An extension of the half-

Neveka M. Olmos ∗ H´ectorVarela† H´ectorW. G´omez‡ Heleno Bolfarine§

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 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 fitting 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] †Departamento de Matem´aticas,Facultad de Ciencias B´asicas,Universidad de Antofa- gasta, Antofagasta, Chile. e-mail: [email protected] ‡Departamento de Matem´aticas,Facultad de Ciencias B´asicas,Universidad de Antofa- gasta, Antofagasta, Chile. e-mail: [email protected] §Departamento de Estat´ıstica,IME, Universidad de Sao Paulo, Sao Paulo, Brasil. e- mail: [email protected] X CONGRESO LATINOAMERICANO DE SOCIEDADES DE ESTAD´ISTICA

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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 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 → ∞, the standard normal distribution follows. On the other hand, if q = 1, we obtain the canonic (standard) slash distribution, with density function given by   φ(0)−φ(x) x 6= 0 p(x) = x2 (2)  1 2 φ(0) x = 0, where φ 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 σqΓ((q + 1)/2)z−(q+1)G z2, (q + 1)/2, (4) Z π 2σ2 where σ > 0, q > 0, z > 0 and G is the cumulative cdf of the . X CONGRESO LATINOAMERICANO DE SOCIEDADES DE ESTAD´ISTICA

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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 µ = E(Zr) = Γ σr (5) r q − r π 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) + ψ( ) + 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 ψ 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

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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

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