their various generalizations. (When k ¼ 1 we obtain 32 the classical (symmetric) Laplace density.) When we apply (1.2) and 33 (1.3) to the above distribution, we obtain a wrapped Laplace distribu- 34 tion, denoted by WLðl; kÞ. This is defined in Table 2 below, which 35 provides the ch.f., the p.d.f., and the c.d.f. of the wrapped Laplace 36 distribution WLðl; kÞ, l 2 R; k > 0. 37 The p.d.f. should be extended in a periodic fashion for the values of y 38 outside of the interval ½0; 2pÞ. Observe that the p.d.f. in Table 2 is positive 39 and integrates to 1 on ½0; 2pÞ for any value of l, including l < 0, and we 40 have 41 42 WLðl; kÞ¼WLðl; 1=kÞ: ð2:4Þ ORDER REPRINTS

200026570_LSTA33_09_R2_090704

Wrapped Distributions 2063

1 Table 2. The wrapped Laplace distribution WLðl; kÞ. 2 1 1 ch.f. f ¼ =ð Þ þ =ð = Þ ; p ¼ 0; 1; 2; ... 3 p 1 ip lk 1 ip l k ð Þ¼ lk elky þ eðl=kÞy ; 2½ ; Þ 4 p.d.f. fw y 1þk2 1e2plk e2pl=k1 y 0 2p 5 ð Þ¼ 1 1elky þ k2 eðl=kÞy1; 2½ ; Þ c.d.f. Fw y 1þk2 1e2plk 1þk2 e2pl=k1 y 0 2p 6 7 8 It is also easy to see that 9 10 1 11 Y WLðl; kÞ if and only if 2p Y WL l; : 12 k 13 14 In case of a symmetric Laplace distribution with k ¼ 1, the wrapped 15 Laplace density simplifies to 16 17 l eð2pyÞl þ ely f ðyÞ¼ ; ð2:5Þ 18 w 2 e2pl 1 19 20 and we have 2p Y ¼d Y in this case, where ¼d denotes distributional 21 equivalence. 22 23 Remark 2.2. The Laplace distribution is a mixture of a positive and a 24 negative exponential distributions, since f in (2.2) can be written as 25 26 fðxÞ¼pf ðxÞþð1 pÞf ðxÞ; x 2 R; ð2:6Þ 27 1 2 28 f ðxÞ¼l e l1x ðx > 0Þ; f ðxÞ¼l el2x ðx < 0Þ; ð2:7Þ 29 1 1 2 2 30 31 and 32 33 1 p ¼ ; l1 ¼ lk; l2 ¼ l=k: ð2:8Þ 34 k2 þ 1 35 36 Consequently, the wrapped Laplace distribution is a mixture of 37 two wrapped exponential distributions, as the p.d.f. in Table 2 can be 38 written as 39 40 l1y l2y 41 ð Þ¼ l1e þð Þ l2e ; 2½ ; Þ; ð : Þ fw y p 1 p y 0 2p 2 9 42 1 e 2pl1 e2pl2 1 ORDER REPRINTS

200026570_LSTA33_09_R2_090704

2064 Jammalamadaka and Kozubowski

1 with the parameters as in (2.8). The corresponding wrapped Laplace 2 WLðl; kÞ r.v. Y admits the representation 3 4 d Y ¼ IY þð1 IÞY ; ð2:10Þ 5 1 2 6 Y Y ð Þ 7 where 1 and 2 are independent wrapped exponential WE lk and ð = Þ 8 WE l k r.v.’s, and I is an indicator random variable (independent of Y Y =ð þ 2Þ 9 1 and 2) taking on the values 1 and 0 with 1 1 k 2=ð þ 2Þ 10 and k 1 k , respectively. 11 12 Remark 2.3. By the factorization of the asymmetric Laplace ch.f., the 13 corresponding r.v. has the same distribution as the difference of two inde- 14 pendent exponential random variables (see, e.g., Kotz et al., 2001). Since 15 the wrapped Laplace ch.f. given in Table 2 admits a similar factorization, 16 we obtain an analogous representation for the wrapped Laplace r.v. Y ð ; Þ 17 WL l k viz. 18 d 19 Y ¼ Y1 þ Y2 ðmod 2pÞ; ð2:11Þ 20 21 where Y1 and Y2 are independent wrapped exponential WEðlkÞ and 22 WEðl=kÞ r.v.’s, mentioned before. 23 24 Remark 2.4. The WLðl; kÞ distribution converges weakly to the circular þ 25 uniform distribution as l ! 0, or k ! 0 ,ork !1. 26 27 Remark 2.5. A three-parameter class of distributions can be defined by 28 introducing a location parameter Z 2½0; 2pÞ and shifting the wrapped 29 Laplace p.d.f. (defined on R by a periodic extension) by Z with the result- 30 ing densities of the form 31 32 gðyÞ¼fwðy ZÞ 33 34 with fw given in Table 2. Parameter Z clearly corresponds to the mode for 35 such a family. 36 37 38 2.1. Characterization of the Densities 39 40 Note that the p.d.f. of the wrapped exponential distribution WEðlÞ is 41 strictly decreasing on the interval ½0; 2pÞ if l > 0, and strictly increasing 42 on ½0; 2pÞ if l < 0. ORDER REPRINTS

200026570_LSTA33_09_R2_090704

Wrapped Distributions 2065

1 In case of the wrapped Laplace distributions, the densities are 2 unimodal with a sharp peak at the mode. Basic properties of the densities 3 are described in the following result taken from Jammalamadaka and 4 Kozubowski (2003). 5 6 Proposition 2.1. Let fð; l; kÞ be the density of the WLðl; kÞ distribution 7 with l > 0. Then 8 (i) fð; l; kÞ is strictly decreasing on ð0; y Þ and strictly increasing 9 on ðy ; 2pÞ, where 10 11 1 2pl=k 12 ¼ l þ 2 e 1 : ð : Þ y lk ln k 2 12 13 k 1 e 2plk 14 15 Moreover 16 y > p for k < 1; y ¼ p for k ¼ 1; and y < p for k > 1: 17 ð : Þ 18 2 13 19 20 (ii) The maximum and the minimum values of fð; l; kÞ are 21 22 ð ; ; Þ¼ ð ; ; Þ¼ lk 1 þ 1 f 0 l k lim f y l k = 23 y!2p 1 þ k2 e2pl k 1 1 e 2plk 24 ð2:14Þ 25 26 and 27 k2 1 28 2pl=k þ 2 2plk þ 2 2plk e 1 1 k e 1 1 k 29 fðy ; l; kÞ¼e1þk2 ; ð2:15Þ 2pl=k 2plk 30 31 respectively. 32 33 (iii) For any given l 0 and k 0, we have 34 35 fðy; l; 1=kÞ¼fð2p y; l; kÞ; y 2½0; 2pÞ: ð2:16Þ 36 37 38 2.2. Trigonometric Moments and Related Parameters 39 40 Computation of the trigonometric moments and related parameters 41 of the wrapped exponential distribution is straightforward. For the 42 convenience of the reader we summarize some common parameters in ORDER REPRINTS

200026570_LSTA33_09_R2_090704

2066 Jammalamadaka and Kozubowski

1 Table 3. Note that since the fp’s are the Fourier coefficients, we have the 2 following Fourier representation of the wrapped exponential density: 3 X 4 ð Þ¼ ipy f y fpe 5 p "# 6 1 1 X l2 lp 7 ¼ 1 þ 2 cos py þ sin py : ð2:17Þ 2p 2 þ 2 2 þ 2 8 p¼1 l p l p 9 10 Similarly the trigonometric moments of the wrapped Laplace distribution 11 are also easy to derive, utilizing the mixture representation (2.10) and 12 formulas for the trigonometric moments of the wrapped exponential 13 distribution given in Table 3. Here, the density of the WLðl; kÞ 14 distribution admits the Fourier representation 15 "# 16 1 1 X kl½klðp2 þ l2Þ cos py þ pl2ð1 k2Þ sin py 17 fðy; l; kÞ¼ 1 þ 2 : 2p ð 2 2 þ 2Þð 2 2 þ 2Þ 18 p¼1 l k p k p l 19 ð2:18Þ 20 21 Other common parameters of the wrapped Laplace laws can be 22 obtained with ease, perhaps with the exception of the circular median. 23 They are summarized in Table 4. Note that the mean direction lies in 24 the interval ½0; p=2Þ for k 1 and in the interval ½3p=2; 2pÞ for k > 1 (this 25 restriction is caused by the fact that the mode is equal to 0). 26 A median direction x of a with density f is any 27 0 solution (in the interval ½0; 2pÞ)of 28 29 Z þ Z þ 30 x0 p x0 2p 1 fðyÞdy ¼ fðyÞdy ¼ ; ð2:19Þ 31 x0 x0þp 2 32 33 34 where the density f satisfies 35 ð Þ > ð þ Þ; ð : Þ 36 f x0 f x0 p 2 20 37 38 see, e.g., Mardia and Jupp (2000). Clearly, the median direction of the 39 wrapped symmetric Laplace distribution WLðl; 1Þ is equal to zero (and 40 coincides with the mean direction and the mode). In the following result, 41 proved in Jammalamadaka and Kozubowski (2003), we describe the 42 median direction for the general case. ORDER REPRINTS

200026570_LSTA33_09_R2_090704

Wrapped Distributions 2067 1 l

1 1 l 1 1 2 0 0 tan 3 tan p p > < l l 4 : 0 0 Þ l p l p l 2 = 1 ð

5 1 3 > < ; 0 0 Þ 2 l l p l l jÞ l > 6 < 1 WE 2 l j l l ÀÁ 2 ÀÁ þ ; ; ;

7 l ffiffiffiffiffiffiffiffi 1 Þ 0 p l p þ tan ; 2 ffiffiffiffiffiffiffiffi 1 p sin tan 1 cos tan l 1 l l 8  2 2 Þð 2 2 þ 2 p = 2 1 p p p l l 2 1 þ Þð 2 j j 3 j þ þ þ p 2 p l l þ 2 2 p 9 l 2 j j j l j 2 tan 2 l ; 4 ffiffiffiffiffiffiffiffiffiffi l ffiffiffiffiffiffiffiffiffiffi l ffiffiffiffiffiffiffiffiffiffi l lp l þ þ þ p l ð þ 2 2 1 l 2 4 tan 2 e j 4 l = p p l  l p 10 1 1 1 2 1 þ ð Þ Þð þ l 2 2 1 þ ffiffiffiffiffiffiffiffi 1 þ j ¼ ¼ ¼ ¼ ¼ ¼ þ 2 l l l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln

11 j ffiffiffiffiffiffiffiffi l ffiffiffiffiffiffiffiffi 1 p p þ þ p p p 0 p ln p p 1 1 2 tan b a r q ð a p p ð 1 l b 12 m 13 14 15 16 17 18 Þ Þ Þ ; 0 0

19 0 0 m m V p p

20 ... ; p 1 2 0 p 4 0 p

21 ð p Þ m 2 0 m b = ð ; ð V i ; 3 0 0 p 2 0

22 1 2ln r m 1 V V i þ sin cos e = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð p 1 ; p 2

23 p p 0 1 2 a 0 r r b a 1 r m p 24 r ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ p 0 1 0 2 0 p 0 0 p p p r g 25 g a f V m b s b 26 f 27 28 29 30 31 32 33 34

35 Trigonometric moments and related parameters of the wrapped exponential distribution 36 37 38 Table 3.

39 0 p 40 m

41 and p Central trigonometric moments Kurtosis Median See (2.19)–(2.20) ParameterTrignometric moments Definition Value Circular variance r Skewness Mean direction Circular standard deviation 42 Resultant length ORDER REPRINTS

200026570_LSTA33_09_R2_090704

2068 Jammalamadaka and Kozubowski

1 2 3

4 1 . ¼ Þ k

5 k ; l

6 ð 2 7 2 2 WL l p p 1 2 2 2 2 þ þ l þ l l þ 2 2 l 2 2 0 0 0 l 0 8 p l l ¼ ¼ ¼ ¼ ¼ ¼ ¼ 9 ¼ p p p 0 p 0 p p r b a r m a b 10 m 11 12 13 14 15 16 17 1 1 1 1 > 18 > k k k 19 k ; 0 ; k

20 l p l k 1 1

21 ; p ; k l p l k

22 tan tan ; 1 1 2 Þ Þ p

23 1 0 0 Þ Þ þ p 1 þ 2 m lk 2 2 lk m 2 tan l tan l k 1 p k 1

24 p = þ þ = Þ Þ 2 2 2 2 2 2 l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k l ffiffiffiffiffiffiffiffiffiffiffiffiffi k l k 2 25 2 2 p 1 þ 2 0 p p lk p lk p 0 p p l 2 tan tan l 2 1 2 m Þð 1 Þð p ð 1 m p k 2 ð 2 ð 3 ð

26 2 2 þ þ p þ p l l 2 kl þ 2 þ k p þ p p 2 2 k sin 2 cos

27 ffiffiffiffiffiffiffiffiffiffiffi 1 k tan 2 k tan 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 2 p p l l p r p ð  ð r 28  ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

29 r p p p 0 p 0 p p m a b b r m 30 a 31 32 33 34

35 Trigonometric moments and related parameters of the wrapped Laplace distribution 36 37 38 Table 4.

39 0 p 40 m

41 and trigonometric moments p Mean direction Parameter(s) Value(s) Case Trigonometric moments 42 Central r Resultant length ORDER REPRINTS

200026570_LSTA33_09_R2_090704

Wrapped Distributions 2069

1 2 Þ

2 2 l 6 l

3 þ 1

4 ð 2 1 l

5 2 2 þ 1. l l Þ 1

6 2 ¼ þ l 4 k

7 2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ln 1 ! l 1 1 þ 1 r 0 8 0 ¼ð ¼ ¼ ¼ 9 ¼ 0 2 0 0 0 0 1 g g s V 10 x 11 12 13 14 Þ 2

15 k = 2

16 l 4 l 17 þ 4 4 p 18 Þð ; 2 Þþ 0 k 3 1 =

19 where Þ 2 2 2½ þ < l k ; 2 Þ 2 1 = l

20 3 1 k 2 x k þ l þ 2 = þ þ > 1 < k 21 2 4 2 = 1 l ; k 2 Þð Þð 0 k l 2 1 2 2 2 Þð

22 k Þð k ; k ; = 3 = k 2 where 2 ln ¼ 0 0 2 = l l l 3

23 þ ; þ l þ þ > > ð 1 k k 4 2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 = = Þ Þð k l l

24 2 2 lp 2 0 4 p 1 Þ 2 lx l e k l 2 e 3 V 2 l k þ 2 Þð l þ ð 2 1 k

25 2 þ 2 ; l 4 þ 2 k , where 0 k 4 p 2 l þ 2 2 þ 2 1 ffiffiffiffiffiffiffiffiffiffiffi 1 V l Þð k 26 = ð l 2 3 þ 0 k k ; p lkp 3 2 V ð þ l l lkx

27 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln e x x = 2 þ 2 1 2 þ e 1 a ð 1 1 r ð  28 b ¼ ¼ ¼ 2 ¼ ¼ ¼ ¼ k 0 2 2

29 1 0 0 2 0 0 1 þ g ^ a V s 1 x b 30 g 31 32 33 34 35 36 37 38 39

40 For the definitions see Table 3. The entries in the last column correspond to the symmetric case with 41 standard deviation Circular variance Circular Kurtosis Note: Median 42 Skewness ORDER REPRINTS

200026570_LSTA33_09_R2_090704

2070 Jammalamadaka and Kozubowski

1 Proposition 2.2. Let Y WLðl; kÞ. Then, Y admits a unique median 2 direction given by 3  4 x ; for l > 0; 0 < k < 1; x ¼ ð2:21Þ 5 0 x þ p; for l > 0; k > 1; 6 7 where x 2½0; p is the unique solution of the equation 8 = 9 1 e lkx elx k 1 þ k2 ¼ : ð2:22Þ 10 1 þ k2 1 þ elkp 1 þ elp=k 2 11 12 Clearly, the median is less than p for k < 1 and greater than p for k > 1. 13 14 3. SOME IMPORTANT PROPERTIES 15 16 In this section we show that wrapped exponential and Laplace distri- 17 butions share some of the well-known properties of their analogues on 18 the real line. 19 20 21 3.1. Infinite Divisibility 22 23 An angular r.v. Y (and its ) is infinitely divi- Y ; ...; Y 24 sible if for any integer n 1 there exist i.i.d. angular r.v.’s 1 n 25 such that 26 d 27 Y1 þþYnðmod 2pÞ¼Y: ð3:1Þ 28 29 Since a circular variable obtained by wrapping an infinitely divisible 30 random variable is infinitely divisible (see Mardia and Jupp, 2000), the 31 infinite divisibility of the wrapped exponential and Laplace distribu- 32 tions follow from that of the linear exponential and Laplace distributions. 33 34 Proposition 3.1. If Y WEðlÞ with l 2 R, then Y is infinitely divisible. 35 Moreover, for any positive integer n 1 the equality in distribution (3.1) 36 holds where the Yi’s have the uniform circular distribution for l ¼ 0 37 and the wrapped with the ch.f. 38 = 39 1 1 n f ¼ ð3:2Þ 40 p 1 ip=l 41 42 for l 6¼ 0. ORDER REPRINTS

200026570_LSTA33_09_R2_090704

Wrapped Distributions 2071

1 Proposition 3.2. If Y WLðl; kÞ, where l; k 0, then Y is infinitely 2 divisible. Moreover, for any positive integer n 1 the equality in distri- 3 bution (3.1) holds with uniform circular variable Y1 if either l ¼ 0 or 4 k ¼ 0, and otherwise with 5 6 d 0 00 Y1 ¼ Y þ Y ðmod 2pÞ; ð3:3Þ 7 8 where Y0 and Y00 are independent wrapped gamma r.v.’s with ch.f.’s 9 = = 10 1 1 n 1 1 n and ; ð3:4Þ 11 1 ip=ðlkÞ 1 þ ip=ðl=kÞ 12 13 respectively. 14 15 16 3.2. Geometric Infinite Divisibility 17 18 Motivated by the stability property of the exponential distribu- 19 tion with respect to geometric compounding, Jammalamadaka and 20 Kozubowski (2001) introduced a notion of geometric infinite divisibility 21 for angular distributions. 22 Y 23 Definition 3.1. An angular r.v. is said to be geometric infinitely 2ð ; Þ Y ; Y ; ... 24 divisible if for any q 0 1 there exist i.i.d. angular r.v.’s 1 2 25 such that 26 d Y þþYn ðmod 2pÞ¼Y; ð3:5Þ 27 1 q 28 where n has the 29 q 30 k1 Pðnq ¼ kÞ¼ð1 qÞ q; k ¼ 1; 2; 3; ...; ð3:6Þ 31 32 The following properties of wrapped exponential and Laplace distribu- 33 tions, established in Jammalamadaka and Kozubowski (2001, 2003), 34 follow from analogous results of classical exponential and Laplace laws. 35 36 Proposition 3.3. If Y WEðlÞ, where l 2 R, then Y is geometric infi- 37 nitely divisible. Moreover, for any q 2ð0; 1Þ the equality in distribution 38 (3.5) holds, where the Yi’s have the uniform circular distribution for 39 l ¼ 0 and the WEðl=qÞ distribution for l 6¼ 0. 40 41 Proposition 3.4. If Y WLðl; kÞ, where l; k 0, then Y is 42 geometric infinitely divisible. Moreover, for any q 2ð0; 1Þ the equality ORDER REPRINTS

200026570_LSTA33_09_R2_090704

2072 Jammalamadaka and Kozubowski

1 in distribution (3.5) holds, where the Yi’s have the uniform circular dis- ¼ ¼ ð ; Þ ; > 2 tribution for lpffiffiffi 0 or k 0, and the WL lq kq distribution for l k 0. 3 Here, lq ¼ l= q and kq is the unique solution of the equation 4 5 1 pffiffiffi 1 kq ¼ q k : ð3:7Þ 6 kq k 7 8 3.3. Maximum Entropy Property 9 10 Recall that the entropy of a r.v. Y with p.d.f. f is defined as 11 Z 12 2p 13 HðYÞ¼ fðyÞ ln fðyÞdy; ð3:8Þ 14 0 15 16 and provides a measure of uncertainty. Jaynes (1957) proposed general 17 inference procedures of finding a distribution that maximizes the entropy, 18 and this method has been applied in a variety of fields including statistical 19 mechanics, stock-market analysis, queuing theory, and reliability (see, 20 e.g., Kapur, 1993). It is well-known that if the mean direction and circular 21 variance are fixed, then the entropy is maximized by the von Mises 22 distribution, and under no restrictions on f the entropy is maximal for 23 the circular uniform distribution (see, e.g., Kapur, 1993). As shown in 24 Jammalamadaka and Kozubowski (2001), the wrapped exponential ð Þ 25 distribution WE l maximizes the entropy under the condition Z 26 2p 27 yfðyÞdy ¼ m; 0 < m < 2p: ð3:9Þ 28 0 29 Proposition 3.5. Consider the class C of all circular r.v.’s with density f 30 satisfying the condition (3.9). Then, the maximum entropy is attained by 31 the WEðlÞ distribution, where l ¼ð2pxÞ 1 and x satisfies the equation 32 33 m 1 ¼ x : ð3:10Þ 34 2p e1=x 1 35 36 Moreover, the maximal entropy is 37 1 e2pl 1 2pe2pl 38 max HðYÞ¼ln þ : ð3:11Þ 39 Y2C l l 1 e2pl 40 41 Remark 3.1. By the monotonicity properties of the function gðxÞ¼ 42 x ðe1=x 1Þ 1, it follows that the distribution maximizing the entropy ORDER REPRINTS

200026570_LSTA33_09_R2_090704

Wrapped Distributions 2073

1 under the restriction (3.9) is wrapped exponential (l > 0) for 0 < m < p, 2 wrapped negative exponential (l < 0) for p < m < 2p, and circular 3 uniform (l ¼ 0) for m ¼ p. 4 5 Remark 3.2. This above result is analogous to the well-known property 6 of the exponential distribution (which maximizes the entropy among all 7 continuous probability distributions on ð0; 1Þ with a given mean, see, 8 e.g., Kapur, 1993). 9 10 11 SUMMARY 12 13 In this paper, we discuss circular distributions resulting from 14 wrapping the exponential and Laplace distributions on the real line 15 around the circle. The densities and distribution functions of wrapped 16 exponential distributions admit explicit forms, as do trigonometric 17 moments and related parameters. Wrapped exponential distributions 18 retain the important properties of infinite divisibility and maxim entropy 19 of the corresponding exponential distributions. The mixture of two 20 wrapped exponential distributions leads to a wrapped Laplace distribu- 21 tion, so that the properties of the former distribution are useful in study- 22 ing the latter one. Both distributions allow for skewness, and are 23 promising for modeling asymmetric directional data. 24 25 26 ACKNOWLEDGMENTS 27 28 We thank the anonymous referee and the editor for their comments. 29 30 31 32 REFERENCES 33 34 Barlow, R. E., Proschen, R. (1996). Mathematical Theory of Reliability. 35 Philadelphia: SIAM. 36 Batschelet, E. (1981). Circular Statistics in Biology. London: Academic 37 Press. 38 Bruderer, B. (1975). Variation in direction and spread of directions of 39 bird migration in northern Switzerland. Der Ornithologische 40 Beobachter 72:169–179 (in German). 41 Fisher, N. I. (1993). Statistical Analysis of Circular Data. Cambridge: 42 Cambridge University Press. ORDER REPRINTS

200026570_LSTA33_09_R2_090704

2074 Jammalamadaka and Kozubowski

1 Gatto, R., Jammalamadaka S. Rao. (2003). Inference for wrapped 2 a-stable circular models. Sankhya¯ 65:333–355. 3 Jammalamadaka S. Rao, Kozubowski, T. J. (2001). A wrapped exponen- 4 tial circular model. Proc. AP Acad. Sci. 5:43–56. 5 Jammalamadaka S. Rao, SenGupta, A. (2001). Topics in Circular 6 Statistics. New York: World Scientific. 7 Jammalamadaka S. Rao, Kozubowski, T. J. (2003). A new family of 8 circular models: The wrapped Laplace distributions. Adv. Appl. 9 Statist. 3:77–103. 10 Jander, R. (1957). Die optische Richtungsorientierung der roten 11 Waldameise. Formica rufa L. Z. vergl. Physiologie 40:162–238. 12 Jaynes, E. T. (1957). Information theory and statistical mechanics. 13 Phys. Rev. 106:620–630; 108:171–190. 14 Kapur, J. N. (1993). Maximum Entropy Models in Science and 15 Engineering. New York: Wiley, Revised Edition. 16 Kotz, S., Kozubowski, T. J., Podgo´rski, K. (2001). The Laplace Distribu- 17 tion and Generalizations: A Revisit with Applications to 18 Communications, Economics, Engineering, and Finance. Boston: 19 Birkhauser. 20 Le´vy, P. L. (1939). Addition des variables ale´atoires de´finies sur une 21 circonfe´rence. Bull. Soc. Math. France 67:1–41. 22 Mardia, K. V. (1972). Statistics of Directional Data. London: Academic 23 Press. 24 Mardia, K. V., Jupp, P. E. (2000). . 2nd ed. 25 New York: Wiley. 26 Matthews, G. V. T. (1961). ‘‘Nonsense’’ orientation in mallard Anas 27 Platyrhynchos and its relation to experiments on birds navigation. 28 Ibis 103a:211–230. 29 Pewsey, A. (2002). Testing circular symmetry. Canad. J. Statist. 30: 30 591–600. 31 Schmidt-Koenig, K. (1964). Uber die orientierung der vogel; Experimente 32 und probleme. Naturwissenschaften 51:423–431. 33 34 35 36 37 38 39 40 41 42

Request Permission or Order Reprints Instantly!

Interested in copying and sharing this article? In most cases, U.S. Copyright Law requires that you get permission from the article’s rightsholder before using copyrighted content. All information and materials found in this article, including but not limited to text, trademarks, patents, logos, graphics and images (the "Materials"), are the copyrighted works and other forms of intellectual property of Marcel Dekker, Inc., or its licensors. All rights not expressly granted are reserved.

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly. Simply click on the "Request Permission/ Order Reprints" link below and follow the instructions. Visit the U.S. Copyright Office for information on Fair Use limitations of U.S. copyright law. Please refer to The Association of American Publishers’ (AAP) website for guidelines on Fair Use in the Classroom.

The Materials are for your personal use only and cannot be reformatted, reposted, resold or distributed by electronic means or otherwise without permission from Marcel Dekker, Inc. Marcel Dekker, Inc. grants you the limited right to display the Materials only on your personal computer or personal wireless device, and to copy and download single copies of such Materials provided that any copyright, trademark or other notice appearing on such Materials is also retained by, displayed, copied or downloaded as part of the Materials and is not removed or obscured, and provided you do not edit, modify, alter or enhance the Materials. Please refer to our Website User Agreement for more details.

Request Permission/Order Reprints

Reprints of this article can also be ordered at http://www.dekker.com/servlet/product/DOI/101081STA200026570