ON THE SHAPES OF BILATERAL GAMMA DENSITIES

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1 ON THE SHAPES OF BILATERAL GAMMA DENSITIES UWE KÜCHLER, STEFAN TAPPE Abstract. We investigate the for parameter family of bilateral Gamma distribtions. The goal of this paper is to provide a thorogh treatment of the shapes of their densities, which is of importance for assessing their fitting properties to sets of real data. This incldes appropriate representations of the densities, analyzing their smoothness, nimodality and asymptotic behavior. Key Words: bilateral Gamma distribtions, selfdecomposability, nimodality, asymptotic behavior, density shapes. Introdction In many fields of applications it is important to find appropriate classes of distribtions for fitting observed data. For this isse, normal distribtions often provide only a poor fit. Specific examples are given by the logarithmic retrns of stock prices, becase their empirical densities typically possess heavier tails and mch higher located modes than normal distribtions. Ths, several athors have looked for other appropriate classes of distribtions. We mention the generalized hyperbolic distribtions [] and their sbclasses, which have been applied to finance in [4], the Variance Gamma distribtions [7] and CGMY-distribtions [3]. Recently, another family of distribtions was proposed in [6]: Bilateral Gamma distribtions. In the mentioned article, bilateral Gamma distribtions are fitted to observed stock prices and compared to other classes of distribtions considered in the literatre. In order to provide a general overview abot their fitting properties also in view of other applications than finance we present a thorogh treatment of the shapes of their densities. After recalling the basic properties of bilateral Gamma distribtions in Section, we provide sitable representations of the densities in Section 3, which we can se in order to obtain density plots with a compter program. Afterwards, the investigation of the shapes of bilateral Gamma distribtions starts: Section 4 concerns the smoothness of the densities, Section 5 the nimodality and Section 6 is devoted to the asymptotic behavior of the densities near zero and for x ±. In Section 7 we characterize typical shapes of the densities and draw implications concerning the fitting properties of bilateral Gamma distribtions.. Bilateral Gamma distribtions In this section, we define bilateral Gamma distribtions and review some of their properties. For details and more informations, we refer to [6]. A bilateral Gamma distribtion with parameters,,, > is defined as the distribtion of X Y, where X and Y are independent, X Γ, ) and Y Γ, ). Date: Febrary 5, 8.

2 UWE KÜCHLER, STEFAN TAPPE The characteristic fnction of a bilateral Gamma distribtion is ) ).) ϕz) = iz, z R iz where the powers stem from the main branch of the complex logarithm. If X is bilateral Gamma distribted with parameters, ;, ), then for any c > the random variable cx has, by.), again a bilateral Gamma distribtion with parameters, c ;, c ). Note that, also by.), the sm of two independent bilateral Gamma random variables with parameters, ;, ) and, ;, ) has again a bilateral Gamma distribtion with parameters, ;, ). In particlar, bilateral Gamma distribtions are stable nder convoltion, and they are infinitely divisible. It follows from [8, Ex. 8.] that both, the drift and the Gassian part in the Lévy-Khintchine formla with trncation fnction h = ), are eqal to zero, and that the Lévy measre is given by ).) F dx) = x, ) x) x,) x) dx. x e x e Ths, we can also express the characteristic fnction ϕ as ϕz) = exp e izx ) ) kx).3) x dx, z R where k : R R is the fnction R.4) kx) = e x, ) x) e x,) x), x R which is decreasing on each of, ) and, ). It is an immediate conseqence of [8, Cor. 5.] that bilateral Gamma distribtions are selfdecomposable, and hence of class L in the sense of [9] and []. This is a key property for analyzing their densities, which is exploited in Sections 4, 5 and 6. Using the characteristic fnction.), we can specify the following qantities. Mean:, Variance: ) / Skewness: ) 3 ) 3 ) / Krtosis: 3 6 ) 4 ) 4 ) ), ) 3/, ) ) ) ) ). 3. Representations of the densities Bilateral Gamma distribtions are absoltely continos with respect to the Lebesge measre, becase they are the convoltion of two Gamma distribtions. Since the densities satisfy the symmetry relation 3.) fx;,,, ) = f x;,,, ), x R \ {} it is sfficient to analyze the density fnctions on the positive real line. As the convoltion of two Gamma densities, they are for x, ) given by 3.) fx) = ) ) ) Γ )Γ ) e x ) v v x e v dv.

3 ON THE SHAPES OF BILATERAL GAMMA DENSITIES 3 We can express the density f by means of the Whittaker fnction W,µ z) [5, p. 4]. According to [5, p. 5], the Whittaker fnction has the representation 3.3) W,µ z) = z e z Γµ ) From 3.) and 3.3) we obtain for x > 3.4) fx) = t µ e t t ) µ dt for µ > z. ) ) ) ) Γ ) x ) e x ) W ), )x )). By [5, p. 4], we can express the Whittaker fnction W,µ z) by the Whittaker fnctions M,µ z), namely it holds W,µ z) = Γ µ) Γµ) Γ µ )M,µz) Γ µ )M, µz). For the Whittaker fnction M,µ z) the identity [5, p. 4] M,µ z) = z µ e z Φµ, µ ; z) is valid, with Φ, γ; z) denoting the conflent hypergeometric fnction [5, p. 3] 3.5) Φ, γ; z) = γ z! ) z γγ )! ) ) z 3 γγ )γ ) 3!... Becase of the series representation 3.5) of Φ, γ; z), we can se 3.4) in order to obtain density plots with a compter program, which is done in Section 7. If one of, is an integer, the representation becomes more convenient at one half of the real axis. 3.. Proposition. Assme N = {,,...}. Then it holds for each x, ) ) ) fx) = a ) k x k e x, )! where the coefficients a k ) k=,..., are given by ) a k = k ) k k= k l= l). Proof. Since is an integer, we can compte the integral appearing in 3.) by sing the binomial expansion formla. The calclations are obvios. The symmetry relation 3.) and the identity [5, p. 7] z z ) W,µ z) = π K µ, where K µ z) denotes the Bessel fnction of the third kind, imply that in the case = =: the density 3.4) is of the form 3.6) fx) = Γ) ) x e x ) x ) π K ) x )

4 4 UWE KÜCHLER, STEFAN TAPPE for x R \ {}. The density of a Variance Gamma distribtion V Gµ, σ, ν) is, according to [7, Sec. 6..5], given by ) exp µx σ ) x ν 4 ) ) σ hx) = ν /ν πσγ ν ) K σ ν µ ν σ x 3.7) ν µ. Inserting the parametrization 3.8) µ, σ, ν) :=,, ) into 3.7), we obtain the density 3.6), showing that bilateral Gamma distribtions with = =: are Variance Gamma with parameters given by 3.8). Conversely, for a bilateral Gamma distribtion which is Variance Gamma it necessarily holds =, see [6, Thm. 3.3]. 4. Smoothness As we have pointed ot in Section, bilateral Gamma distribtions are selfdecomposable. Therefore, we may se the reslts of [9] and [] in the seqel. The smoothness of the density depends on the parameters and. Let N N = {,,,...} be the niqe nonnegative integer satisfying N < N. 4.. Theorem. It holds f C N R \ {}) and f C N R) \ C N R). Proof. This is an immediate conseqence of [9, Thm..]. Ths, the N-th order derivative of the density f is not continos. The only point of discontinity is zero. In Section 6, we will explore the behavior of f N) near zero. The densities of bilateral Gamma distribtions satisfy the following integrodifferential eqation. 4.. Proposition. f satisfies for x R \ {} xf x) = )fx) fx )e d fx )e d. Proof. The assertion follows from Cor.. of [9]. 5. Unimodality Bilateral Gamma distribtions are strictly nimodal, which is the content of the next reslt. 5.. Theorem. There exists a point x R sch that f is strictly increasing on, x ) and strictly decreasing on x, ). Proof. The existence of the mode x is a direct conseqence of [9, Thm..4], becase neither the distribtion fnction of a bilateral Gamma distribtion nor its reflection is of type I 4 in the sense of [9]. We emphasize that the mode x from Theorem 5. can, in general, not be determined explicitly. However, we get the following reslt, which narrows the location of the mode.

5 ON THE SHAPES OF BILATERAL GAMMA DENSITIES Proposition. If,, then x =. Presmed > and, it holds x, ). In the case, > we have x, ), and it holds x = if and only if =, x > if and only if >, x < if and only if <. Proof. The first statement is a conseqence of parts viii) and ix) of [9, Thm..3]. Since the mode of a Γ, )-distribtion with > is given by, parts ii) and iii) of [9, Thm. 4.] yield the second assertion. In the case, >, part iv) of [9, Thm. 4.] shows that x, ). According to Theorem 4., the density f is continosly differentiable. Using the representation 3.) and Lebesge s dominated convergence theorem, we obtain the first derivative for x, ) f ) ) x) = ) Γ )Γ ) )e x v [ e x x v v x ) e v dv v ]. ) e v dv Applying Lebesge s dominated convergence theorem again, by the continity of f and the fact Γx ) = xγx), x > we get f ) = ) ) Γ [ ) ) Γ )Γ ) ], which yields the remaining statement of the proposition. A particlar conseqence of Proposition 5. is that for and the mode x is necessarily close to zero. 6. Asymptotic behavior We have seen in Section 4 that for N := the N-th order derivative of the density f is not continos. The only point of discontinity is zero. We will now explore the behavior of f N) near zero. For the proof of the pcoming reslt, Theorem 6., we need the following properties of the Exponential Integral [, Chap. 5] E x) := e xt t dt, x >. The Exponential Integral has the series expansion ) n 6.) E x) = γ ln x n n! xn, where γ denotes Eler s constant γ = lim n [ n k= n= ] k lnn). The derivative of the Exponential Integral is given by 6.) x E x) = e x x. De to symmetry relation 3.) it is, concerning the behavior of f N) near zero, sfficient to treat the case x.

6 6 UWE KÜCHLER, STEFAN TAPPE 6.. Theorem. Let N :=. ) lim x f N) x) is finite if and only if N = {,,...}. ) If / N and / N, then f N) x) C x as x for constants C,, ). 3) Let / N be sch that N. Then f N) x) Mx) as x, where M is a slowly varying fnction as x satisfying lim x Mx) =. Moreover, it holds lim x f N) x) f N) x)) = C R. The constants in Theorem 6. are given by = N, C = ) ) sin π) Γ N) sin )π), C = ) ) ) ) N cos π) cos π). Proof. For N we conclde the finiteness of the limit in the first statement from [, Thm. 3], since for each β, ) recall that the fnction k was defined in.4)) lim β k)) = lim β e ) =. In order to prove the rest of the theorem, we evalate expressions.8).) in [9], and then we apply [9, Thm..7]. The constant c in [9, eqn..8)] is in the present sitation 6.3) c = exp ) e d ) e e ) d. e d The first integral appearing in 6.3) is by 6.) and the series expansion 6.) e [ ] = d = lim E ) ln = E ) γ. x =x Using 6.) and the fact lim x E x) =, see [, Chap. 5], for each constant > the identity e [ ] =x d = lim E ) = E ) x = is valid. Ths, we obtain 6.4) c = e )γ E ) E ). The fnction Kx) in [9, eqn..9)] is in the present sitation ) e e Kx) = exp d. Since by 6.) we obtain 6.5) e x x d = E x ) E ) for >, Kx) = e E ) E ) x ) e E x ) E x ).

7 ON THE SHAPES OF BILATERAL GAMMA DENSITIES 7 alphaalpha Figre. The shapes of f for =. Different choices of and may shift the mode and change the skewness. Using the series expansion 6.), we get 6.6) lim x Kx) = ) ) e )γ E ) E ), showing that for the slowly varying fnction in [9, eqn..)] it holds 6.7) Lx) = x K) d lim Lx) =. x Applying [9, Thm..7] and relations 6.4) 6.7) completes the proof. The asymptotic behavior of the Whittaker fnction for large vales of z is, according to [5, p. 6], W,µ z) e z z Hz) with H denoting the fnction [ µ Hz) = )] [ µ 3 )] [µ k )] k!z k. k= Obviosly, it holds Hz) for z. Taking into accont 3.) and 3.4), for x ± the density has the asymptotic behavior fx) C 3 x e x fx) C 4 x e x where the constants C 3, C 4 > are given by C 3 = ) ) ) Γ ) as x, as x, and C 4 = ) ) ) Γ ). As a conseqence, we obtain for the logarithmic density fnction ln f ln fx) lim = ln fx) and lim =. x x x x In particlar, the density of a bilateral Gamma distribtion is semiheavy tailed.

8 8 UWE KÜCHLER, STEFAN TAPPE 7. Shapes of the densities The shapes of bilateral Gamma distribtions can have remarkable differences. Using the reslts of the previos sections, we characterize typical shapes of their densities. If, then f is not continos at zero by Theorem 4.. According to Theorem 6., it holds 7.) lim fx) = and lim fx) =. x x We infer that the density has a pole at the mode x =. Notice that in the special case = the difference fx) f x) tends to a finite vale as x by the third statement of Theorem 6.. We observe that densities with 7.) are appropriate for fitting data sets with many observations accmlating closely arond zero. If <, then, by Theorem 4., f is continos on R, bt its derivative is not continos at zero. Let s have a closer look at the behavior of f near zero in this case. If,, ) and, ), it holds, according to Theorem 6., lim f x) = and lim f x) =. x x Hence, we have a steep mode of the density at zero with exploding slope from the left and from the right. This shape is also sitable for sets of data with many observations being close to zero. If < <, applying Theorem 6. yields lim f x) = and lim f x) =. x x Hence, the mode x is located at the positive half axis and f has infinite slope at zero. We remark that in the special case = the difference f x) f x) tends to a finite vale as x by the third statement of Theorem 6.. If, =, we have a two-sided exponential distribtion, which is in particlar Variance Gamma, as we have shown at the end of Section 3. We obtain lim f x) = C and lim f x) = C x x with finite constants C, C, ). Conseqently, we have a peak mode of the density at zero with finite slope from both sides. If >, then the density is smooth, that is at least of type C R) by Theorem 4.. Choosing and, the mode x is necessarily close to zero by Proposition 5.. Sch shapes are in particlar applicable for observations of financial data. We refer to [6, Sec. 9], where for a specific data set of stock retrns the maximm likelihood estimation =.55, = 33.96, =.94, = 88.9 provided a good fit. Smmarizing the preceding reslts, Figre provides an overview abot typical shapes of bilateral Gamma densities. References [] Abramowitz, M. and Stegn, I. A. 97) Handbook of mathematical fnctions. Dover Pblications, New York. [] Barndorff-Nielsen, O. E. 977) Exponentially decreasing distribtions for the logarithm of particle size. Proceedings of the Royal Society London Series A, Vol. 353, [3] Carr, P., Geman, H., Madan, D. and Yor, M. ) The fine strctre of asset retrns: an empirical investigation. Jornal of Bsiness 75),

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