We introduce a new class of the slash distribution using the epsilon half normal distribution. The newly defined model extends the slashed half normal distribution and has more kurtosis than the ordinary half normal distribution. We study the characterization and properties including moments and some measures based on moments of this distribution. A simulation is conducted to investigate asymptotically the bias properties of the estimators for the parameters. We illustrate its use on a real data set by using maximum likelihood estimation.
Epsilon Half Normal Distribution; Slash Distribution; Kurtosis; Skewness; Maximum Likelihood Estimation1. Introduction
The epsilon half normal distribution, proposed by Castro et al. [1], is widely used for nonnegative data modeling, for instance, to consider the lifetime process under fatigue. We say that a random variable has an epsilon half normal distribution with parameters and, denoted by, if its density function is given by, for,
where denotes the standard normal density function. When, the epsilon half normal distribution reduces to the half normal distribution investigated in [2-4].
Castro et al. [1] provided mathematical properties of the epsilon half normal distribution and discussed some inferential aspects related to the maximum likelihood estimation.
On the other hand, a random variable has a standard slash distribution with parameter, introduced in [5], if can be represented as
where and are independent. It generalizes normality and has been much studied in the statistical literature.
For the limit case, yields the standard normal distribution. Let, the canonical slash distribution follows, see [6]. It is well known that the standard slash density has heavier tails than those of the normal distribution and has larger kurtosis. It has been very popular in robust statistical analysis and studied by some authors.
The general properties of this canonical slash distribution were studied in [5,7]. Kafadar [8] investigated the maximum likelihood estimates of the location and scale parameters.
Gómez et al. [9] replaced standard normal random variable by an elliptical distribution and defined a new family of slash distributions. They studied its general properties of the resulting families, including their moments. Genc [10] proposed the univariate slash by a scale mixtured exponential power distribution and investigated asymptotically the bias properties of the estimators. Wang et al. [11] introduced the multivariate skew version of this distribution and examined its properties and inferences. They substituted the standard normal random variable by a skew normal distribution studied in [12] to define a skew extension of the slash distribution.
Olmos et al. [3] introduced the slashed half normal distribution by a scale mixtured half normal distribution and showed that the resulting distribution has more kurtosis than the ordinary half normal distribution. Since the epsilon half normal distribution is an extension of the half normal distribution, it is naturally to define a slash distribution based on it in which skewness and thick tailed situations may exist. It leads to a new model on nonnegative measurements with more flexible asymmetry and kurtosis parameters.
The paper is organized as follows: in Section 2, we introduce the new slash distribution and study its relevant properties, including the stochastic representation etc. In Section 3 we discuss the inference, moments and maximum likelihood estimation for the parameters. Simulation studies are performed to investigate the behaviors of estimators in Section 4. In Section 5, we give a real illustrative application and report the results. Section 6 concludes our work.
2. Epsilon Half Normal Slash Distribution2.1. Stochastic Representation
Definition 2.1 A random variable has an epsilon half normal slash distribution if it can be represented as the ratio
where defined in (1) and
are independent, , ,. We denote it as.
Proposition 2.2 Let. Then, the density function of is given by, for,
where, ,.
Proof. From (1), the joint probability density function of and is given by, for,
Using the transformation:
the joint probability density function of and is given by, for,
The marginal density function of is given by
After changing the variable into, the dendity function will be obtained as stated.
Remark 2.3 If, the density function (4) reduces to
which is the density function for the slashed half normal distribution studied by [3]. As,
The limit case of the epsilon half normal slash distribution is the epsilon half normal distribution. For, the canonical case follows.
Figure 1 shows some plots of the density function of the epsilon half normal slash distribution with various parameters.
The cumulative distribution function of the epsilon half normal slash distribution is given as follows. For,
where is the cumulative distribution function for the standard normal random variable.
Proposition 2.4 Let and, then.
Proof.
Remark 2.5 Proposition 2.4 shows that the epsilon half normal slash distribution can be represented as a scale mixture of an epsilon half normal distribution and uniform distribution. The result provides another way besides the definition (3) to generate random numbers from the epsilon half normal slash distribution
.
2.2. Moments and Measures Based on Moments
In this section, we derive the moment generating function,the k-th moment and some measures based on the moments.
Proposition 2.6 Let, then the moment generating function of is given by for,
where and. Proof. See [1].
Proposition 2.7 Let, then the moment generating function of is given by, for,
where, and.
Proof. From Proposition 2.4 and using properties of the conditional expectation, we have
Making the transformation and the result follows.
Proposition 2.8 Let, then the non-central moments are given by
for. where and.
Proof. See [1].
Proposition 2.9 Let, where, and. For and, the non-central moment of is given by
Proof. From the stochastic representation defined in (3) and the results in (9), the claim follows in a straightforward manner.
The following results are immediate consequences of (10).
Corollary 2.10 Let, where, and. The mean and variance of are given by
and for,
For the standardized skewness
and kurtosis coefficients
we have the following results.
Corollary 2.11 Let, where, and. The skewness and kurtosis coefficients of are given by
where
Remark 2.12 As, the skewness coefficient converges.
and the kurtosis coefficient converges as well
which are the corresponding skewness and kurtosis coefficients for the epsilon half normal distribution.
Figure 2 shows the skewness and kurtosis coefficients with various parameters for the model. The skewness and kurtosis coefficients decrease as increases. The parameter does not affect the two coefficents.
3. Maximum Likelihood Inference
In this section, we consider the maximum likelihood estimation about the parameters of the model defined in (4). For,
where.
Suppose is a random sample of size from the epsilon half normal slash distribution
. Then the log-likelihood function can be written as
The estimates of the parameters maximize the likelihood function. By taking the partial derivatives of the log-likelihood function with respect to respectively and equalizing the obtained expressions to zero, we obtain the following maximum likelihood estimating equations.
The maximum likelihood estimating equations above are not in a simple form. In general, there are no implicit expression for the estimates. The estimates can be obtained through some numerical procedures such as Newton-Raphson method. Many programs provide routines to solve such maximum likelihood estimating equations.
In this paper, all the computations are performed using software R. The MLE estimators are computed by the optim function which uses L-BFGS-B method. In the following section, a simulation is conducted to illustrate the behavior of the MLE.
For asymptotic inference of, the Fisher information matrix plays a key role. It is well known that its inverse is the asymptotic variance matrix of the maximum likelihood estimators. For the case of a single observation, we take the second order derivatives of the log-likelihood function in (14) and the Fisher information matrix is defined as
for and.
Proposition 3.1 Let is a random sample of size from the distribution, where and is the maximum likelihood estimator of, we have
Proof. It follows directly by the large sample theory for maximum likelihood estimators and the Fisher information matrix given above.
4. Simulation Study4.1. Data Generation
In this section, we present how to generate the random numbers from the epsilon half normal slash distribution.
Proposition 4.1 Let be two independent random variables, where and is such that
then, where and .
Proof. For,
The density function of is
which proves the result.
Using the definition in (3) and the results in Proposition 4.1, we can generate variates from the epsilon half normal slash distribution with the following algorithm.
Algorithm 4.2 Using the definition in (3) to generate data
• Generate, and
•
• Let if. Otherwise
• Set
• Set
4.2. Behavior of MLE
In this section, we perform a simulation study to illus trate the behavior of the MLE estimators for parameters, and.
It is known that as the sample size increases, the distribution of the MLE tends to the normal distribution with mean and covariance matrix equal to the inverse of the Fisher information matrix. However, the log-likelihood function given in (14) is a complex expression. It is not generally possible to derive the Fisher information matrix. Thus, the theoretical properties (asymptotically normal, unbiased etc) of the MLE estimators are not easily derived. We study the properties of the estimators numerically.
We first generate 500 samples of size and from the distribution for fixed parameters. The estimators are computed by the optim function which uses L-BFGS-B method in software R. The empirical means and standard deviations(SD) of the estimators are presented in Table 1.
It can be seen from Table 1 that the parameters are well estimated and the estimates are asymptotically un-
Table 1. Empirical means and SD for the MLE estimators of, and.
Table 2. Summary for the life of fatigue fracture.
Table 3. Maximum likelihood parameter estimates(with (SD)) of the HN, EHN and EHNS models for the stressrupture data.
biased. The empirical mean square errors decrease as sample size increases as expected.
5. Real Data Illustration
In this section, we consider the stress-rupture data set, the life of fatigue fracture of Kevlar 49/epoxy that are subject to the pressure at the 90% level. The data set has been previously studied in [1,3,13,14].
Table 2 summarizes descriptive statistics of the data set where and are sample asymmetry and kurtosis coefficients, respectively. This data set indicates non negative asymmetry.
We fit the data set with the half normal, the epsilon half normal and the epsilon half normal slash distributions using maximum likelihood method. The results are reported in Table 3. The usual Akaike information criterion (AIC) and Bayesian information criterion (BIC) to measure of the goodness of fit are also computed. and. where is the number of parameters in the distribution and is the maximized value of the likelihood function. The results indicate that model fits best. Figures 3(a) and (b) display the fitted models using the MLE estimates.
6. Concluding Remarks
In this article, we have studied the epsilon half normal slash distribution. It is defined to be the quotient of two independent random variables, an epsilon half normal random variable and a power of the uniform distribution. This nonnegative distribution extends the
epsilon half normal, the half normal distribution etc. Probabilistic and inferential properties are derived. A simulation is conducted and demonstrates the good performance of the maximum likelihood estimators. We apply the model to a real dataset and the results demonstrate that the proposed model is very useful and flexible for non negative data.
7. Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their valuable comments and suggestions to improve the quality of the paper.
REFERENCESNOTESReferencesL. Castro, H. Gómez and M. Valenzuela, “Epsilon Half-Normal Model: Properties and Inference,” Computational Statistics & Data Analysis, Vol. 56, No. 12, 2012, pp. 4338-4347. Hdoi:10.1016/j.csda.2012.03.020A. Pewsey, “Improved Likelihood Based Inference for the General Half-Normal Distribution,” Communications in Statistics—Theory and Methods, Vol. 33, No. 2, 2004, pp. 197-204. Hdoi:10.1081/STA-120028370N. Olmos, H. Varela, H. Gómez and H. Bolfarine, “An Extension of the Half-Normal Distribution,” Statistical Papers, Vol. 53, No. 4, 2011, pp. 1-12.M. Wiper, F. Girón and A. Pewsey, “Objective Bayesian Inference for the Half-Normal and Half-t Distributions,” Communications in Statistics—Theory and Methods, Vol. 37, No. 20, 2008, pp. 3165-3185.
Hdoi:10.1080/03610920802105184W. Rogers and J. Tukey, “Understanding Some Long- Tailed Symmetrical Distributions,” Statistica Neerlandica, Vol. 26, No. 3, 1972, pp. 211-226.
Hdoi:10.1111/j.1467-9574.1972.tb00191.xN. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” Wiely, Hoboken, 1995.F. Mosteller and J. Tukey, “Data Analysis and Regression: A Second Course in Statistics,” Addison-Wesley Pub. Co., Boston, 1977.K. Kafadar, “A Biweight Approach to the One-Sample Problem,” Journal of the American Statistical Association, Vol. 77, No. 378, 1982, pp. 416-424.
Hdoi:10.1080/01621459.1982.10477827H. Gómez, F. Quintana and F. Torres, “A New Family of Slash-Distributions with Elliptical Contours,” Statistics & Probability Letters, Vol. 77, No. 7, 2007, pp. 717-725.
Hdoi:10.1016/j.spl.2006.11.006A. Genc, “A Generalization of the Univariate Slash by a Scale-Mixtured Exponential Power Distribution,” Communications in Statistics—Simulation and Computation, Vol. 36, No. 5, 2007, pp. 937-947.
Hdoi:10.1080/03610910701539161J. Wang and M. Genton, “The Multivariate Skew-Slash Distribution,” Journal of Statistical Planning and Inference, Vol. 136, No. 1, 2006, pp. 209-220.
Hdoi:10.1016/j.jspi.2004.06.023A. Azzalini, “A Class of Distributions Which Includes the Normal Ones,” Scandinavian Journal of Statistics, Vol. 12, No. 2, 1985, pp. 171-178.D. Andrews and A. Herzberg, “Data: A Collection of Problems from Many Fields for the Student and Research Worker,” Vol. 18, Springer-Verlag, New York, 1985.R. Barlow, R. Toland and T. Freeman, “A Bayesian Analysis of the Stress-Rupture Life of Kevlar/Epoxy Spherical Pressure Vessels,” In: C. Clarotti and D. Lindley, Eds., Accelerated Life Testing and Experts Opinions in Reliability, Elsevier Science Ltd., Amsterdam, 1988, pp. 203-236.
has an epsilon half normal distribution with parameters 
and
, denoted by
, if its density function is given by, for
,
denotes the standard normal density function. When
, the epsilon half normal distribution reduces to the half normal distribution investigated in [2-4].
has a standard slash distribution 
with parameter
, introduced in [
can be represented as
and 
are independent. It generalizes normality and has been much studied in the statistical literature.
, 
yields the standard normal distribution
. Let
, the canonical slash distribution follows, see [
by an elliptical distribution and defined a new family of slash distributions. They studied its general properties of the resulting families, including their moments. Genc [
by a skew normal distribution studied in [
is an extension of the half normal distribution, it is naturally to define a slash distribution based on it in which skewness and thick tailed situations may exist. It leads to a new model on nonnegative measurements with more flexible asymmetry and kurtosis parameters.
has an epsilon half normal slash distribution if it can be represented as the ratio
defined in (1) and
are independent, 
, 
,
. We denote it as
.
. Then, the density function of 
is given by, for
,
, 
,
.
and 
is given by, for
,
the joint probability density function of 
and 
is given by, for
,
is given by
, the dendity function will be obtained as stated.
, the density function (4) reduces to 
,
, the canonical case follows.
is given as follows. For
,
is the cumulative distribution function for the standard normal random variable.
and
, then
.
.
, then the moment generating function of 
is given by for
,
and
. Proof. See [
, then the moment generating function of 
is given by, for
,
, 
and
.
and the result follows.
, then the 
non-central moments are given by
. where 
and
.
, where
, 
and
. For 
and
, the 
non-central moment of 
is given by
, where
, 
and
. The mean and variance of 
are given by
,
we have the following results.
, where
, 
and
. The skewness and kurtosis coefficients of 
are given by
, the skewness coefficient converges.
.
model. The skewness and kurtosis coefficients decrease as 
increases. The parameter 
does not affect the two coefficents.
model defined in (4). For
,
.
is a random sample of size 
from the epsilon half normal slash distribution
. Then the log-likelihood function can be written as
respectively and equalizing the obtained expressions to zero, we obtain the following maximum likelihood estimating equations.
, the Fisher information matrix 
plays a key role. It is well known that its inverse is the asymptotic variance matrix of the maximum likelihood estimators. For the case of a single observation
, we take the second order derivatives of the log-likelihood function in (14) and the Fisher information matrix is defined as
and
.
is a random sample of size 
from the distribution
, where 
and 
is the maximum likelihood estimator of
, we have
.
be two independent random variables, where 
and 
is such that
, where 
and 
.
,
is
with the following algorithm.
, 
and
if
. Otherwise 
, 
and
.
and covariance matrix equal to the inverse of the Fisher information matrix. However, the log-likelihood function given in (14) is a complex expression. It is not generally possible to derive the Fisher information matrix. Thus, the theoretical properties (asymptotically normal, unbiased etc) of the MLE estimators are not easily derived. We study the properties of the estimators numerically.
and 
from the 
distribution for fixed parameters. The estimators are computed by the optim function which uses L-BFGS-B method in software R. The empirical means and standard deviations(SD) of the estimators are presented in 
, 
and
.
and 
are sample asymmetry and kurtosis coefficients, respectively. This data set indicates non negative asymmetry.
and
. where 
is the number of parameters in the distribution and 
is the maximized value of the likelihood function. The results indicate that 
model fits best. Figures 3(a) and (b) display the fitted models using the MLE estimates.
. It is defined to be the quotient of two independent random variables, an epsilon half normal random variable and a power of the uniform distribution. This nonnegative distribution extends the