On some properties of the new Sine-skewed Cardioid Distribution
Cherif Mamadou Moctar Traoré, Moumouni Diallo, Gane Samb Lo, Mouhamad Ahsanullah, Okereke Lois Chinwendu
OON SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION
Abstract. The new Sine Skewed Cardioid (ssc) distribution been just introduced and char-acterized by Ahsanullah (2018). Here, we study the asymptotic properties of its tails bydetermining its extreme value domain, the characteristic function, the moments and likeli-hood estimators of the two parameters, the asymptotic normality of the moments estimatorsand the random generation of data from the ssc distribution. Finally, we proceed to a sim-ulation study to show the performance of the random generation method and the quality ofthe moments estimation of the parameters. (1)
Cherif Mamadou Moctar Traor´e.Universit´e des Sciences, des Technique et des Technologies, Bamako, MaliEmail : [email protected] (2)
Moumouni DialloUniversit´e des Scences Economiques et de Gestion (USEG), Bamako, Mali.Email : [email protected]. (3)
Gane Samb Lo.LERSTAD, Gaston Berger University, Saint-Louis, S´en´egal (main affiliation).LSTA, Pierre and Marie Curie University, Paris VI, France.AUST - African University of Sciences and Technology, Abuja, [email protected], [email protected], [email protected] address : 1178 Evanston Dr NW T3P 0J9,Calgary, Alberta, Canada. (4)
Mouhamad AhsanullahDepartment of Management Sciences. Rider University. Lawrenceville, New Jersey, USAEmail : [email protected] (5)
Okereke Lois ChinwenduAUST - African University of Sciences and Technology, Abuja, NigeriaEmail : [email protected].
1. IntroductionAhsanullah (2018) introduced the new distribution with probability dis-tribution function ( pdf ) f ( x ) = 12 π (1 + λ sin x )(1 + ρ cos x )1 [ − π,π ] , associated with the parameters ( λ, ρ ) ∈ [ − , and named as the sine-skewed cardioid distribution. a r X i v : . [ s t a t . O T ] N ov ON SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION
In this note, we will state a number properties of that new law. In par-ticular, we are going asymptotic properties of its tails by determining itsextreme value domain, the moments and likelihood estimators of the twoparameters, the asymptotic normality of the moments estimators and therandom generation of data from the ssc distribution. But, before we pro-ceed, we recall for Ahsanullah (2018) that the cumulative distributionfunction (cdf) is F ( x ) = 12 + 12 π (cid:18) x − λ (cos x + 1) + ρ sin x − λρ x − (cid:19) , x ∈ [ − π, π ] . To prepare studying the upper tail (in a neighborhood of π ) and the lowertail (in a neighborhood of − π ), we may write, respectively, for x ∈ [ − π, π ] ,(1.1) F ( x ) = 1 − π (cid:18) ( π − x ) − λ (cos x + 1) + ρ sin( π − x ) − λρ x − (cid:19) and(1.2) F ( x ) = 12 π (cid:18) ( x + π ) − λ (cos x + 1) − ρ sin( π + x ) − λρ x − (cid:19) . We will have to deal with some statistical properties. So, we suppose thatwe have a sequence X , X , X , · · · , etc. of real-valued random variablesdefined on the same probability space (Ω , A , P with cdf F , then supportedby [ − π, π ] . We define the sequence of empirical maxima and the minim X ,n = min( X , · · · , X n ) and X n,n = max( X , · · · , X n ) , n ≥ and the empirical moments X n = 1 n (cid:88) ≤ j ≤ n X j , S n = 1 n − (cid:88) ≤ j ≤ n (cid:0) X j − X n (cid:1) , n ≥ and the non-centered second moment m ,n = 1 n (cid:88) ≤ j ≤ n X j . N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 3
Figure 1. PDF of f sc ( x, , λ ) , Black- λ = − . , Red- λ = − . , Green- λ = 0 . and Brown- λ = 0 . Some graphical illustrations of the pdf for some values of the parameters ( λ, ρ ) are also given by Ahsanullah (2018) in Figure 2The rest of the paper is organized as follows. Section 2 is devoted a studyof the extreme behavior the the tails of the ssc cdf . In Section 3, we dealwith the moments and likelihood estimation of the parameters λ and ρ anddetermine the asymptotic laws of the moments estimators. In Section 4,we provide the characteristic function. Finally, in Section 5, we proposea method of generating samples from the ssc model. Simulation studiesare undertaken to test the generation algorithm and next to test the per-formance of the moments estimators. VB6 Subroutines for the generationmethods are given in the appendix. The paper ends by a conclusion sec-tion. 2. Asymptotic Properties of the tails Theorem 1.
Define the constants πα = 1 + ρ, πα = λ − ρ ) , πα = − ρ πα = λ
24 (4 ρ − , πα = ρ , πα = λ
126 (1 − ρ ) . ON SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION
We have, as x → π,α α ( π − x ) (cid:18) α α ( π − x ) (cid:18) α α ( π − x ) (cid:18) α α ( π − x ) (cid:18) α α ( π − x ) (cid:18) − F ( x ) α ( π − x ) (cid:19) − (cid:19) − (cid:19) , − (cid:19) − (cid:19) − O ( π − x ) and as x → − π , α α ( π + x ) (cid:18) α α ( π + x ) (cid:18) α α ( π + x ) (cid:18) α α ( π + x ) (cid:18) α α ( π + x ) (cid:18) F ( x ) α ( π + x ) (cid:19) − (cid:19) − (cid:19) − (cid:19) − (cid:19) − O ( π + x ) . Remark : Such an expansion is limited to an order 6. But it might begiven an any order k ≥ . Proof of Theorem 2 . We only give elements for the proof of the first. Weuse the following elementary expansions sin( π − x ) = ∞ (cid:88) k =0 ( − k +1 (2 k + 1)! ( π − x ) k +1 cos x + 1 = cos x − cos π = ∞ (cid:88) k =1 ( − k +1 (2 k )! ( π − x ) k cos 2 x − x − cos 2 π = ∞ (cid:88) k =1 ( − k k (2 k )! ( x − π ) k . We consider the limited expansion at order 6 at π to have − F ( x ) = α ( π − x ) + α ( π − x ) + α ( π − x ) + 2 πα ( π − x ) + α ( π − x ) + α ( π − x ) + O (( π − x ) . This justifies the expansion at + π . The situation is the same at − π fromFormula . The reason is that + π and − π play the same roles in the devel-opments and the two terms − ρ sin( π + θ ) and ρ sin( π − θ ) are expanded in N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 5 the same way with respect to π − x and π + x respectively. (cid:3) From Theorem 2, we directly get the extreme law of X and Y = 1 / ( π + X ) . Theorem 2.
We have the following properties(a) F (cid:63) ( x ) = F ( π − x ) , x > belong to the Frechet extreme value Domain D ( H ) that is ∀ γ > , lim x → + ∞ − F (cid:63) ( γx )1 − F (cid:63) ( x ) = γ − and the second order condition ∀ γ > , lim x → + ∞ S ( x ) (cid:18) − F (cid:63) ( γx )1 − F (cid:63) ( x ) − γ − (cid:19) = 0 for S ( x ) = α x/α , x > .b) As a consequence F is in the Weibull extreme value Domain D ( H − ) andwe have n (1 + ρ ) ( X n,n − π ) (cid:32) H − , as n → ∞ . (c) To have the extreme lower law is found by using X ,n = − ( − X ) n,n and ( − X ) n,n and X n,n have the same limit in type, that is n (1 + ρ ) (( − X ) n,n − π ) (cid:32) H − , as n → ∞ . Proof of Theorem 2 . Point (a) is a direct consequence of Theorem at thefirst order. By Theorem 8 in Lo (2018b), we have that F ∈ D ( H ) if andonly if F ∗ ∈ D ( H − ) . So (b) holds from (a). Furthermore, by Proposition 8in Lo (2018b), we also have(2.1) X n,n − uep ( F ) uep ( F ) − F − (1 − /n )) (cid:32) H − , as n → ∞ . Now from the expansions in Theorem , we have for any − < u < − F ( x ) = u ⇔ α ( π − x ) + α ( π − x ) + ... + α ( π − x ) + O (( π − x ) ) . We get that, as x → + π , ( π − x ) ∼ u/α , ON SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION and x = F − (1 − u ) = π − α u + α α u (1 + ε (1)) + · · · + α α u (1 + ε (1)) + O ( u ) , where the function ε h ( u ) go to zero as u ↓ . In particular, we have π − F − (1 − u ) ∼ α u = u ρ and, as n → + ∞ π − F − (1 − /n ) ∼ α = 1 n (1 + ρ ) , which combined with Formula 2.1 concludes Point (b) of the proof.Point (c) is based the cdf of − X , which is F ⊥ ( x ) = 1 − F ( − x ) , for − π ≤ x ≤ π .And the expansion of − F ⊥ ( x ) gives the same expansion as in Formula at + π . We get the same conclusion as for X . (cid:4) Interesting Pedagogical Example . As we can find in Lo (2018b), page133, a criterion of belonging of F to D ( H ) , when uep ( F ) = π is that F admits a derivative in a right neighborhood of π and that(2.2) lim x → π ( π − x ) F (cid:48) ( x )1 − F ( x ) = 1 . It is also stated that the condition 2.2 holds if F ∈ D ( H ) and F (cid:48) is ul-timately non-increasing as x (cid:37) π . Here, the limit holds for all values of λ and ρ in ] − , although F (cid:48) is ultimately not non-increasing for somevalues of λ and ρ for example, for λ = − . and ρ = − . , for which F (cid:48) isincreasing as shown in Figure 2With theses values, if the practitioner concludes that F ∈ D ( H ) , he iswrong in the use of the rule but his conclusion is correct by coincidence.So it is important to check the ultimate decreasingness of F (cid:48) , which is usedin the proof of the rule. N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 7
Figure 2. Ssc pdf for λ = − . and ρ = − .
3. Parameters estimation3.1.
Moments estimation.A. Point Estimation .The kth moment of X is given by E X k = 12 π (cid:90) π − π x k (1 + λ sin x + ρ cos x + ( λρ/
2) sin 2 x ) dx. From the following facts (cid:90) π − π x cos x dx = 0 , (cid:90) π − π x sin x dx = 2 π, (cid:90) π − π x cos x sin x dx = − π/ (cid:90) π − π x cos x dx = − π, (cid:90) π − π x sin x dx = 0 , (cid:90) π − π x sin x cos x dx = 0 , we get m = E X = λ (1 − ρ/ , m = E X = π / − ρ. ON SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION
The moments estimators are solutions of the system of two equations : m = X n and m = m ,n . We immediately have, for n ≥ ,(3.1) ˆ ρ n = ( π / − m ,n ) / and ˆ λ n = 4 X n − ˆ ρ n = 8 X n − π / m ,n . We may need to check such estimation by a simulation study. We will dothis in Section 4 where we propose a simple method for generating data forthe ssc distribution. For now we want to do more on the moment problem.In the appendix, we study the integrals, for n ≥ , I n = (cid:90) π − π x n cos x dx, J n = (cid:90) π − π x n sin x dx, and H n = (cid:90) π − π x n sin 2 x dx. We established the following recurrence formula(3.2) I = 0 , ∀ n ≥ , I n = − nπ n − − n (2 n − I n − and ∀ n ≥ , I n +1 = 0 , , (3.3) ∀ n ≥ , J n = 0 and ∀ n ≥ , J n +1 = 2 π n +1 − n (2 n + 1) J n − , and(3.4) ∀ n ≥ , H n = 0 and ∀ n ≥ , H n +1 = − π n +1 − n (2 n + 1) H n − . and for the needs of that paper, we have computed I = 0 , I = (2 π )( − π ) , I = 0 , I = 4(2 π )(6 − π )) J = 2 π, J = 0 , J = (2 π )( π − , J = 0 H = (2 π )( − / , H = 0 , H = (2 π )((3 − π ) / , H = 0 . With such facts, we easily have E X = λ ( π −
6) + λρ − π ) E X = π − ρ ( π − . (3.5) N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 9
We will see how we need these parameters for the asymptotic laws of themoment estimators. 4. The characteristic function
Proposition 1.
The characteristic function (fc) of X is given, for t / ∈ {− , − , , , } ,bt ψ ( t ) = 1 π (cid:18) t + λi ( t − − ρtt − − λρi ( t − (cid:19) sin ( πt ) and is extended to values in {− , − , , , } by continuity of the fc. Proof . We may write for t ∈ R and x ∈ [ − π, π ] , e it f ( x ) = 12 π e itx (cid:20) λ i (cid:0) e ix − e − ix (cid:1) + ρ (cid:0) e ix + e − ix (cid:1) + λρ (cid:0) e i x − e − i x (cid:1)(cid:21) = 12 π (cid:20) e itx + λ i (cid:0) e ix ( t +1) − e ix ( t − (cid:1) + ρ (cid:0) e ix ( t +1) + e ix ( t − (cid:1) + λρ (cid:0) e ix ( t +2) − e − ix ( t − (cid:1)(cid:21) . Integrating from − π to + π leads to the announced results. (cid:3) B. Asymptotic Normality of the moment estimators and Statisticaltests .The moments estimators are treated by using the function empirical pro-cess defined, for any n ≥ and h ∈ L ( P X ) , by G n ( h ) = 1 √ n n (cid:88) j =1 ( h ( X j ) − E ( h ( X j ))) , as explained in Lo (2016) and Lo (2018a). For a more general source, thebook by van der Vaart and Wellner (1996) is one the best in the field butwe no need the great artillery provided there. We have the following result. Theorem 3. we have the asymptotic laws of the moment estimators, as n → + ∞ , (cid:16) √ n ( ˆ ρ n − ρ ) , √ n (ˆ λ n − λ ) (cid:17) (cid:32) N (0 , Σ) , with Σ = V ar ( X )4 , Σ = V ar ( X + λX ) µ and Σ = Σ = − C ov ( X , X + λX )2 µ . Remark . As anticipated in Formula 3.5, the asymptotic laws need the firstfour moments of X . Proof . Let us set h (cid:96) ( x ) = x (cid:96) for x ∈ R . By applying the methods in Lo(2016), we get √ n ( ˆ ρ n − ρ ) = G n ( − h /
2) + o P (1) . Next, we have ˆ λ n = 8 (cid:16) m + √ n G n ( h ) (cid:17) − π / m + √ n G n ( h ) . By Lemma 2 in Lo (2016), we get that for µ = 8 − π / m , √ n (ˆ λ n − λ ) = G n (( h + λh ) /µ ) + o P (1) . By using the functional Brownian stochastic process G , which is the weaklimit of G n and defined by the variance-covariance function Γ( h, k ) = (cid:90) R ( h ( x ) − E h ( X ))( k ( x ) − E k ( X )) d P X ( x ) , where ( h, k ) ∈ L ( P X ) , we get that (cid:16) √ n ( ˆ ρ n − ρ ) , √ n (ˆ λ n − λ ) (cid:17) (cid:32) N (0 , Σ) , with Σ = V ar ( h ( X )4 , Σ = V ar ( h ( X ) + λh ( X ) µ and N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 11 Σ = Σ = − C ov ( h ( X ) , h ( X ) + λh ( X ))2 µ . (cid:4) Maximum Likelihood Estimators. we are going to see that the ML -estimators are not defined here. We begin the remark that the first threelinear differential operators in ( λ, ρ ) of f are πDf ( u, v ) = sin x (1 + ρ cos x ) u + cos x (1 + λ sin x ) v, πD f ( u, v ) = (cos x sin x ) uv, ( u, v ) ∈ R ,D f ( u, v ) = 0 . By applying the Taylor-Lagrange-Cauchy formula (see Valiron (1946), page233) : for ( λ, λ , ρ, ρ ) ∈ [ − , , for | θ i | < , i = 1 , f ( x, λ, ρ ) = f ( x, λ , ρ ) + Df ( λ − λ , ρ − ρ ) + 12 D f ( λ − λ , ρ − ρ )+ D f ( θ ( λ − λ ) , θ ( ρ − ρ )) , and we get f ( x, λ, ρ ) = f ( x, λ , ρ ) + sin x (1 + ρ cos x )( λ − λ ) + cos x (1 + λ sin x )( ρ − ρ )+ (cos x sin x )( λ − λ )( ρ − ρ ) First, for x / ∈ {∓ π, ± π/ } , the zeros of f are − / sin x and − / cos x whichdo not belong to [ − , . Even on R , not requiring that f be non-negativeto make it a probability density function, Formula which becomes at anycritical point ( λ , ρ ) of this C -function (in λ and in ρ ), f ( x, λ, ρ ) = (cos x sin x )( λ − λ )( ρ − ρ ) . So, for a fixed x / ∈ {∓ π, ± π/ } , there can be an extremum point ( λ , ρ ) forthe likelihood function. 5. GenerationSince the cdf F is explicitly known and is strictly increasing and contin-uous, we may use the dichotomous algorithm to find the inverse of F . Itworks as follows. Given v ∈ [0 , , to find x such that F ( x ) = v , we fix thenumber of decimals of the solution x denoted nbrDec . In the Appendix, be-ginning by page 19, the VB 6 code for computing the cdf F is given in page λ ρ mean (E) mean (M) 2nd moment (E) 2nd moment (M) quotient (Q)0.9 -0.9 1.1025 1.2322 5.0898 5.0281 1.01220.9 -0.6 1.035 1.0641 4.4898 4.4796 1.00220.9 -0.3 0.9675 0.9396 3.8898 3.8987 0.99770.9 0.1 0.8775 0.8796 3.0898 3.2221 0.95890.9 0.4 0.81 0.8048 2.4898 2.5953 0.95930.9 0.7 0.7425 0.7536 1.8898 1.8690 1.01110.9 0.9 0.6975 0.7077 1.4898 1.4810 1.0059-0.9 0.9 -0.6975 -0.6972 1.4898 1.5574 0.9566-0.9 0.7 -0.7425 -0.7773 1.8898 1.8280 1.0338-0.9 0.4 -0.81 -0.8218 2.4898 2.2965 1.0841-0.9 0.1 -0.8775 -0.9273 3.0898 3.2964 0.9373-0.9 -0.3 -0.9675 -1.0273 3.8898 3.9613 0.9819-0.9 -0.6 -1.035 -0.9654 4.4898 4.4845 1.0011-0.9 -0.9 -1.1025 -1.1300 5.0898 5.1902 0.9806 Table 1. Legend : (E) : Exact, (M) empirical, (Q) : quotient exact secondmoment to empirical second moment
21, the Dichotomous algorithm is described in page 19 and implementedin VB6 in 19. Finally, the VB 6 subroutine which generates a sample of anarbitrary size is given in 21.
A. Numerical test of the computer programs .For different values for λ and ρ in ] − , , samples of size n = 1000 aregenerated and the comparison between the exact means and the secondmoments as given in Formua are compared with the sample counterparts.Table 1 demonstrates the quality of the generation. B. Moment estimation .Based on the generation techniques as introduced above, we also reportinteresting performances for the estimation of the parameter when thesizes varies in Table 2. The Mean Absolute Error (MAE) and the MeanSquare-root Quadratic Error (MSQE) are reported both for λ and ρ . Thesesimulations have been for λ = ρ = − . .Figure 3 shows how both errors decreases to zero. N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 13
Error n 10 50 100 200 300 400 500 750 1000MAE- λ λ ρ ρ Table 2. Errors given in multiplesFigure 3. Red : related to ρ . Blue : errors related to ρ
6. ConclusionThis is an immediate contribution of the study of the Sine Skewed Cae-dioid distribution. Further deeper properties will be addressed later.ReferencesAhsanullah M. (2018). Sine-Skewed Cardioid Distribution. In
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Weak Convergence and Em-pirical Processes With Applications to Statistics . Springer, New-York.
N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 15
Appendix . I. Integral Computations .Let us define, for n ≥ , I n = (cid:90) π − π x n cos x dx, (cid:90) π − π x n sin x dx, (cid:90) π − π x n cos 2 x dx and (cid:90) π − π x n sin 2 x dx. A - Computation of I n . For n = 0 , I = (cid:90) π − π cos x dx = (cid:90) π − π d (sin x ) = 0 and for n = 1 , I = (cid:90) π − π xd (sin x ) = [ x sin x ] − ππ − (cid:90) π − π − d (cos x ) = 0 and for n = 2 I = (cid:90) π − π x d (sin x ) =] x sin x [ − ππ − (cid:90) π − π − xd (cos x )= (cid:18) x cos x ] − ππ − (cid:90) π − π cos x ) (cid:19) = 4[ x cos x ] − ππ = − π. Now, the general guess is that I n +1 = 0 for n ≥ . Since this already holds n = 0 , let us remark that for n ≥ , I n +1 = (cid:90) π − π x n +1 d (sin x ) =] x n +1 sin x [ − ππ − (2 n + 1) (cid:90) π − π − x n d (cos x )= (2 n + 1) (cid:90) π − π x n d (cos x ) = (2 n + 1)[ x n cos x ] − ππ − (2 n )(2 n + 1) (cid:90) π − π − x n − cos x dx = − (2 n )(2 n + 1) I n − . By an descendent induction, we will have I n +1 = C n I = 0 . Now, we haveto find I n , for n ≥ , which is I n = (cid:90) π − π x n d (sin x ) =] x n sin x [ − ππ − (2 n ) (cid:90) π − π ( − x n − ) d (cos x )= (2 n ) (cid:90) π − π x n − d (cos x ) = [ x n − cos x ] − ππ − (2 n )(2 n − (cid:90) π − π − x n − cos x dx = − nπ n − − n (2 n − I n − . For example, we find again I = − π for n = 1 . B - Computation of J n . For n = 0 , we have J = (cid:90) π − π sin x dx = (cid:90) π − π d ( − cos x ) = 0 and for n = 1 , J = (cid:90) π − π xd ( − cos x ) = − [ x cos x ] − ππ + (cid:90) π − π − d (sin x ) = 2 π. Our guess is that J n = 0 for n ≥ and we have for n ≥ , J n = (cid:90) π − π x n d ( − cos x ) =] x n cos x [ − ππ +(2 n ) (cid:90) π − π x n − d (sin x )= (2 n ) (cid:90) π − π x n − d (sin x ) = (2 n )[ x n − sin x ] − ππ − (2 n )(2 n − (cid:90) π − π x n − sin x dx = − (2 n )(2 n − J ( n − . By an descendent induction, we will have J n = C n J = 0 . Now, he have tofind J n +1 , for n ≥ , which is J n +1 = (cid:90) π − π x n +1 d ( − cos x ) = − ] x n +1 cos x [ − ππ +(2 n + 1) (cid:90) π − π x n d (sin x )= 2 π n +1 + (2 n + 1)[ x n sin x ] − ππ − (2 n )(2 n + 1) (cid:90) π − π x n − sin x dx = 2 π n +1 − n (2 n + 1) J n − . C - Computation of H n . For n = 0 , we have H = (cid:90) π − π sin xd (sin x ) = (1 / (cid:90) π − π d (sin x ) = 0 , and for n = 1 , N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 17 H = (1 / (cid:90) π − π x sin 2 x dx = − (1 / (cid:90) π − π xd (cos 2 x )= − (1 / (cid:18) [ x cos 2 x ] − ππ − (cid:90) π − π (1 / d (sin 2 x ) (cid:19) = − π/ . We still guess that H n = 0 for n ≥ and find, for n ≥ , H n = (1 / (cid:90) π − π x n sin 2 x dx = − (1 / (cid:90) π − π x n d (cos 2 x )= − (1 / (cid:18) [ x n cos 2 x ] − ππ − (1 / n ) (cid:90) π − π x n − d (sin 2 x ) (cid:19) = (1 / n (cid:90) π − π x n − d (sin 2 x ) = (1 / n (cid:18) [ x n − sin 2 x ] − ππ − (2 n − (cid:90) π − π x n − sin 2 x dx (cid:19) = − (1 / n (2 n − H n − , and we conclude that H n = 0 for all n ≥ , based on that H = 0 and theunveiled descendent induction formula above. Nor for all n ≥ , H n +1 = (1 / (cid:90) π − π x n +1 sin 2 x dx = − (1 / (cid:90) π − π x n +1 d (cos 2 x )= − (1 / (cid:18) [ x n +1 cos 2 x ] − ππ − (1 / n + 1) (cid:90) π − π x n d (sin 2 x ) (cid:19) = − (1 / (cid:18) (2 π n +1 ) − (1 / n + 1) (cid:18) [ x n sin 2 x ] − ππ − (2 n ) (cid:90) π − π x n − sin 2 x ) (cid:19)(cid:19) = − (1 / π n +1 − (1 / / n + 1)(2 n ) (cid:90) π − π x n − sin 2 x )= − (1 / π n +1 − (1 / n (2 n + 1) H n − We conclude(6.1) I = 0 , ∀ n ≥ , I n = − nπ n − − n (2 n − I n − and ∀ n ≥ , I n +1 = 0 , , (6.2) ∀ n ≥ , J n = 0 and ∀ n ≥ , J n +1 = 2 π n +1 − n (2 n + 1) J n − , and (6.3) ∀ n ≥ , H n = 0 and ∀ n ≥ , H n +1 = − π n +1 − n (2 n + 1) H n − . For the needs of this paper, we have I = 0 , I = (2 π )( − π ) , I = 0 , I = 4(2 π )(6 − π ) ,I = 0 , I = − π )( π − π + 120) , I = 0 , I = − π )( π − π + 840 π − J = 2 π, J = 0 , J = (2 π )( π − , J = 0 ,J = 2 π ( π − π + 120) , J = 0 , J = 2 π ( π − π + 840 π − , J = 0 ,H = (2 π )( − / , H = 0 , H = (2 π )((3 − π ) / , H = 0 ,H = − π (2 π − π + 15)8 , H = 0 , H = − π (4 π − π + 210 π − . E X = λ ( π − π + 120) − λρ
16 (2 π − π + 15) E X = π − ρ ( π − π + 120) E X = λ ( π − π + 840 π − − λρ
32 (4 π − π + 210 π + 315) E X = π − ρ ( π − π + 840 π − . N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 19
II. Samples generations . Description of the dichotomous algorithm .0. Fix the count index o f the decimals countDec to − − < v , then3a . Fix x=1 and set h=1.3b . P r o g r e s s i v e l y increment x by h and t e s tand compare F ( x ) and v .3bA . IF F ( x )= v . x i s the searched number . Stop3bB. IF F ( x) < v , Goto 3bA and continuethe incrementation .3bC. IF F ( x) > v , then Decrement by h : x=x − hIncrement countDecI f countDec < nbrDectake h=h/10Goto 3bI f countDec=nbrDec . Stop3. IF F(1) > v , then4a . Fix x=1 and set h=0.14b . P r o g r e s s i v e l y decrement x by h : x=x − h and t e s t and compareF ( x ) and v .4bA . IF F ( x )= v . x i s the searched number . Stop4bB. IF F ( x) > v , Goto 4bA and continuethe decrementation4bC. IF F ( x) < v , then increment by h : x=x − hIncrement countDecI f countDec < nbrDec − Implementation of the dichotomous algorithm in Visual Basic 6 .Function sscinv ( v As Double , lambda As Double , rho As Double , nbrDec ) As Double
Dim dich As Double , count As Integer , h As DoubleDim countDecsscinv = 1dich = ssc ( sscinv , lambda , rho )I f ( dich = v ) ThenExit FunctionEnd I fI f ( dich < v ) Thenh = 1count = 0countDec = − < nbrDec ) And ( count < > v ) ThencountDec = countDec + 1sscinv = sscinv − hh = h / 10End I fWendEnd I fI f ( dich > v ) Thenh = 0.1count = 0countDec = 0While ( ( countDec < nbrDec ) And ( count < − hI f ( ssc ( sscinv , lambda , rho ) = v ) ThenExit FunctionEnd I fI f ( ssc ( sscinv , lambda , rho ) < v ) ThencountDec = countDec + 1 N SOME PROPERTIES OF THE NEW SINE-SKEWED CARDIOID DISTRIBUTION 21 sscinv = sscinv + hh = h / 10End I fWendEnd I fEnd Function
VB6 Subroutine for cumputing the cdf F .P r i v a t e Function ssc ( x As Double , lambda As Double , rho As Double ) As DoubleDim pi As Doublepi = 4 ∗ Atn ( 1 )ssc = (1 / 2) + (1 / (2 ∗ pi ) ) ∗ ( x − ( lambda ∗ ( Cos ( x ) + 1 ) ) + rho ∗ Sin ( x ) − ( ( ( lambda ∗ rho ) / 4) ∗ ( Cos(2 ∗ x ) − rssc : VB6 Subroutine for generating samples from the ssc distribution, which thesample in the file ssc100 .Sub rssc ( t a i l As Integer , lambda As Double , rho As Double , nbrDec As Integer )Dim i i As Integer , cheminD As StringcheminD = ”G: \ backup \ DataGslo \ gslo \ TveSimulation \ ssc \\