Wrapped Generalized Gompertz Distribution

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s Roy and Shams Adnan, J Biom Biostat 2012, 3:6

c J s Journal of Biometrics & Biostatistics DOI: 10.4172/2155-6180.1000153 ISSN: 2155-6180

OpenOpen Acc Accessess Wrapped Generalized Gompertz distribution: an Application to Ornithology Shongkour Roy1* and Mian Arif Shams Adnan2 1Center for Nutrition and Food Security (CNFS), International Center for Diahorreal Diseases Research (ICDDRB), Mohakhali, Dhaka, Bangladesh 2Department of Statistics, Jahangirnagar University, Dhaka, Bangladesh

Abstract In many scientific fields such as biology (orientation of birds), geology (orientations of feldspar laths)and meteorology (wind direction and ozone concentration), data occur as angular forms. In this paper, a class of wrapped distribution called wrapped generalized Gompertz distributions are introduced (WGG). Characteristic function and fundamental properties of this distribution are described. Some theorems that relate the distribution to some other circular distributions are established. Applications to density estimation and goodness-of-fit tests are used to analyze data on the heading of orientation of the nest of noisy scrub birds, and it is shown that the model fits much better than some other existing circular symmetric and non-symmetric models.

Keywords: Directional data; Generalized Gompertz distribution; The main objective of this article is to introduce a class of Trigonometric moments; Wrapped distribution; Watson statistics distribution that wrapped generalized Gompertz distribution, and provide an application in ornithology. Introduction The rest of the paper is organized as follows: Sect. 3.1 provides the Levy [1] proposed a novel method for introducing a wrapped probability and cumulative density function of the wrapped generalized variable to the symmetric and non-symmetric distribution, which is Gompertz distribution. Sect. 3.2 makes explicit the characteristic known in the literature as a wrapped distribution. Several authors have function and the fundamental properties with related parameters. done the extensive work on introducing wrapped distribution such as Maximum likelihood estimation is given in Sect. 3.5. In Sect. 3.6, an wrapped exponential, wrapped Laplace, wrapped Cauchy, wrapped application in ornithology has been discussed briefly, illustrating the Chi-square and wrapped weighted exponential distribution. Moreover, modeling potential of wrapped generalized Gompertz distribution. their statistical properties and inference procedures have been Finally, the conclusion of the paper appears in Sect. 4. investigated thoroughly in the literature; for example Jammalamadaka and Kozubowski [2,3], Levy [1], Roy and Adnan [4-6]. However, Figure 1 depicts some wrapped generalized Gompertz densities. it seems that no attempts were made to implement Levy [1] idea for Derivation of the density non-symmetric distribution, in case of wrapped generalized Gompertz distributions, which produce equally interesting circular models. In this section, the definition of the wrapped WGG distribution was introduced, denoted by WGG (a, b, c), and cumulative density function For the real line, Willemse and Kaas [7] first introduced the of the same distribution. generalized Gompertz distribution and its different properties. Furthermore, they showed that generalized Gompertz distribution Let us consider θ= θ(x)=X(mod 2π) (3) posses some good properties and can be used as a good fit to survival Then, θ is wrapped (around the circle) or a generalized Gompertz model in biology, compared to the other popular distributions such as circular random variable, that for θ∈(0, 2π) has the probability density exponential, Gamma or logistic distributions. function ∞ In this paper, a class of wrapped generalized Gompertz distribution g()θ=∑ fx ( θ + 2 k π ) is introduced. This class of distribution is found with wrapping a k =−∞ generalized Gompertz distribution, introduced by Willemse and Kaas 1 ∞  θ+2 π k − a θ+2 π k − a  [7] and Coelho [8] around the circumference of a unit circle, and GG g()θ = ∑ expc − e b  ; (4) bΓ() c k =−∞ b distribution defined as follows: a random variable X is said to have a   generalized Gompertz distribution with a shape parameter b>0, scale θ∈[0,2 π ) ,b > 0, c > 0,θ + 2 π k > a , a ∈[ 0,2π ) parameter c>0, and location parameter -∞

And cumulative density function (CDF) Received August 30, 2012; Accepted October 29, 2012; Published November  x− a  03, 2012 Γc,exp   b  F() x =1 − , -∞

J Biom Biostat ISSN:2155-6180 JBMBS, an open access journal Volume 3 • Issue 6 • 1000153 Citation: Roy S, Shams Adnan MA (2012) Wrapped Generalized Gompertz distribution: an Application to Ornithology. J Biom Biostat 3:153. doi:10.4172/2155-6180.1000153

Page 2 of 4 which is the probability density function of wrapped generalized ∞ Γ()itb + c ita itb ita itbc+ −1 − y ita . Gompertz distribution. The random variable θ has a wrapped =e E() y = e∫ y e dy= e 0 Γ()c generalized Gompertz distribution, with a parameter and the above Hence, using the Eq. (5), the characteristic function of the wrapped PDF as follows: θ ~WGG(a,b,c). generalized Gompertz distribution is obtained as

1 Figure 1 depicts the probability density function of the wrapped ipa Γ()ipb + c φ ()p= e ;p = ± 1, ± 2, ± 3,..., where i =() −1 2 generalized Gompertz distribution. Γ()c µ Now, according to Mardia and Jupp [9], Jammalamadaka and i p φ()p= ρ p e (6) SenGupta [10] cumulative density function of wrapped generalized Gompertz distribution can be obtained as Γ()ipb + c Where ρ = ,.µ = pa ∞ p Γ()c p G()θ=∑ F()() θ +2 π k − F 2 π k ;0 ≤ θ < 2 π . m=0 Trigonometric moments and related parameters Using the immediate equation, the cumulative density function Using the definition of the trigonometric moment, (CDF) of wrapped generalized Gompertz distribution could be got. φp= α p + i β p ; p = ±1, ± 2,... .   θ +2 π k − a    2π k − a      Γc,exp   Γ c,exp   ∞   b   b  And therefore, the non-central trigonometric moments of the G(θ )=  1 − − 1 −   ∑   respective distribution are: k =−∞  Γ()c   Γ()c          α= ρcos( µ )       p p p And βp= ρ psin( µ p ) . Γ()ipb + c ∞ 1   2π k− a   θ+2 π k − a  So, the equation comes to be as α = cos(pa ) and = Γ c,exp − Γ c,exp . p Γ()c ∑        Γ()ipb + c k =−∞ Γ()c   b    b  β p = sin(pa ) . Γ()c Fundamental properties of the density Now, according to Jammalamadaka and SenGupta [10], an alternative expression for the PDF of the wrapped generalized Gompertz The trigonometric moment of order, p, for a wrapped circular distribution can be obtained, using the trigonometric moments as distribution corresponds to the value of the characteristic function of 1  ∞  the unwrapped r.v. X say φ x ()t at the integer value p [10]. g(θ )= 1 + 2∑ (αp cosp θ+ β p sin p θ ) ; θ∈[0,2 π ) . 2π  p=1  φ()()p= φ t . (5) x As such substituting the value of αp and βp in the immediate above On the real line, the characteristic function of the generalized equation, the equation got is: Gompertz distribution is 1  ∞  Γ()ipb + c Γ()ipb + c  g(θ )= 1 + 2 ∑  cos(pa )cos pθ + sin(pa )sin pθ  2π  p=1 Γ()c Γ()c  Use the fact that X bln Y+ a for a Gamma(c, 1) random variable Y   1  ∞ Γ()ipb + c  φ ()t= E eitx = E eit() bln Y+ a  =1 + 2 ∑ (cos(pa )cos pθ + sin( pa )sin pθ ) x ()   2π  p=1 Γ()c  1  ∞ Γ()ipb + c  ∴g(θ ) = 1 + 2 ∑ cosp (θ − a ); θ∈[]0,2 π ,b > 0, θ + 2 π k > a, 2π  p=1 Γ()c  Wrapped generalized Gompertz distribution   a ∈[]0,2π .

a=1, b=1.25, c=9 This is the alternative PDF of wrapped generalized Gompertz distribution. Now, the central trigonometric moment is defined as a=1, b=1, c=8 α= ρ µ − µ p pcos( p p 1 ) And βp= ρ psin( µ p − p µ1 ). Thus, the central trigonometric moments of the same distribution will be g(th) Γ()ipb + c Γ()ipb + c Therefore,α p = cos(ap− pa ) and β = sin ()ap− pa (7) Γ()c p Γ()c Γ()ipb + c a=1, b=1.75, c=6 ∴α p = And β = 0 . Γ()c p Γib + c a=1, b=2, c=4 () The resultant length, ρ= ρ1 thus ρ = , Γ()c The Mean direction, µ= µ1 so that µ1 = a

0.00 0.05 0.10 0.15 0.20 0.25 0.15 0.20 0.25 0.100.30 0.35 0.00 0.05 The circular variance is Γ()ib + c 0 1 2 3 4 5 6 V =1 − ρ , 0 V0 =1 − , Γ()c th And the circular standard deviation is

Figure 1: Some wrapped generalized Gompertz densities. σ 0 = −2log (1 −V0 )

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= −2log (1 − 1 + ρ ) n  θ  n  n c ∑ i  ∑ θi  nac 2π nck na i=1 2 π nk    − i=1 ∞  − ∞  Γ()ib + c 1 b b b  eb e b e b  = −2log l =   e e e − e e e  , Γ ∑  ∑  ()c bΓ() c   k =−∞   k =−∞  α −(1 −V )4     ς 0 = 2 0    The kurtosis is 2 2 given by V0 4 The log likelihood function L = ln(l) is given by n n Γ()2ib + c Γ() ib + c  θ θ ∑ i −   c i ∞ i=1 na ∞ 2π nk   ∑ π − ΓΓ()c ()c nac i=1 2 nck b b b ς 0 =   L= − nlog()() b − n log() Γ c − + +∑ −e − e − ∑ e 2 2 b b=−∞ b =−∞ .  Γ()ib + c  k k 1−   Γ()c  The first partial derivatives of the log likelihood L with respect to Γ()2ib + c a,b and c are obtained as follows α = . na Since from (7) for p=2, 2 ∂Lnc n − Γ()c = − + e b , (10) Theorem: The difference between two wrapped generalized ∂a b b n n n θ Gompertz distributed angular variables are also a wrapped generalized c θ θ ∑ i ∑i ∞ ∑ i i=1 na ∞ 2π nk ∂L n nac 2π nck na − 2π n = − + −i=1 − +i=1 b −b + b Logistic distribution. 2 2 ∑ 2 2 e2 e 2 ∑ Ke , ∂b b b bk =−∞ b b b b k =−∞ Proof: Let us consider two independent angular variables with n (11) generalized Gompertz distributions are θ and θ its difference θ -θ . θ 1 2 1 2 ∂L n na ∑ i ∞ 2π nk = − Ψ() Γ()c − +i=1 + ∑ . (12) i.e. θ 1  WGG() a1,, b 1 c 1 and θ 2  WGG() a2,, b 2 c 2 , providing in ∂c Γ() c b bk=−∞ b fact a generalization of the wrapped logistic distribution, assuming The likelihood equations can be obtained by setting the first partial a1= a, a 2 = 0, equal parameters c1 and c2, the characteristic function of derivatives of L to zeros. That is, the likelihood equations take the θ1-θ2, with following form: na nc n − θ 1  WGG() a,, b c1 and θ 2  WGG()0, b , c2 is − +e b = 0 , (13) b b Γ + Γ − n n n iap ()()c1 ibp c2 ibp ∑θi φθ− θ ()()()p= φθ p φ θ − p = e c θ θ na 2π nk 1 2 1 2 (8) ∑i ∞ ∑ i i=1 − ∞ ΓΓ()()c c n nac = 2π nck = na 2π n 1 2 − + −i 1 − +i 1 b −b + b = 2 2 ∑ 2 2 e2 e 2 ∑ Ke 0, Since, we know that ch.. f satisfies uniquely any distribution function, b b bk =−∞ b b b b k =−∞ thus θ -θ is also wrapped generalized Logistic distribution [11]. n (14) 1 2 θ n na ∑ i ∞ 2π nk Wrapped logistic distribution − Ψ() Γ()c − +i=1 +∑ = 0. (15) Γ()c b bk=−∞ b In special case, wherein (8) a1= a 2 =0, b 1 = b 2 = 1 and c1= c 2 = 1, the To derive the MLEs of the unknown parameters a, b, and c, the difference between two standard wrapped Gompertz angular variables is above non-linear system of equations in a, b, and c has to be solved. got, such an angular variable has a wrapped logistic distribution, i.e.

iap Γ(1 +ip ) Γ (1 − ip ) From (13), the MLE of as a function of estimators of the rest of the φθ− θ ()p= e bˆ, cˆ 1 2 ΓΓ(1) (1) parameters () as aˆ can be got [13]. =eiap B(1 + ip ,1 − ip ) For ip < 1 An application in ornithology which is the ch.f. of wrapped logistic distribution, and its density The following application shows the effectiveness of the WGG function is given below model when applied to some data from a real problem. Such data 1   ∞  frequently form orientation in biology [14-18]. The data set was g(θ )= 1 + 2∑ β ()() 1 −ip , i + ip cosθ − a  ; θ ∈[ 0,2 π) ,a ∈[ 0,2 π ) . taken from Fisher [19]. The main goal is to provide for monitoring 2π   p=1  the orientations of the nest of noisy scrub birds, along the bank of a Maximum Likelihood Estimation creek bed. The species of birds in this region are more important for In this section, the maximum likelihood estimators of the unknown the biodiversity. This study is important due to the growing evidence of the vulnerability of the noisy scrub birds to chance in biodiversity. parameters (a,b,c) of the WGG (a,b,c) are derived. Assume θ1, θ 2 ,..., θn be a random sample of size n from WGG (a,b,c), then the likelihood To demonstrate the modeling behavior of the wrapped generalized functions l is [12] Gompertz distribution, we consider nest orientations of noisy of scrub n birds, discussed by Fisher [19]. For the orientation of the nests of 50 l= ∏ g()θi ;,, a b c . (9) noisy scrub birds along the bank of a creek bed, the data (degree) are i=1 as given below: Substituting from (9) into (4) to be get 160°, 145°, 225°, 230°, 295°, 295°, 140°, 140°, 140°, 205°, 215°, 135°, n 1 ∞  θ+2 π k − a θ+2 π k − a  110°, 240°, 230°, 250°, 30°, 215°, 215°, 135°, 110°, 240°, 105°, 125°, 125°, l = ∏ ∑ expc − e b , i=1 bΓ() c k =−∞  b  130°, 160°, 105°, 90°, 130°, 200°, 240°, 105°, 125°, 125°, 125°, 130°, 160°, 160°, 250°, 200°, 200°, 240°, 240°, 240°, 250°, 250°, 250°, 140°, 140° n  θ+ π − n  n ∑()i 2 k a ∑ ()θi +2 π k − a  ∞ i=1 ∞ i=1  Assuming a symmetric wrapped generalized Gompertz model 1  c b l =    ∑e b − ∑ ee , (with b=1), the first step is to find the maximum likelihood estimates bΓ() c   k =−∞ k =−∞      of the parameters a and c for a WGG distribution. This was carried out   with maple in the following results of radian measure over (0, 2π). Again

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Distributions MLEs Log-likelihood AIC BIC 9. Mardia KV, Jupp PE (1972) Directional Statistics. John Wiley & Sons, Academic press, New York. vM(µ,k) (3.0640, 1.3090) -22.8794 57.7589 69.2310 WN(µ,ρ) (3.0900, 0.5773) -22.9907 57.9814 69.4535 10. Jammalamadaka SR, SenGupta A (2001) Topics in circular Statistics. World scientific, Singapore. WC(µ,ρ) (2.7420, 0.4080) -23.1427 58.3054 69.7776 WGG(a,c) (4.4600, 0.0570) -22.6904 57.2189 68.6910 11. Dattatreya RAV, Ramabhadra SI, Girija SVS (2007) On wrapped version of some life testing models. Commun Stat Theory Methods 36: 2027-2035. Table 1: Summary of fits for the noisy scrub bird’s data. 12. Gatto R, Jammalamadaka SR (2002) Inference for wrapped symmetric -stable using R with circstat package, the MLE of the other circular models circular models. Sankhya (In press). is fit therein, i.e. vonMises (vM), wrapped Cauchy distribution (WC) 13. Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood and wrapped normal (WN). Table 1 provides the relevant numerical estimators and likelihood ratio tests under nonstandard conditions. J Am Stat Assoc 82: 605-610. summaries for the four fits. 14. Jander R (1957) The optical directional orientation of the red wood ant (Formica To check the goodness of fit of the distributions, WGG, vM, WC Rufa L.). Cf Z Physiol 40: 162-238. and WN, both the kuiper and Watson statistics for each model are 15. Matthews GVT (1961) “Nonsense” orientation in mallard Anas Platyrhynchos computed. These tests may be found in reference Jammalamadaka and its relation to experiments on bird navigation. Ibis 103a: 211-230. and SenGupta [10]. Using the values of aˆ and cˆ in table 1, the kuiper 16. Schmidt-Koenig K (1964) Uber die orientierung der vogel; Experimente und statistic and the Watson statistic for a WGG distribution has the values, probleme. Naturwissenschaften 51: 423-431. 2 V50 = 6.5548 and W50 = 3.7983 , respectively [20]. Again using the values of ˆ 17. Bruderer B (1975) Variation in direction and spread of directions of bird migration µˆ and k in table 1, the kuiper statistic and the Watson statistic for a vM in northern Switzerland. Der Ornithologische Beobachter 72: 169-179. distribution has the values, V = 6.9323 and W 2 = 3.9240 respectively. 50 50 18. Edward B (1981) Circular Statistics in Biology. Academic Press, London. For WC and WN distribution, the kuiper and Watson statistic values, 2 2 19. Fisher NI (1993) Statistical analysis of circular data. Cambridge University V50 = 6.9096 , W50 = 3.7995 and V50 = 6.8604, W50 = 3.8805, respectively. The smaller value of the both statistic, AIC and BIC for a WGG distribution press, Melbourne. indicate a better fit. Thus, it is confirmed that the wrapped generalized 20. Umbach D, Jammalamadaka SR (2009) Building asymmetry into circular Gompertz distribution provides a superior model for these data than distributions. Stat Probab Lett 79: 659-663. the vonMises, wrapped Cauchy and wrapped normal distributions. Conclusions In this paper, an expression for the probability density function of wrapped generalized Gompertz distribution is suggested. Some of the other wrapped distributions have also been unfolded through proper specification of the parameters. The alternative form of the PDF of the same distribution is also obtained using trigonometric moments. A class of basic properties is also found here. Modules programmed in or with circstat package, of maple-12 for the concern computations and graphics are available. Finally, effectiveness of this model with real data from ornithology was depicted. Acknowledgments The author thanks the editor and two anonymous referees for thoughtful comments and corrections. The paper have been substantially improved due to their constructive feedbacks.

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J Biom Biostat ISSN:2155-6180 JBMBS, an open access journal Volume 3 • Issue 6 • 1000153