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On the Gamma-Half Normal Distribution and Its Applications Journal of Modern Applied Statistical Methods Volume 12 | Issue 1 Article 15 5-1-2013 On The aG mma-Half Normal Distribution and Its Applications Ayman Alzaatreh Austin Peay State University, Clarksville, TN Kristen Knight Austin Peay State University, Clarksville, TN Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Alzaatreh, Ayman and Knight, Kristen (2013) "On The aG mma-Half Normal Distribution and Its Applications," Journal of Modern Applied Statistical Methods: Vol. 12 : Iss. 1 , Article 15. DOI: 10.22237/jmasm/1367381640 Available at: http://digitalcommons.wayne.edu/jmasm/vol12/iss1/15 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright © 2013 JMASM, Inc. May 2013, Vol. 12, No. 1, 103-119 1538 – 9472/13/$95.00 On The Gamma-Half Normal Distribution and Its Applications Ayman Alzaatreh Kristen Knight Austin Peay State University, Clarksville, TN A new distribution, the gamma-half normal distribution, is proposed and studied. Various structural properties of the gamma-half normal distribution are derived. The shape of the distribution may be unimodal or bimodal. Results for moments, limit behavior, mean deviations and Shannon entropy are provided. To estimate the model parameters, the method of maximum likelihood estimation is proposed. Three real-life data sets are used to illustrate the applicability of the gamma-half normal distribution. Key words: T-X families; gamma-X family; unimodal; bimodal; Shannon entropy. Introduction in the literature (e.g., Alzaatreh, et al. (2013a); In recent years, advancements in technology and Alzaatreh, et al. (2013b); Alzaatreh, et al. science have resulted in a wealth of information, (2012a); Alzaatreh, et al. (2012b); Lee, et al. which is expanding the level of knowledge (2013)). across many disciplines. This information is One well-known distribution is the half- gathered and analyzed by statisticians, who hold normal distribution, which has been used in the responsibility of accurately assessing the variety of applications. Previous work by Bland data and making inferences about the population and Altman (1999) used the half-normal of interest. Without this precise evaluation of distribution to study the relationship between data, each field remains limited to its current measurement error and magnitude. Bland (2005) state of knowledge. In the last decade, it has extended the work of Bland and Altman (1999) been discovered that many well-known by using the distribution to estimate the standard distributions used to model data sets do not offer deviation as a function so that measurement enough flexibility to provide an adequate fit. For error could be controlled. In his work, various this reason, new methods are being proposed exercise tests were analyzed and it was and used to derive generalizations of well- determined that variability of performance does known distributions. With these distributions, decline with practice (Bland, 2005). strong applications have been made to real-life Manufacturing industries have utilized the half- scenarios. normal distribution to model lifetime processes Alzaatreh, et al. (2013b) proposed the T- under fatigue. These industries often produce X families of distributions. These families of goods with a long lifetime need for customers, distributions were used to generate a new class making the cost of the resources needed to of distributions which offer more flexibility in analyze the product failure times very high. To modeling a variety of data sets. Several save time and money the half normal members of the T-X families have been studied distribution is used in this reliability analysis to study the probabilistic aspects of the product failure times (Castro, et al., 2012). Due to the fact that the half-normal Ayman Alzaatreh is an Assistant Professor in the distribution has only one shape, various Department of Mathematics & Statistics. Email generalizations of the distribution have been him at: [email protected]. Kristen Knight is derived. These generalizations include the a student in the Department of Mathematics and generalized half-normal distribution (Cooray, et Statistics. Email her at: al., 2008), the beta-generalized half-normal [email protected]. (Pescrim, et al., 2010) and the Kumaraswamy 103 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS generalized half-normal (Cordeiro, et al., 2012). (2009). When α =1 and 1/β =∈n , the Several of the corresponding applications gamma-X family reduces to the distribution of include the stress-rupture life of kevlar 49/epoxy the first order statistics of the random variable strands placed under sustained pressure (Cooray, X . et al., 2008), failure times of mechanical If X is the half normal random variable components and flood data (Cordeiro, et al., with the density function 2012). In this article the gamma and half normal 2 − 22θ distributions are combined to propose a new fx()=> ex /2 , x 0, then (1.3) gives generalization of the half-normal distribution, θπ namely, the gamma half-normal distribution. Let Fx() be the cumulative distribution gx()= 2 function (CDF) of any random variable X and − x 1 −−−1 2 2 xxα − rt() be the probability density function (PDF) e 2θ (−Φ log(2 ( )))1 (2 Φ ( ))β , α of a random variable T defined on [0,∞ ) . The πθΓ() α β θθ CDF of the T-X family of distributions defined αβθ,,>> 0;0x by Alzaatreh, et al. (2013b) is given by (1.4) −−log() 1Fx ( ) Φ Gx()==−− rtdt () R{} log(1 Fx () where is the CDF of the standard normal 0 distribution. (1.1) A random variable X with the PDF g(x) in (1.4) is said to follow the gamma-half normal When X is a continuous random variable, the distribution with parameters α , β and θ . From probability density function of the T-X family is (1.1), the CDF of the gamma- half normal distribution is obtained as fx() gx()=−− r() log1() Fx () − −1 1()Fx Gx()=−γ {,α β log(2( Φ−Γ x /θα ))}/( ), = hxr()() Hx (), (1.5) (1.2) t α −− where γα(,)tuedu= 1 u is the incomplete 0 where hx() and H ()x are the hazard and the gamma function. cumulative hazard functions of the random A series representation of Gx() in (1.5) variable X associated with f ()x . can be obtained by using the series expansion of If a random variable T follows the the incomplete gamma function from Nadarajah α β gamma distribution with parameters and , and Pal (2008) as − ααβ1 −− rt()=Γ()βα ( ) t1/ et , t ≥ 0, then the ∞ (1)− kkxα + γα(,)x = . (1.6) definition in (1.2) leads to the gamma-X family α + with the PDF k =0 kk!( ) gx()= From (1.6), the CDF of the gamma half-normal distribution can be written as 1 1 α −1 −1 −− − β α fx()() log1() Fx ()() 1 Fx () . Γ()αβ 1∞ (−− 1)kk [ log(2 Φ− (x /θ ))]α + Gx()= . (1.3) Γ+ααβ α +k ()k=0 kk !( ) (1.7) β = When 1, the gamma-X family in (1.3) reduces to the gamma-generated distribution The hazard function associated with the gamma- introduced by Zografos and Balakrishnan half normal distribution is 104 AYMAN ALZAATREH & KRISTEN KNIGHT gx() Lemma 2 Proof hx()= 1()− Gx Since the random variable X is defined on (0,∞ ) , this implies limgx ( )= 0 . Using 1 →∞ −1 x −−22θα 2exxx /2 (−Φ− log(2 ( /θθ ))) 1 (2 Φ− ( / )) β L’Hôpital’s rule it can be shown that = , πθβα Γ− α γ α − β−1 Φ− θ limhx ( ) =∞. Now, hx()[1−= Gx ()] gx () [() {, log(2(x / ))}] x→∞ x > 0. = implies that lim++gx ( ) lim hx ( ). Results in (1.8) xx→→00 (2.1) follow immediately from definition (1.4). Some Properties of the Gamma-Half Normal The modes of the gamma-half normal Distribution distribution can be obtained by taking the Lemma 1 gives the relation between the derivative of gx(). The derivative with respect gamma-half normal distribution and the gamma to x of (1.4) can be simplified to distribution. gx'( ) = 2 Lemma 1 (Transformation) − x 1 −−−2 If a random variable Y follows the 2 θ 2 xxα −2 β ekx2 (−Φ log(2 ( ))) [2 Φ ( )] ( ), α gamma distribution with parameters α and β , πθΓ() α β θθ then the random variable (2.2) −− X =Φθ 1(1 − 0.5e Y ) follows the gamma-half normal distribution with parameters α , β and where θ . kx()= x −xx Lemma 1 (Transformation) Proof log(2Φ+− ( )) (α 1)h ( ) The results follow by using the θθz θ transformation technique. − +−β −1 xx Φ The limiting behaviors of the gamma- ( 1)hz ( )log(2 ( ) half normal PDF and the hazard function are θθ given in Lemma 2. Setting (2.2) to 0, the critical values of Lemma 2 gx() are x = 0 and the solution of the equation The limit of the gamma-half normal kx()= 0. The solution of kx()= 0 is →∞ density function as x is 0 and the limit of equivalent to the equation the gamma-half normal hazard function as x →∞ is ∞. Also, the limit of the gamma-half + 11α − normal and hazard function as x → 0 is given xhx=−−θθ(/)1 , z βθΦ− by log(2 (x / )) 0,α > 1 (2.3) 2 θ =−Φφ θθ limgx ( )== lim hx ( ) ,α = 1 where hxz (/) (/)/(1(/)). x x →→++ xx00 πθβ Corollary 1 α ≤ β ≤ ∞<,1.α If 1 and 1, the gamma-half (2.1) normal distribution is unimodal and the mode is at x = 0 .
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