Slash Distributions and Applications

Transcription

1 CHAPTER 2 Slash Distributions and Alications 2.1 Introduction The concet of slash distributions was introduced by Kafadar (1988) as a heavy tailed alternative to the normal distribution. Further literature are available in Wang and Genton (26), Tan and Peng (25), Arslan (25), Genc (27), Arslan (28) and Arslan and Genc (29). Another imortant symmetric and heavy tailed distribution is the Exonential ower (EP) distribution introduced by Subottin (1923), characterized by a scale arameter σ >, a shae arameter > and a location arameter η R, has the robability density Some results included in this chater form art of a aer short-listed for ISCA Young Scientist Award 28 (Lishamol, 27). See also Jose and Lishamol (27) and Lishamol and Jose (29). 29

2 function ( df ), f(x;, σ, η) = ( 1 1 2σ ( 1 1) Γ( 1) ex x η σ ) ; x R, >, σ >. (2.1.1) (2.1.1) yields the Lalace density for = 1, the normal density for = 2 and the uniform distribution as. Here f(x;, 1, ) corresonds to the standard EP distribution df, for convenience denoted by f(x; ). For < 2 the distributions in (2.1.1) are heavytailed. By changing the shae arameter, the EP distribution describes both letokurtic ( < < 2) and latikurtic ( > 2) distributions. A number of skewed versions of EP distribution are introduced and studied by various authors. Ayebo and Kozubowski (26), based on the findings of Fernandez and Steel (1998), introduced the asymmetric exonential ower (AEP) distribution with shae arameter, scale arameter σ, location arameter η having df, f(x;, σ, η, κ) = κ ( σγ(1/) 1 + κ ex κ 2 σ [(x η)+ ] 1 ) σ κ [(x η) ] with x R, >, σ >, η R and κ > where u; u + if u = ; if u < u; and u if u = ; if u >. (2.1.2) In this chater, slash exonential ower distribution, a family of slash distributions which arises as the ratio of indeendent exonential ower and uniform ower function distributions is introduced and studied. It includes slash normal and slash Lalace as secial cases. Slash Lalace distributions and their roerties in both univariate and multivariate cases are discussed. Skewed versions namely, asymmetric slash exonential ower, asymmetric slash Lalace and its multivariate case are develoed. Discussion on alications of the new slash distributions along with a case study on daily observations of US dollar-indian ruee exchange rate is also carried out. 3

3 2.2 Slash Exonential Power Distribution We are considering the slash version of the standard exonential ower distribution. From standard case, the general slash densities corresonding to the df in (2.1.1) can be easily constructed through a scale multilication and location shift. The standard slash exonential ower (SLEP) distribution can be defined as the distribution of the ratio X = Y U 1/q where Y is a standard EP random variable and U is an indeendent uniform random variable over (,1) and q >. It is denoted by X SLEP (, q). Remark As q the standard SLEP yields the standard EP itself. With simle algebra, df of the standard slash exonential ower variable X can be obtained as g(x;, q) = u 1/q f(xu 1/q ; ) du; < x < where f( ) is the df of standard EPD. The cumulative distribution function ( cdf ) is G(x;, q) = x where F ( ) is the cdf of standard EPD. g(x;, q) dx = F (xu 1/q ; ) du, In terms of v = u 1/q, < v < 1 the df and cdf of standard SLEP can be obtained, resectively as h(x;, q) = q H(x;, q) = q v q f(xv; ) dv; < x <. (2.2.1) v q 1 F (xv; ) dv. For x =, (2.2.1) gives h(;, q) = q 2 ( 1 1) Γ( 1 )(q + 1). Obviously, H(;, q) = 1 2 due to the symmetry of the distributions involved. A closed form 31

4 of the density h( ) is given by, h(x;, q) = q 2 ( 1 ) Γ( 1 ) ( x ) q+1 ( ( q + 1 Γ ) ( )) q + 1 Γ, x ; q = 1, 2,.... (2.2.2) Theorem The ratio W of two indeendently and identically distributed (iid) standard SLEP random variables has the df k(w) = C 2 z 1/q (1 + wz 1/q ) 2/ dz + C 2 1 z (1/q) 2 dz (1 + wz 1/q ) 2/ where C = Γ( 2 ) 2[Γ( 1 )]2. Proof: Here W = X 1/q 1/U 1 X 2 /U 1/q, X 1 and X 2 are indeendent EP random variables and U 1 2 and U 2 are indeendent uniform random variables. Take Z = X 1 X 2. Thereafter using the Jacobian of transformation 1 the df can be derived. This exression can be reduced in z terms of Beta functions also. Remark Let Y 1 and Y 2 be two indeendent standard SLEP random variables. Then the conditional distribution of the ratio of Y 1 and Y 2 given U 1 = U 2 = u yields the standard generalized Cauchy distribution. Proof: It follows directly from the fact that the standard generalized Cauchy distribution given by Sebastian and Preethi (23) having df f(x; ) = Γ( 2) 1 2[Γ( 1 ; >, x R, (2.2.3) )]2 [1 + x ] 2 can be obtained as the distribution of the ratio of two iid EP random variables. Remark For = 2, (2.1.1) yields the normal density, hence (2.2.1) gives the slash normal density of Kafadar (1998) given by the density, h(x; 2, q) = q 32 v q φ(xv) dv

5 where φ( ) denotes the standard normal df. Remark For = 1, (2.2.1) yields the slash Lalace density (See Section 2.3) with df h(x; 1, q) = q where l 1 ( ) denotes the standard Lalace df. 2.3 Slash Lalace Distribution v q l 1 (xv) dv The standard slash Lalace (SLL) distribution can be defined as the distribution of the ratio X = Y U 1/q where Y is a standard Lalace random variable and U is an indeendent uniform random variable over the interval (,1) and q >. It is denoted by X SLL(, 1, q) or SLL(q) (for convenience). The df of the standard slash Lalace variable X can be obtained as g(x; q) = u 1/q f(xu 1/q ) du; < x < where f( ) is the standard Lalace df given by, f(x) = 1 2 e x, < x <. Then its cdf is G(x; q) = x g(x; q) dx = F (xu 1/q ) du, where F ( ) denote the standard Lalace cdf. In terms of v = u 1/q the df and cdf of standard slash Lalace distribution can be obtained resectively as h(x; q) = q H(x; q) = q v q f(xv) dv; < x <. (2.3.1) v q 1 F (xv) dv. In articular, h(, q) = q v q q f() dv = 2(q + 1) and H(, q) = 1. As q, the 2 standard slash Lalace yields the Lalace density itself. Figure 2.1(a) gives the density curves of the standard Lalace(highest one), slash 33

6 (a) (b) Figure 2.1: (a) Comarison of the density curves of slash Lalace distributions (b) Comarison between slash Lalace, slash normal and slash t distributions Lalace with q = 1 (lowest one) and slash Lalace with q = 2 (middle one). A comarative study between the density curves of slash Lalace (highest one), slash normal (middle one) and the slash t (lowest one) distributions, for q = 1 where the arent distribution is standard, is given in Figure 2.1(b). It shows that the tail of slash Lalace takes an intermediate osition between that of slash normal and slash t distributions. For q = 1 we obtain the density (2.3.1) as, h(x, 1) = g() g(x) x g(x) ; x 2 x g() 2 x =. For q = 2, h(x, 2) = 4g() 2g(x) 4g(x) 4g(x) ; x 3 x x 2 x 3 x 2g() ; 3 x =. In a similar way for different values of q, closed-form exressions for the comuted. df can be 34

7 Now let us examine the tail behaviour of the slash Lalace distribution. Let the survival function of standard Lalace distribution be F = 1 F. Then the survival function of the slash Lalace is, H(x; q) = 1 H(x; q) = q v q 1 F (xv) dv. The survival function of Lalace distribution decays at the rate of the ower function, F (x) e x, as x. Hence it follows that H(x; q) e x, as x. Then H(x; q) = P [X > x] = P [Y > U 1/q x] = P [Y > u 1/q x] du P [Y > x] du = F (x). In a similar way, a study on the tail behaviour when x has negative values can be done with the hel of cdf. On this basis we are led to the following remark. Remark The slash Lalace has heavier tails than the Lalace distribution. From Figure 2.1(a) also, it can be seen that Remark holds. From (2.3.1), the density of the general slash Lalace distribution denoted by X SLL(a, b, q) can be given as h(x; a, b, q) = q v q f(xv; a, b) dv (2.3.2) where f(x; a, b) denotes the df of a Lalace random variable with characteristic function ( cf ), φ Y (t) = eita 1 + b 2 t 2, t R. Figure 2.2(a) gives the robability density curves of the classical slash Lalace with a = 2, b = 3 for q =.1, 1, 2, 5, 8. The Figure reveals that, for higher values of q, the eakedness increases and hence the slash Lalace tends to Lalace distribution. The cdf of slash Lalace with a = 2, b = 3 and q = 3 is given in the Figure 2.2(b). 35

8 (a) (b) Figure 2.2: (a) The slash Lalace df for different values of q (b) cdf of the slash Lalace Estimation of Parameters In the standard case, the first and second order moments namely, the mean and variance of the standard slash Lalace distribution can be shown to be E(X) =, for q > 1 and V ar(x) = 2q, for q > 2. q 2 Hence by roceeding with the usual method of moments, the estimate of q can be obtained n as ˆq = 2 1 (x n i x) 2. i=1 The moment estimates of a, b and q in SLL(a, b, q) can be obtained by solving the three normal equations, aq q 1 = m 1 (2b 2 + a 2 )q (q 2) = m 2 (6ab 2 + 5a 3 )q (q 3) = m 3 36

9 where m r = 1 n x r is the r th raw moment and n is the samle size. From (2.3.2), the likelihood function of a random samle x 1, x 2,..., x n of size n from SLL(a,b,q) can be obtained as, L(x 1,..., x n ; a, b, q) = n i=1 q 2b v q e x i v a b dv. Then by solving the normal equations logl a =, logl b = and logl q = where logl is the log-likelihood function, the maximum likelihood estimates of the arameters can be obtained. For = 1, (2.2.2) yields the closed form of SLL df. function, the likelihood function is, Hence in terms of gamma L(x 1, x 2,..., x n ; q) = q 2 ( x 1x 2 x n ) (q+1) n i=1 (Γ (q + 1) Γ (q + 1, x i )). 2.4 Multivariate Slash Lalace Distribution A random vector X R has a -dimensional slash Lalace (SLL ) distribution if X = where Y is a Lalace random variable with cf given by φ Y (t) = t Σt, t R and U U(, 1) indeendent of Y. Y U 1/q Throughout this section, we have been considering the multivariate Lalace distribution as a distribution with location at zero. Here Σ is a non-negative definite symmetric matrix. The df of the SLL random variable can be shown to be g (x;, Σ, q) = q v q+ 1 f (xv;, Σ)dv, x R (2.4.1) where f ( ) is the density function of the -dimensional Lalace random variable, which is 37

10 given by f (y) = 1 (2π) /2 Σ 1/2 ( ) y Σ 1 ν/2 y k ν ( 2y Σ 1 y) 2 where ν = (2 ) Kotz et al., 21). and k ν (.) is the modified Bessel function of the third kind (for details see The cdf of the SLL random variable can be obtained as G (x;, Σ, q) = q v q+ 2 F (xv;, Σ)dv, x R where F ( ) is the cdf of the -dimensional Lalace random variable. Theorem If X SLL (, Σ; q) then the linear transformation Z = b + AX SLL (b, AΣA T ; q), where A is a non-singular matrix of order, A T reresents the transose of A and b is a vector in R. Proof: Consider the transformation Z = b + AX where X SLL (, Σ; q). Using the Jacobian determinant of transformation A 1, the df of Z can be obtained as A 1 f ( A 1 (z b);, Σ, q ). Hence Z has a multivariate slash Lalace distribution SLL (b, AΣA T ; q). Hence slash Lalace distribution is invariant under linear transformations. Remark When A is a full row rank matrix only, then also Theorem holds, since the marginal distributions of multivariate Lalace are still Lalace. Theorem If X SLL (, Σ; q), then E(X) = ; q > 1 and D(X) = where D(X) denotes the disersion matrix of X. q (q 2) Σ ; q > 2 38

11 Proof: The roof follows from the fact that E(U k/q ) = where U U(, 1) and X can be considered as X = Σ 1/2 where Y is standardized Lalace random vector. q q k ; q > k Theorem The marginal distributions of a slash Lalace distribution are again slash Lalace. Proof: It is sufficient to show that Y U 1/q g (x 1,..., x ; q) dx r+1 dx = g r (x 1,..., x r ; q); x 1,..., x r R and r. Using the df defined in (2.4.1) in the left hand side, and then by the substitution z r+1 = vx r+1,..., z = vx, v > followed by the fact that marginals of multivariate Lalace are again Lalace, yields the theorem. For the multivariate case, the tail behaviour can be examined in a way similar to that of univariate case and the same conclusion can be obtained. 2.5 Asymmetric Slash Distributions Asymmetric Slash Exonential Power Distribution The skewed version of SLEP distribution can be obtained as the distribution of the random variable X = Y U 1/q and is called the asymmetric slash exonential (ASLEP) distribution, denoted by X ASLEP (x; q) where Y is distributed as AEP, U U(, 1) and q >. 39

12 The df of ASLEP can be obtained as, g(x; q) = q v q f(xv) dv where f( ) is given by (2.1.2). The arameter art of the reresentation deends on the definition of AEP considered Asymmetric Slash Lalace Distribution The asymmetric slash Lalace distribution is defined as the distribution of the ratio X = Y U 1/q where Y is distributed as asymmetric Lalace (AL) and U U(, 1), indeendent of Y. The distribution can be denoted as ASLL( ;q), where the arameter art is taken according as the definition of AL considered. The df of ASLL can be shown to be, g(x; q) = q v q f(xv) dv where f( ) is the AL density considered. As q, the ASLL becomes AL Multivariate Asymmetric Slash Lalace Distribution This section deals with the multivariate slash Lalace distribution and its df. A random vector X R has a -dimensional asymmetric slash Lalace (ASLL ) distribution if X = Y U 1/q where Y follows AL (m, Σ) with cf given by φ Y (t) = t Σt im t where m R and Σ is a non-negative definite symmetric matrix and U U(, 1) indeendent of Y. It may be denoted by ASLL (m, Σ; q). Note that for m =, the distribution of Y reduces to symmetric Lalace and hence X reduces to SLL. The density function of ASLL can be shown to be, h (x; q) = q v q+ 1 l (xv;, Σ) dv 4

13 where l (x;, Σ) is the density of -variate AL considered. Here we considered the more general non-central -dimensional AL random vector with location centered at m. 2.6 Slash Generalized Cauchy Distributions The distribution of the random variable defined as X = Y U 1/q is called the slash generalized Cauchy distribution, denoted by X SLGC(x; ϑ, τ,, q) where Y is distributed as generalized Cauchy with df f(x;, τ, ϑ) = Γ( 2) 1 2[Γ( 1 )]2 τ [ 1 + x ϑ ] 2 ; x R, τ where < ϑ <, τ >, >. U U(, 1), q > and Y and U are indeendent. When considering the standard case, where Y is with df in (2.2.3), the distribution is denoted as Y SLGC(x;, q). For = 2, the SLGC yields the slash Cauchy (SLC) distribution. Theorem The distribution of the ratio of two iid EP random variables to the ratio of uniform ower function random variables defines the slash generalized Cauchy distributions. Proof: The roof can be done by following the concet from Remark The density function of standard SLGC can be obtained as, where f( ) is defined in (2.2.3). g(x;, q) = q v q f(xv; ) dv Closed form exressions for standard SLGC df can be obtained as, g(x;, q) = Γ( 2 ) 2[Γ( 1 )]2 ( q q + 1 q F 1, 2 ; 1 + q + 1 ; x 41 ) ; q = 1, 2,...

14 We know that Student s t with one degree of freedom gives standard Cauchy distribution. This result leads us to the following remark. Remark The univariate slash t distribution of Tan and Peng (25) with one degree of freedom can be obtained as a articular case of standard SLGC for = Case Study We consider a data of daily observation of US dollar-indian ruee exchange rate during 2-28 (See Usually the currency exchange rates are fat-tailed and shar eaked at the origin. The basic assumtion of financial modelling has been the normality assumtion for log X t+1 X t. However, to make the modelling more aroriate, the focus of the basic assumtion should have been shifted to the non-normality for log X t+1 X t. Here we do aly the SLL distribution to met this urose. The histogram of the transformed data (n log X t+1 X t ) is lotted along with df s of distributions of interest. A comarison of slash Lalace (a =.3715, b =.97814, q = ) (solid line) with the Lalace distribution (a =.551, b = ) (dashed lines) and normal distribution (µ =.551, σ = ) (dotted lines) is made by fitting the robability distribution over the data, using moment estimates. From Figure 2.4, we can claim that the slash Lalace distribution gives a better model than the Lalace and normal distributions. Again, the existing slash distributions namely, slash normal and slash t are comared with slash Lalace in Figure 2.5. The slash Lalace (a =.3715, b =.97814, q = ) (solid line) gives better fit to the data than slash normal (µ =.3674, σ = , q = 3.175) (dashed lines) and slash t (m = , q = ) (dotted lines). 2.8 Alications Slash exonential ower distribution is a family of slash distributions. For different values of the arameters, it generalizes various slash models to a common set u. Hence it gives wide flexibility in tail behaviour and eakedness and can be alied to data from different 42

15 Figure 2.3: Time series lot of daily exchange rate of US Dollar to Indian Ruees during 2-28 Figure 2.4: Embedded df s of the Slash Lalace, normal and Lalace distributions on the histogram for currency exchange rates Figure 2.5: Comarison of the Slash Lalace with Slash normal and Slash t distributions for financial modelling 43

16 contexts. The slash Lalace distribution introduces flexibility to Lalace distribution in tail behaviour by the addition of a new arameter through a transformation of variables. It gives a good model for data which exhibits heavy tails and eak center, where Lalace distribution is not a good fit. Slash distributions based on normal and student s t are alied to AIS and fiber glass data by Wang and Genton (26) and Tan and Peng (25). Our study on slash Lalace distribution, based on grahs, reveals the fact that slash Lalace distribution has more eaked center than slash normal and slash t and have tails at intermediate ositions. For financial modelling, the data on currency exchange rate is observed to have better fit with slash Lalace than normal, Lalace, slash normal and slash t distributions. The asymmetric slash Lalace distribution allows flexibility to Lalace distribution both in skewness and in tail behaviour. Recently multivariate Lalace distributions are in exloring stage and have received great interest from various areas. Hence its generalizations, multivariate slash Lalace and multivariate asymmetric slash Lalace distributions having heavy tails and skewness are also of good interest. References Arslan, O. (25). A new class of multivariate distributions: scale mixture of Kotz- tye distributions, Statist. Probab. Lett., 75, Arslan, O. (28). An alternate multivariate skew slash distribution, Statist. Probab. Lett., 78(16), Arslan, O., Genc, A.I. (29). A generalization of the multivariate slash distribution, J. Statist. Plan. Infer., 139(3), Ayebo, A., Kozubowski, T.J. (26). Lalace laws, Pre-rint. An asymmetric generalization of Gaussian and 44

17 Fernandez, C., Steel, M.F.J. (1998). On Bayesian modelling of fat tails and skewness, J. Amer. Statist. Assoc., 93, Genc, A.I., (27). A generalization of the univariate slash by a scale- mixtured exonential ower distribution, Comm. Statist. Simulation Comut., 36(4-6), Jose, K.K., Lishamol, T. (27). Slash Lalace distributions and their alications, STARS Int. Journal (Sciences), 1(2), Kafadar, K. (1988). Slash Distribution, Encycloedia of Statistical Sciences, 8, (Johnson, N.L., Kotz, S., Read, C., Eds.). Wiley: New York, Kotz, S., Kozubowski, T.J., Podgorski, K. (21). The Lalace Distribution and Generalizations: A Revisit with Alications to Communications, Economics, Engineering and Finance. Birkhäuser, Boston. Lishamol, T. (27). Slash exonential ower distributions and alications, Paer shortlisted for the ISCA Young Scientists Award-28 in Mathematical Sciences section of The Indian Science Congress held at Visakhaattanam during October 27. Lishamol, T., Jose, K.K. (29). Generalized slash Lalace distributions and alications in financial modelling, Comm. Statist. Theory Methods, Submitted. Sebastian, G., Preethi, S. G. (23). Random variate generation from a generalized Cauchy distribution, STARS Int. Journal, 4(2), 1-9. Subottin, M.T. (1923). On the law of frequency of error, Mathematicheskii Sbornik, 31, Tan, F., Peng, H. (25). The slash and skew-slash student t distributions, Pre-rint. Wang, J., Genton, M. G. (26). The multivariate skew-slash distribution, J. Statist. Plann. Infer., 136,