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An Extension of the Half-Normal Distribution An extension of the half-normal distribution Neveka M. Olmos ¤ H¶ectorVarelay H¶ectorW. G¶omezz Heleno Bolfarinex Abstract In this work we introduce a new distribution, namely, the slashed half-normal distribution and it can be seen as an extension of the half-normal distribution. It is shown that the resulting distribution has more kurtosis than the ordinary half-normal distribution. Mo- ments and some properties are derived for the new distribution. Mo- ment estimators and maximum likelihood estimators can computed using numerical procedures. Results of two real data application are reported where model ¯tting is implemented by using maximum like- lihood estimation. KEY WORDS: Half-normal distribution, slash distribution, slashed half-normal distribution, kurtosis. ¤Departamento de Matem¶aticas,Facultad de Ciencias B¶asicas,Universidad de Antofa- gasta, Antofagasta, Chile. e-mail: [email protected] yDepartamento de Matem¶aticas,Facultad de Ciencias B¶asicas,Universidad de Antofa- gasta, Antofagasta, Chile. e-mail: [email protected] zDepartamento de Matem¶aticas,Facultad de Ciencias B¶asicas,Universidad de Antofa- gasta, Antofagasta, Chile. e-mail: [email protected] xDepartamento de Estat¶³stica,IME, Universidad de Sao Paulo, Sao Paulo, Brasil. e- mail: [email protected] X CONGRESO LATINOAMERICANO DE SOCIEDADES DE ESTAD¶ISTICA CORDOBA,¶ ARGENTINA, 16 A 19 DE OCTUBRE DE 2012 1 Introduction A distribution closely related to the normal distribution is the slash distribu- tion. It can be represented as the quotient between to independent random variables, a normal one (numerator) and the power of a uniform (0; 1) dis- tribution (denominator). Hence, we can say that a random variable S has a slash distribution if it can be represented as 1 S = Z=U q ; (1) where Z » N(0; 1), independent of U » U(0; 1) and q > 0. In particular, as q ! 1, the standard normal distribution follows. On the other hand, if q = 1, we obtain the canonic (standard) slash distribution, with density function given by 8 < Á(0)¡Á(x) x 6= 0 p(x) = x2 (2) : 1 2 Á(0) x = 0; where Á represents the density function of the standard normal distribution (Johnson, Kotz and Balakrishnan 1995). It is well known that this dis- tribution presents heavier tails than the normal one that is, it has greater kurtosis. Properties of this family are studied in Rogers and Tukey (1972) and Mosteller and Tukey (1977). Maximum likelihood estimators for the location scale case are studied in Kafadar (1982). Wang and Genton (2006) developed a multivariate version and also an asymmetric multivariate ver- sion studying its properties and inference. G¶omez et al. (2007) and G¶omez and Venegas (2008) extends the slash distribution by introducing the slash- elliptical family. Asymmetric versions of this family are discussed in works of Arslan (2008). Arslan and Genc (2009) discussed a symmetric extension of the multivariate slash distribution and Genc (2007) discussed a symmetric X CONGRESO LATINOAMERICANO DE SOCIEDADES DE ESTAD¶ISTICA CORDOBA,¶ ARGENTINA, 16 A 19 DE OCTUBRE DE 2012 univariate generalization of the slash distribution. Arslan (2009) has also studied the ML estimation for the parameters of the skew slash distribution introduced in Arslan (2008). G¶omez et al. (2009) use the slash-elliptical family to extend the Birnbaum-Saunders distribution. 2 Incorporating Kurtosis 2.1 Stochastic representation We consider that the random variable Z has a slashed half-normal distribu- tion with parameters σ and q if it can be represented as the ratio X Z = (3) Y 1=q where X » HN(σ) and Y » U(0; 1) are independent, σ > 0, q > 0. We denote this by writing Z » SHN(σ; q). 2.2 Density function Let Z » SHN(σ; q). Then, the pdf of Z is given by r µ ¶ 2q 1 f (z; σ; q) = q σq¡((q + 1)=2)z¡(q+1)G z2; (q + 1)=2; (4) Z ¼ 2σ2 where σ > 0, q > 0, z > 0 and G is the cumulative cdf of the gamma distribution. X CONGRESO LATINOAMERICANO DE SOCIEDADES DE ESTAD¶ISTICA CORDOBA,¶ ARGENTINA, 16 A 19 DE OCTUBRE DE 2012 2.3 Moments Let Z » SHN(σ; q). Then, for r = 1; 2; ::: and q > r it follows that the r-th distributional moment is given by r µ ¶ q 2r r + 1 ¹ = E(Zr) = ¡ σr (5) r q ¡ r ¼ 2 3 Inference 3.1 Maximum Likelihood Given a random sample Z1;:::;Zn from the distribution of SHN(σ; q), the maximum likelihood equations are given by Xn G (z2) nq 1 i = ¡ (6) G(z2) σ i=1 i n n n q + 1 Xn G (z2) Xn + log(2) + nlog(σ) + Ã( ) + 2 i = log(z ) (7) q 2 2 2 G(z2) i i=1 i i=1 2 2 q+1 1 2 d 2 2 d 2 where G(zi ) = G(zi ; 2 ; 2σ2 ), G1(zi ) = dσ G(zi ), G2(zi ) = dq G(zi ) and à is the digamma function. 4 Illustration We consider a data set of the life of fatigue fracture of Kevlar 49/epoxy that are subject to constant pressure at the 90% stress level until all had failed, so we have complete data with the exact times of failure. For previous studies with the data sets see Andrews and Herzberg (1985) and Barlow et al. (1984). X CONGRESO LATINOAMERICANO DE SOCIEDADES DE ESTAD¶ISTICA CORDOBA,¶ ARGENTINA, 16 A 19 DE OCTUBRE DE 2012 Table 1: Parameter estimates (with (SD) indicating standard deviation) and log-likelihood values for HN and SHN models for the stress-rupture life data set. Parameter estimates HN(SD) SHN(SD) σ 1:514(0:107) 0:802(0:111) q ¡ 2:567(0:691) AIC 232:34 210:26 References [1] Arslan, O. 2008. \An Alternative Multivariate Skew-Slash Distribution". Statistics and Probability Letters 78(16), 2756-2761. [2] Arslan, O. 2009. \Maximum likelihood parameter estimation for the mul- tivariate skew-slash distribution". Statistics and Probability Letters 79, 2158-2165. [3] Arslan, O. and Genc, A.I. 2009. \A Generalization of the Multivariate Slash Distribution". Journal of Statistical Planning and Inference 139(3), 1164-1170. [4] Barlow, R. E., Toland, R. H., Freeman, T. 1984. A Bayesian analysis of stress-rupture life of kevlar 49/epoxy spherical pressure vessels, In: Proc. Canad. Conf. Appl. Statist. New York: Marcel Dekker. [5] Genc, A.I. 2007. \A Generalization of the Univariate Slash by a Scale- Mixture Exponential Power Distribution". Communications in Statistics - Simulation and Computation 36(5), 937-947. X CONGRESO LATINOAMERICANO DE SOCIEDADES DE ESTAD¶ISTICA CORDOBA,¶ ARGENTINA, 16 A 19 DE OCTUBRE DE 2012 [6] G¶omez,H. W., Olivares-Pacheco, J. F., Bolfarine, H. 2009. An extension of the generalized Birnbaum-Saunders distribution. Statistics and Proba- bility Letters. 79(3), 331-338. [7] G¶omez,H. W., Quintana, F. A. and Torres, F. J. 2007. New Family of Slash-Distributions with Elliptical Contours. Statistics and Probability Letters. 77(7), 717-725. [8] G¶omez,H.W. and Venegas, O. 2008. \Erratum to: A New Family of Slash-Distributions with Elliptical Contours [Statist. Probab. Lett. 77 (2007) 717-725]". Statistics and Probability Letters. 78(14), 2273-2274. [9] Jonhson, N.L., Kotz, S., and Balakrishnan, N. 1995. Continuous univari- ate distributions, Vol 1, 2nd edn. New York: Wiley. [10] Kafadar, K. 1982. A biweight approach to the one-sample problem. J. Amer. Statist. Assoc. 77, 416-424. [11] Mosteller, F., Tukey, J.W. 1977. Data analysis and regression. Addison- Wesley, Reading, MA. [12] Rogers, W.H., Tukey, J.W. 1972. Understanding some long-tailed sym- metrical distributions. Statist. Neerlandica. 26, 211-226. [13] Wang, J., Genton, M.G. 2006. The multivariate skew-slash distribution. Journal Statistical Planning and Inference. 136, 209-220..
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