On general error distributions

Transcription

1 ProbStat Forum, Volume 6, October 3, Pages ISSN ProbStat Forum is an e-journal. For details please isit On general error distributions R. Vasudea, J. Vasantha Kumari Department of Studies in Statistics, Uniersity of Mysore, Mysore-576, India Abstract. General error distributions are used in statistical modeling, where in the errors are not necessarily normally distributed. In this paper, we establish some interesting properties of these distributions and obtain a characterization.. Introduction In statistical modeling, such as linear models, the difference between the obsered alue and the expected alue is called an error. For many decades, it was assumed that the error random ariable r.. follows a normal distribution with mean zero. In many situations, it was obsered that the normality is not an appropriate assumption. As alternaties, Subbotin 93 introduced a class of distributions that are symmetric, but with ariation in urtosis. He noted that these distributions hae many structural properties close to a normal distribution. This class of distributions is called the general error distributions. Nelson 99 has deeloped linear regression models and time series models with heay tails, assuming the underlying distribution as general error distribution GED. Ley 4 and Nadarajah 5 discuss many distributional properties of a GED. In our paper, we call this class of distributions as general error distributions of the first ind, denoted by GED-I, as we will be discussing one more class of general error distributions asymmetric. The probability density function pdf of GED-I is gien by f x exp{ x λ } λ +/ Γ, >, x R, [ where λ Γ ] Γ 3 and Γ denotes the Gamma function. It is well nown that if X is GED-I, EX, EX and EX < for all >. When, GED-I reduces to a standard normal distribution and when, it reduces to a double exponential distribution. Peng et al. 9 established that the tail of the distribution function d.f. F of GED-I has the asymptotic relation, F x x fx, as x, where means asymptotically equal. One can see that the tail of the d.f. is asymptotically Weibullian. Another class of error distributions was introduced, by allowing the tail to be highly sewed see, Wiipedia. We call this class as the general error distributions of the second ind, denoted as Keywords. General Error Distributions, Moment generating function, Characterization Receied: 8 December ; Accepted: 7 October 3 Visiting Professor, Sir M. Viseswaraiah chair, Uniersity of Mysore, Mysore addresses: asudea.rasbagh@gmail.com R. Vasudea, asanthaumarisj@gmail.com J. Vasantha Kumari

2 exp gx Vasudea and Kumari / ProbStat Forum, Volume 6, October 3, Pages GED-II. The pdf of GED-II is gien by { log xξ π xξ }, with x, ξ + if >, x ξ +, if <, where ξ is a real constant and a positie constant. When, g reduces to the pdf of a normal random ariable r.. with mean ξ and ariance. The d.f. G has the tail exp Gx { log xξ π log xξ }, as x, when < and as x ξ + /, when >. The form of Gx is well nown when. In this paper, we establish that the moment generating function mgf of a GED-I exists when and fails to exist when < <. We obtain a characterization and establish some additie properties of a GED-I. This is done in the next section. In Section 3, we show that the mgf of a GED-II fails to exist wheneer. Also, we show that the tail Gx is asymptotically, sandwiched between a Weibullian tail and a regularly arying tail. Throughout the paper, Mt Ee tx, < t < denotes a mgf.. General error distribution I Theorem.. Let a random ariable X hae GED-I with parameter >. Then the mgf Mt exists for all t, when >, exists in the region, when, and fails to exist for any t > when < <. Proof. We hae for any > that Mt c e tx e c x dx, < t <, where c λ +/ Γ and c λ. Suppose that < <. Let t >. Then for any x > e tx e c x e tx c tx. Let x > be such that e tx c tx tx e c tx for all x x. Consequently, x e tx e c x dx, < for all x x, so that which implies that Mt fails to exit for any t >, wheneer < <. When, the pdf fx e x, < x <. Here Mt e tx x dx I + I,

3 Vasudea and Kumari / ProbStat Forum, Volume 6, October 3, Pages where I e tx+ x dx and I e tx x dx. Put x y. Then I Hence Mt Mt fails to exit. Let >. Then Since Mt c e yt+ dy t + +t + c and I e tx x dx e x t dx t. t t t, wheneer t <. Also, when t, e txc x dx e txc x dx + c I + I, say. tx c x c x e txcx dx t c x c x + o, as x, one gets I <, for any real t. Similarly, for x < we hae that tx c x c x tx c x c x t + c x c x + o, as x. Consequently, I < for any t. In turn, Mt exists for all t,. Applying Maclaurian expansion and identifying M EX,, one gets Mt t! EX. Since X is symmetric about zero, we hae EX for odd. When is een, m say, Γ EX m m Γ m+ Γ 3 Γ, m. Hence Mt m t m Γ m! Γ 3 m Γ m+ Γ. A closed form expression does not exist for Mt. Theorem.. A real alued random ariable X, symmetric about, is GED-I if and only if for some Γ 3 / >, the random ariable Γ has gamma distribution with parameter /. X

4 Vasudea and Kumari / ProbStat Forum, Volume 6, October 3, Pages Proof. Let X be GED-I with parameter. Put Y and let the pdf of Z be gz. Then gz f z z λ +/ Γ e z λ z, z >. Γ 3 / Γ / Γ. Then X / X Γ Y. Let Z X 3 Γ 3 Denote by h, the pdf of Y. The relation, Y Γ Z gies hy g y / c c y, where c Γ 3 Γ. Obsere that λ / c /. Substituting for g and λ, on simplification, one gets hy Γ ey y, y >, which is the pdf of a gamma r.. with parameter /. Γ 3 / Conersely, let Y Γ be Gamma/ and let H denote the d.f. of Y with pdf h. For any X y >, defining c Γ 3 /Γ, we hae Γ 3 Hy P Y y P y Γ X P X y P X y. c c Differentiating with respect to y and recalling that X is symmetric about, with pdf f, one gets hy f y y. c c Put x y / c. Then gies c fx h c x cx c / x Γ c / x c x ec c x. Γ ec Since λ / Γ / c /, for x >, one gets, fx λ + Γ e x Γ 3 λ. Also, fx fx, x >, implies that fx λ + Γ e x λ, < x <, which is the pdf of GED-I. Theorem.3. Let X, X,..., X n be independent and identically distributed i.i.d. GED-I random ariables with parameter. Then n i X i is a gamma distributed random ariable. Proof. Gien that X is GED-I, proceeding as in the proof of Theorem., one can show that X is Gamma/λ, /. By the closure property of gamma distribution, X, X,..., X n are i.i.d. GED-I implies n j X j is Gamma/λ, n/.

5 Vasudea and Kumari / ProbStat Forum, Volume 6, October 3, Pages Remar.4. From the aboe theorem, we obsere that if X and X are independent r.. s with a common GED-I, then X, X and the conolution X + X are gamma distributed random ariables. Remar.5. Conolution of GED-I, need not necessarily be GED-I. When, X and X are Laplace distributed random ariables, but X + X is not Laplace distributed random ariable. When, X and X are normal distributed random ariables and X + X is also normal distributed random ariable. 3. General error distributions II Theorem 3.. The mgf Mt of a GED-II fails to exist for any t > when < and for any t < when >. Proof. Consider the case <. Then the pdf of GED-II is { } exp log + xξ gx, x ξ + π + xξ, where >, ξ, are constants. The mgf is gien by Mt e tx gxdx, < t <. Put + xξ ξ+ y. Then x y + ξ and hence We hae Mt eξ+ t π yt log y e ξ+ t π e yt log y yt e yt dy log y log y dy. 3 log y log y log y + log y +. yt Hence, for any gien t >, one can find a y > such that yt from 3, Mt eξ+ t π y e yt dy, log y log y yt. Consequently i.e. the mgf fails to exist as it does not exist for any t >. Now consider the case, >. Let X be GED-II with >. Define Y X. The pdf of Y is exp log + yξ hy, y ξ π + + yξ, where ξ ξ and. Hence Y is GED-II with <, consequently, Ee ty fails to exist for any t > and in turn Mt Ee tx fails to exist for any t <.

6 Vasudea and Kumari / ProbStat Forum, Volume 6, October 3, Pages Theorem 3.. Let H x, x >, denote the tail of a Weibullian d.f. and H x, x >, be a d.f. with regularly arying tail. Then for <, the tail G of a GED-II satisfies the relation, H x lim x Gx and where H H. lim H x x Gx, Proof. For <, the tail of GED-II is gien by { } exp log + xξ Gx. 4 π log + xξ Let y + xξ. Then the numerator on the right hand side expression of 4 is exp log ylog y exp log y log y y log y. Obsere that y + xξ x + ξ x and hence log y log x+log +log + ξ x. Since + ξ x as x, gien δ, δ >, one can find a x > such that for all x x, δ x y + δ x and log x + log δ log y log x + log + δ. Hence for all x x, log x+log +δ + δ x y log y δ x Also, the denominator on the right side of 4 satisfies for x x, π log x + log δ π log y log x+log δ. π log x + log + δ. Since log x + log ± δ log x, as x, for a δ 3 >, one can find a x x such that for all x x, δ 3 log x log x + log ± δ + δ 3 log x. Put d δ and d + δ. Then for all x x, the right hand side of 4 satisfies the inequality, +δ 3 log x d x π + δ 3 log x exp log y π log y δ 3 log x d x π δ 3 log x. For a gien δ 4,, one can hence find a x x such that for all x x, +δ 3 log x δ 4 d x π + δ 3 log x Gx δ 3 log x + δ 4 d x. π δ 3 log x

7 Vasudea and Kumari / ProbStat Forum, Volume 6, October 3, Pages Put c 5 π δ 4 +δ 3 and c 6 +δ 4 π δ 3. Then the aboe inequality can be written as +δ 3 log x c 5 d x log x δ 3 log x Gx c 6d x log x, 5 wheneer x x. Let H be a d.f. with Weibullian tail. Then H x e xβ x r +o, as x, for some >, β > and r,. Also, let H be a d.f. with regularly arying tail, i.e. H x x r Lx, for some r > and L a slowly arying function. From 5 one can show that lim completes the proof. x H x Gx and lim x H x Gx, which Remar 3.3. We hae noticed that the tail thicness of GED-II is in between Weibullian and regularly arying tails. Perhaps, one reason for considering such a tail is that sewed distributions with Weibullian tail can be easily constructed from GED-I by truncating to the left right and sewed distributions with regularly arying tail will not hae all moments finite, which is supposed to be an underlying structure of a random ariable with GED. References Ley, G. 4 Computational finance, numerical methods for pricing financial instruments, Elseier. Nadarajah, S. 5 A generalized normal distribution, J. Appl. Stat. 3 7, Nelson, D.B. 99 Conditional heteroscedasticity in asset returns: A new approach, Econometrica 59, Peng, Z., Tong, B., Nadarajah, S. 9 Tail behaiour of the general error distribution, Commun. Stat. - Theory Methods 38, Subbotin, M. 93 On the law of frequency of errors, Mathematichesii Sborni 3, Wiipedia. Generalized normal distribution, Wiipedia.org/Wii/ free encyclopedia.