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Skewed Double Exponential Distribution and Its Stochastic Rep- Resentation

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Skewed Double Exponential Distribution and Its Stochastic Rep- Resentation EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 1, 2009, (1-20) ISSN 1307-5543 – www.ejpam.com Skewed Double Exponential Distribution and Its Stochastic Rep- resentation 12 2 2 Keshav Jagannathan , Arjun K. Gupta ∗, and Truc T. Nguyen 1 Coastal Carolina University Conway, South Carolina, U.S.A 2 Bowling Green State University Bowling Green, Ohio, U.S.A Abstract. Definitions of the skewed double exponential (SDE) distribution in terms of a mixture of double exponential distributions as well as in terms of a scaled product of a c.d.f. and a p.d.f. of double exponential random variable are proposed. Its basic properties are studied. Multi-parameter versions of the skewed double exponential distribution are also given. Characterization of the SDE family of distributions and stochastic representation of the SDE distribution are derived. AMS subject classifications: Primary 62E10, Secondary 62E15. Key words: Symmetric distributions, Skew distributions, Stochastic representation, Linear combina- tion of random variables, Characterizations, Skew Normal distribution. 1. Introduction The double exponential distribution was first published as Laplace’s first law of error in the year 1774 and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error, disregarding sign. This distribution comes up as a model in many statistical problems. It is also considered in robustness studies, which suggests that it provides a model with different characteristics ∗Corresponding author. Email address: (A. Gupta) http://www.ejpam.com 1 c 2009 EJPAM All rights reserved. K. Jagannathan, A. Gupta, and T. Nguyen / Eur. J. Pure Appl. Math, 2 (2009),(1-20) 2 than other symmetric distributions. In particular, the tails are thicker than those of the normal distribution, but not as thick as the Cauchy distribution. This distribution has not gained much exposure, possibly partly due to a lack of available statistical techniques and partly due to the sharpness of the peak. Many applications are concerned with tail probabilities and the double exponential distribution would be a good choice when exponential tails are required. The skewed double exponential distribution (SDE), proposed here, is a good addition to the family of skewed distributions in the sense that, like the skew normal distribution, it is a skewed distribution that possesses many of the properties that symmetric distributions do, while at the same time, being skewed, provides a better fit to real life data that is seldom symmetric in nature. There are many SDE type random variables in literature including those introduced by McGill(1962), Holla and Bhattacharya(1968), Lingappaiah(1988), Poiraud-Cassanova and Thomas-Agnan(2000), Hinkley and Revankar(1977), and Kozubowski and Podgórski(2000) to name a few. One of the approaches that we take in this paper is to look at the mixture of a double exponential random variable and an exponential random variable. The other approach is similar to that of Azzalini(1985) in generating skew normal random variables. There are advantages and disadvantages to both models as will be evident in later sections of this paper. Section 2 deals with preliminary results necessary for results in this paper. Section 3 talks about the SDE1 distribution and its various representations. Section 4 discusses the SDE2 distribution and its characterization in terms of the double exponential distribution. 2. Preliminary Results and Definitions Definiton 1. Consider a random variable U distributed as a double exponential distribution (U DE(η, θ)) if it has probability density function(p.d.f.) given by ∼ 1 u η /θ f (u)= e| − | , u R. U 2θ ∈ K. Jagannathan, A. Gupta, and T. Nguyen / Eur. J. Pure Appl. Math, 2 (2009),(1-20) 3 where η R and θ > 0. The corresponding cumulative distribution function(c.d.f.) is given ∈ by 1 (u η)/θ e − i f u < η 2 FU (u)= 1 (u η)/θ 1 e− − i f u η − 2 ≥ Note that if U DE(0,1), then V = θ(U)+ η has a double exponential (η, θ) distribution, ∼ i.e V DE(η, θ). As a special case, let U be a standard double exponential (DE) random ∼ variable, i.e U DE(0,1). Then, U is distributed as U E x p(1). ∼ | | | |∼ There are two approaches taken in generating the SDE random variables. The first (SDE1) involves the scaled mixture of exponential and double exponential random variables. The second (SDE2) deals with the product of the p.d.f. and scaled c.d.f. of the double exponential distribution(DE(0,1)). The following sections treat each of the two methods in detail. 3. The SDE1 Distribution : Definition and Stochastic Representation In this section, we present the distribution function of the different models of the skewed double exponential(SDE1) distribution. Theorem 3.1. Consider two i.i.d random variables U,V distributed as DE(0,1). Then, the ran- dom variable Y defined as a U + bV, for a > 0 , b R+ , a = b is defined to be distributed as | | ∈ 6 SDE1(a, b) and has the c.d.f. b e y/b, i f y < 0 2(a+b) FY (y)= a2 y/a b y/b 1 + e− e− , i f y 0 b2 a2 2(b a) − − − ≥ and it’s p.d.f. is given by y/b e , i f y < 0 2(a+b) fY (y)= y/b e− a e y/a, i f y 0 . 2(b a) (b2 a2) − − − − ≥ If a = b, then the c.d.f. is 1 e y/a, i f y < 0 4 FY (y)= 3a+2y y/a 1 e− , i f y 0 − 4a ≥ K. Jagannathan, A. Gupta, and T. Nguyen / Eur. J. Pure Appl. Math, 2 (2009),(1-20) 4 and it’s p.d.f. is given by y/a e , i f y < 0 4a fY (y)= a+2y y/a e− , i f y 0 . 4a2 ≥ Proof. We give a proof that is similar to the one given by Henze(1986). Consider ∞ y au P P [Y y] = [V − ] f U (u) du ≤ ≤ · | | Z0 b y au − ∞ b 1 v u = e−| | dv e− du. (3.1) Z0 Z 2 −∞ y au If y < 0, then, since a > 0 and the support of U is (0, ), − < 0 , u. Hence, (1) | | ∞ b ∀ becomes y au − ∞ b 1 v u P[Y y] = e dv e− du ≤ Z0 Z 2 −∞ b = e y/b , y < 0. 2(a + b) y au y au On the other hand, if y 0, then either − < 0 for u > y/a or − > 0 for u < y/a. ≥ b b So, (1) is equivalent to y y au − a b 1 1 v u P[Y y] = + e− dv e− du ≤ Z 2 Z 2 0 0 (i) y au − | ∞ b {z } 1 v u + e dv e− du . (3.2) Z y Z 2 a −∞ (ii) | {z } K. Jagannathan, A. Gupta, and T. Nguyen / Eur. J. Pure Appl. Math, 2 (2009),(1-20) 5 Consider : y y y au − a a b 1 u 1 v u (i) = e− du + e− d v e− du Z0 2 Z0 Z0 2 y a y au 1 y/a 1 − u = 1 e− + 1 e− b e− du 2 − Z 2 − 0 y/a b y/b b y/a = 1 e− e− + e− . (3.3) − − 2(b a) 2(b a) − − Also, y au − ∞ b 1 v u (ii) = e dv e− du Z y Z 2 a −∞ b y/a = e− . (3.4) 2(a + b) Adding the right hand sides of (3) and (4), as required by (2), we get via some elementary algebra that the c.d.f. of the SDE1(a, b) is as follows: b e y/b, i f y < 0 2(a+b) FY (y)= (3.5) a2 y/a b y/b 1 + e− e− , i f y 0 . (b2 a2) 2(b a) − − − ≥ Differentiating (3.5) with respect to y, we get the p.d.f. of an SDE1(a, b) random variable to be y/b e , i f y < 0 2(a+b) fY (y)= y/b e− a y/a e− , i f y 0 . 2(b a) (b2 a2) − − − ≥ A similar approach can be taken in the case when a = b to find the c.d.f. and p.d.f. of the SDE1(a, b) random variable. Note. In the above theorem the value of "b" is restricted to the positive half line. Since d V DE(0,1), bV = bV . Thus, we can restrict our attention to the case when b 0 as the ∼ − ≥ results will be the same when b < 0. K. Jagannathan, A. Gupta, and T. Nguyen / Eur. J. Pure Appl. Math, 2 (2009),(1-20) 6 The following result given by Theorem(3.2) is obtained by using the fact that F (y)= P(a U + bV y)= P( a U + bV y)= 1 F ( y) Ya,b Y a,b | | ≤ − | | ≥− − − − and the result of Theorem(3.1). Theorem 3.2. Consider two i.i.d random variables U,V distributed as DE(0,1). Then, the ran- dom variable Y defined as a U + bV, for a < 0 , b R+ , a = b has the c.d.f. | | ∈ 6 − b y/b a2 y/a e e− , i f y < 0 2(a+b) − b2 a2 FY (y)= − 1 b e y/b, i f y 0 2(b a) − − − ≥ and it’s p.d.f. is given by e y/b a y/a + e− , i f y < 0 2(a+b) (b2 a2) fY (y)= − y/b e− , i f y 0 .
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