On Wrapping of Quasi Lindley Distribution

mathematics

Article On Wrapping of Quasi Lindley Distribution

Ahmad M. H. Al-khazaleh 1,* and Shawkat Alkhazaleh 2

1 Department of Mathematics, Al-Albayt University, Al-Mafraq 25113, Jordan 2 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13132, Jordan; [email protected] or [email protected] * Correspondence: [email protected] or [email protected]

Received: 21 August 2019; Accepted: 26 September 2019; Published: 9 October 2019

Abstract: In this paper, as an extension of Wrapping Lindley Distribution (WLD), we suggest a new circular distribution called the Wrapping Quasi Lindley Distribution (WQLD). We obtain the probability density function and derive the formula of a cumulative distribution function, characteristic function, trigonometric moments, and some related parameters for this WQLD. The maximum likelihood estimation method is used for the estimation of parameters.

Keywords: circular statistics; wrapping; quasi lindley; trigonometric moments

1. Introduction Directional data have many new and individual characteristics and tasks in modelling statistical analysis. It occurs in many miscellaneous fields, such as Biology, Geology, Physics, Meteorology, Psychology, Medicine, Image Processing, Political Science, Economics, and Astronomy Mardia and Jupp [1]. The wrapping of a linear distribution around the unit circle is one of the various ways in which to find a circular distribution. There are a lot of studies that have been done in this area. L’evy [2] presented wrapped distributions. Jammalamadaka and Kozubowski [3] studied circular distributions derived by wrapping the classical exponential and Laplace distributions on the real line around the circle. In 2007, Rao et al. [4] discussed wrapping of the lognormal, logistic, Weibull, and extreme-value distributions for the life testing models. Meanwhile, the wrapped weighted exponential distribution was introduced by Roy and Adnan [5]. Rao et al. [6] obtained the characteristics of a wrapped gamma distribution. Joshi and Jose [7] introduced the wrapped Lindley distribution, and they also studied the properties of the new distribution, such as characteristic function and trigonometric moments. Adnan and Roy [8] introduced the wrapped variance gamma distribution and applied it to wind direction. Lindley [9,10] suggested the Lindley distribution (LD) and he defined the probability density function (pdf) and cumulative distribution function (CDF) as follows:

α2 f (x, α) = (1 + x) e−αx ; x > 0, α > 0. (1) α + 1

αx F(x, α) = 1 − 1 + e−αx ; x > 0, α > 0. (2) α + 1 while the wrapped Lindley distribution was introduced by Joshi and Jose (2018) [11]. They deﬁned the PDF and the CDF of the wrapped LD, respectively, as follows: " # λ2 1 + θ 2πe−2πλ ( ) = −λθ + ∈ [ ) > g θ e − 2 , θ 0, 2π , λ 0. (3) λ + 1 1 − e 2πλ 1 − e−2πλ

Mathematics 2019, 7, 930; doi:10.3390/math7100930 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 930 2 of 9

1 λθ G(θ) = 1 − e−λθ − e−λθ − 1 − e−2πλ λ + 1 ! (4) 2πλ e−2πλ − −λθ ∈ [ ) > 1 e 2 , θ 0, 2π , λ 0. λ + 1 1 − e−2πλ Shanker and Mishra [12] derived the Quasi Lindley distribution (QLD). They introduced the QLD with two parameters, one for scale (α) and one for shape (β). The (PDF) and (CDF) of (QLD), respectively, was deﬁned as follows:

β(α + βx) f (x, α, β) = e−βx ; x > 0, β > 0, α > −1. (5) α + 1

1 + α + βx F(x, α, β) = 1 − e−βx ; x > 0, β > 0, α > −1. (6) α + 1 Ghitany et al. [13] established many properties of LD and proved that the LD was a better model than the exponential distribution in many ways by using a real data set. They utilized the method of maximum likelihood to provide evidence that the ﬁt of the Lindley distribution was better. In this paper, as an extension of the Wrapping Lindley Distribution (WLD), we suggest a new circular distribution called the Wrapping Quasi Lindley Distribution (WQLD). We also obtain the probability density function and derive the formula of the cumulative distribution function, as in Section2. In Section3 we drive the characteristic function, which is trigonometric moments with some related parameters for this WQLD. In Section4, the maximum likelihood estimation method is used for estimation of parameters.

2. Circular Distribution A circular distribution is a probability distribution whose total probability is concentrated on the circumference of a unit circle (see [14]), where the points on the unit circle characterize a direction, each direction representing the values of its probabilities. The range of a circular random variable θ, stated in radians, may take 0 ≤ θ ≤ 2π or −π ≤ θ < π. There are discrete or continuous circular probability distributions which satisfy the property XW = X(mod2π).

Deﬁnition 1 ([14,15]). If θ is a random variable on the real with distribution function F(θ), then the random variable θW of the wrapped distribution satisﬁes the following properties: R 2π 1. 0 f (θ)dθ = 1 and 2.f (θ) = f (θ + 2kπ). for any integer k and f (θ) is periodic.

Thus, we can deﬁne the Wrapped Quasi Lindley Distribution (WQLD) as follows:

Deﬁnition 2. A random variable θ is said to have a Wrapped Quasi Lindley Distribution (WQLD) as follows:

∞ ∞ θ(α + β (θ + 2kπ)) g(θ) = g(θ + 2kπ) = e−β(θ+2kπ) ∑ ∑ + k=0 k=0 α 1 (7) θe−βθ ∞ = (α + β (θ + 2kπ)) e−2βkπ. + ∑ α 1 k=0 Mathematics 2019, 7, 930 3 of 9

It can also be simpliﬁed to: ! θe−βθ 1 1 ∞ k g(θ) = α + βθ + β2π k e−2βπ + −2βπ −2βπ ∑ α 1 1 − e 1 − e k=0 ! (8) θe−βθ 1 1 e2βπ = + + α − βθ − β2π 2 . α + 1 1 − e 2βπ 1 − e 2βπ −1 + e2βπ

The cumulative distribution function of WQLD can be derived as follows:

∞ G(θ) = ∑ {F(θ + 2kπ) − F(2kπ)} k=0

∞ 1 + α + β(2kπ) 1 + α + β(θ + 2kπ) = e−β(2kπ) − e−β(θ+2kπ) . ∑ + + k=0 α 1 α 1

∞ e−β(2kπ) n o G(θ) = (1 + α + β(2kπ)) − (1 + α + β(θ + 2kπ)) e−βθ . (9) ∑ + k=0 α 1

∞ −2βπk Remark 1. We used the ratio test to check whether the series ∑k=0 k e in both the PDF and CDF of WQLD converged as follows:

−2βπk+1 ak+1 (k + 1) e (k + 1) 1 lim = lim = lim = < 1. k 2βπ 2βπ k→∞ ak k→∞ k e−2βπ k→∞ ke e

Figure1 shows the circular representation of the PDF of WQLD for different values of α, keeping the value for the parameter β at 3.0.

Β=3,Α=1 0.08

Β=3,Α=2

0.06 Β=3,Α=3

Β=3,Α=4 0.04

0.02

0.02 0.04 0.06 0.08 0.10 0.12

Figure 1. PDF of the WQLD distribution (Circular Representation), β = 3. Mathematics 2019, 7, 930 4 of 9

The same circular representation for the PDF of WQLD with different values of β, keeping the value for the parameter α at 3.0 as in Figure2.

0.3

Β=1,Α=3 0.2

Β=1.5,Α=3

Β=2,Α=3 0.1

Β=2.5,Α=3

-0.2 -0.1 0.1 0.2

-0.1

Figure 2. PDF of the WQLD distribution (Circular Representation), α = 3.

Figure3 shows the circular representation of the CDF of WQLD for different values of α, keeping the value for the parameter β at 3.0.

1.0

0.5

Β=1,Α=3

Β=1.5,Α=3

- - 1.0 0.5 0.5 1.0 Β=2,Α=3

Β=2.5,Α=3

-0.5

-1.0

Figure 3. CDF of the WQLD distribution (Circular Representation), β = 3. Mathematics 2019, 7, 930 5 of 9

The same circular representation for the CDF of WQLD with different values of β, keeping the value for the parameter α at 3.0, as in Figure4.

1.0

0.5

Β=3,Α=1

Β=3,Α=2

-1.0 -0.5 0.5 1.0 Β=3,Α=3

Β=3,Α=4

-0.5

-1.0

Figure 4. CDF of the WQLD distribution (Circular Representation), α = 3.

3. Characteristic Function of Wrapped Quasi Lindley Distribution

itθ The characteristic function of Xw for the distribution function G(θ) is given by ϕθ(t) = E e . The characteristic function for the Quasi Lindley Distribution given as follows: " # itθ β α (β − it) + β ϕθ(t) = E e = . (10) α + 1 (β − it)2

Now, we can ﬁnd the characteristic function of the circular model: itθ R 2π itθ by ϕθ(t) = E e = 0 e g(θ)dθ

Z 2π itθ itθ (α) −βθ E e = e − θe 0 (α + 1) 1 − e 2βπ ! (β) 2βπ + θ2e−βθ + θe−βθ dθ (α + 1) 1 − e−2βπ (α + 1) e2βπ (11) Z 2π (α) −βθ itθ = − θe e 0 1 − e 2βπ (α + 1) ! (β) β2π + θ2e−βθeitθ + θe−βθeitθ dθ. 1 − e−2βπ (α + 1) e2βπ (α + 1) Mathematics 2019, 7, 930 6 of 9

Rearranging the Equation (11), we have: " # Z 2π itθ (α) β2π −θ(β−it) E e = − + θe dθ 1 − e 2βπ (α + 1) e2βπ (α + 1) 0 Z 2π (β) 2 −θ(β−it) + − θ e dθ. 1 − e 2βπ (α + 1) 0

Assuming the previous integrals consist of two parts, the ﬁrst part can be calculated as follows: ! Z 2π (α) β2π −θ(β−it) I = − + θe dθ 1 − e 2βπ (α + 1) e2βπ (α + 1) 0 ( = − ( − ) = u = = let u θ β it ; θ −(β−it) ; θ 0 u 0 = − ( − ) = du = = − ( − ) du β it dθ; dθ −(β−it) ; θ 2π u 2π β it Z −2π(β−it) u du = eu 0 − (β − it) − (β − it) 1 Z −2π(β−it) = ueudu (β − it)2 0 1 = 1 − e−2π(β−it)(2π (β − it) + 1) (β − it)2 ! −2π(β−it) −2π(β−it) α β2π 1 − [2π (β − it)] e − e = + . 1 − e−2βπ e2βπ (α + 1)(β − it)2

Z 2π (β) 2 −θ(β−it) J = − θ e dθ 1 − e 2βπ (α + 1) 0 h −2π(β−it) i (β) 2 1 − e (2π (β − it)(π (β − it) + 1) + 1) = . 1 − e−2βπ (α + 1) (β − it)3 Now, combining both integrals I and J, we have the characteristic function of the WQLD: ! −2π(β−it) −2π(β−it) α 2βπ 1 − [2π (β − it)] e − e ϕθ(t) = + 1 − e−2βπ e2βπ (α + 1)(β − it)2 h i 2β 1 − e−2π(β−it) (2π (β − it)(π (β − it) + 1) + 1) + . 1 − e−2βπ (α + 1)(β − it)3

We can simplify the characteristic function of the WQLD as follows:

1 ϕθ(t) = (α + 1)(β − it)2 " ! α 2βπ + 1 − [(2π (β − it)) + 1] e−2π(β−it) + 1 − e−2βπ e2βπ (12) h i  2β 1 − [2π (β − it)(π (β − it) + 1) + 1] e−2π(β−it)  . 1 − e−2βπ (β − it) Mathematics 2019, 7, 930 7 of 9

By the trigonometric deﬁnition, we have φp = αp + iβp p = 0, ±1, ±2, ..., where αp = E(cos pθ) and βp = E(sin pθ).

αp = E(cos pθ) " # Z 2π 1 −βθ 2 −βθ = cos pθ − αθe + βθ e dθ+ 0 1 − e 2βπ (α + 1) Z 2π β2π cos pθ θe−βθ dθ. 0 e2βπ (α + 1)

By some simpliﬁcations, we have " # Z 2π α β2π −βθ αp = − + θe [cos pθ] dθ+ 1 − e 2βπ (α + 1) e2βπ (α + 1) 0 (13) Z 2π β 2 −βθ − θ e [cos pθ] dθ. 1 − e 2βπ (α + 1) 0

Since Equation (13) contains two integrals, we can integrate separately, as follows: " # Z 2π α β2π −βθ I = − + θe [cos pθ] dθ 1 − e 2βπ (α + 1) e2βπ (α + 1) 0

e−2πβ[2p(πp2+πβ2+β) sin(2πp)−((2πβ−1)p2+β2(2πβ+1)) cos(2πp)−e2πβ p2+β2e2πβ] = . (14) (p2+β2)2

Z 2π β 2 −βθ J = − θ e [cos pθ] dθ 1 − e 2βπ (α + 1) 0 2βe−2πβ p(((2π2 p4+4π2 β2 p2+4πβp2)−p2)+β2((2π2 β2+4πβ)+3)) sin(2πp) = − (15) (1−e−2βπ )(α+1) (p2+β2)3 ( 2 4− 4+ 2 3 4− 2+ 2 5+ 4+ 3) ( )+ 2πβ 2− 3 2πβ 2π βp 2πp 4π β p 3βp 2π β 2πp β cos 2πp 3βe p β e . (p2+β2)3

Adding both Equations (14) and (15), we have the parameter αp. Similarly, we get and simplify the parameter βp, as follows: " # Z 2π 1 −βθ 2 −βθ βp = E(sin pθ) = sin pθ − αθe + βθ e dθ 0 1 − e 2βπ (α + 1) (16) Z 2π β2π + sin pθ θe−βθ dθ. 0 e2βπ (α + 1)

Rearranging the entire integrals in Equation (16), we have: " # Z 2π α β2π −βθ βp = − + θe [sin pθ] dθ+ 1 − e 2βπ (α + 1) e2βπ (α + 1) 0 (17) Z 2π β 2 −βθ − θ e [sin pθ] dθ. 1 − e 2βπ (α + 1) 0 Mathematics 2019, 7, 930 8 of 9

Since Equation (17) contains two integrals, we integrate separately as follows:

Z 2π β 2 −βθ I = − θ e [sin pθ] dθ 1 − e 2βπ (α + 1) 0 −2βe−2πβ (2π(πβ−1)p4+β(4π2 β2−3)p2+2π2 β5+β3) sin(2πp) = + (18) (1−e−2βπ )(α+1) (p2+β2)3 ( 2( ( 2+ 2+ )− )+ 2( ( + )+ )) ( )− 2 2πβ+ 3 2πβ p p 2π πp 2πβ 2β 1 β 2πβ πβ 2 3 cos 2πp 3β pe p e . (p2+β2)3

" # Z 2π α β2π −βθ − + θe [sin pθ] dθ = 1 − e 2βπ (α + 1) e2βπ (α + 1) 0

(−e−2πβ)[((2πβ−1)p2+2πβ3+β2) sin(2πp)+2p(πp2+πβ2+β) cos(2πp)−2βpe2πβ] α + (19) (1−e−2βπ )(α+1) (p2+β2)2 (−e−2πβ)[((2πβ−1)p2+2πβ3+β2) sin(2πp)+2p(πp2+πβ2+β) cos(2πp)−2βpe2πβ] β2π . e2βπ (α+1) (p2+β2)2

Adding both Equations (18) and (19) to each other resulted in the parameter βp.

4. Maximum Likehood Estimations Here, the maximum likelihood estimators of the unknown parameters (α, β) of the WQLD are derived. Let θ1, θ2, θ3, ..., θn be a random sample of size n from WQLD. Then, the likelihood function is L(θ1, θ2, ..., θn , α, β). We can deﬁne the ML as follows:

( + ( + )) n n −βθi ∞ α β θi 2kπ −2βkπ L(θ1, θ2, ..., θn , α, β) = ∏i g(θi) = ∏i θie ∑k=0 α+1 e . (20)

The log likelihood function is given by h i n n n ∞ (α+β(θi+2kπ)) −2βkπ ln L(θ1, θ2, ..., θn , α, β) = ∑i=1 ln θi − β ∑i=1 θi + ln ∑i=1 ∑k=0 α+1 e .

n n n ∞ −2βkπ −2βkπ −2βkπ ln L = ∑i=1 ln θi − β ∑i=1 θi − ln (α + 1) + ln ∑i=1 ∑k=0(αe + βθie + 2βkπe ). (21)

Equating the partial derivative of the log-likelihood function with respect to α and β to zero, we get ∂ ln L 1 ∑n ∑∞ e−2βkπ = − + i=1 k=0 . (22) + n ∞ −2βkπ −2βkπ −2βkπ ∂α α 1 ∑i=1 ∑k=0(αe + βθie + 2βkπe )

n ∞ −2βkπ 2 2 ∂ ln L ∑ ∑ e −2αkπ − 2βkπθi − 4βk π = −β + i=1 k=0 n ∞ −2βkπ . (23) ∂β ∑i=1 ∑k=0 e (α + βθi + 2βkπ ) Since Equations (22) and (23) cannot be solved analytically, we can therefore use some numerical techniques to get a solution for both parameters α and β.

5. Conclusions In this paper, we introduced and studied a new kind of distribution, namely, the Wrapping Quasi Lindley Distribution (WQLD). The PDF and CDF of (WQLD) were derived and the shapes of the density function and distribution function for different values of the parameters were found by using Mathematics 2019, 7, 930 9 of 9

Mathematica. An expression for the characteristic function resulted. The alternative form of the PDF of the (WQLD) was also obtained by using trigonometric moments. The parameters of (WQLD) were estimated by using the method of maximum likelihood.

Author Contributions: Investigation, A.M.H.A.-k.; Methodology, A.M.H.A.-k.; Project administration, A.M.H.A.-k.; Software, S.A.; Supervision, A.M.H.A.-k.; Writing–original draft, A.M.H.A.-k.; Writing–review and editing, S.A. Funding: This research received no external funding. Acknowledgments: The authors would like to acknowledge the financial support received from Al-Albayt University and Zarqa University. Conflicts of Interest: The authors declare no conflict of interest.

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