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 .
105
ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS
Corollary 1 Proof When α <1, and for certain values of If α <1, then Lemma 2 implies that β , the gamma-half normal distribution becomes x = 0 is a modal point. When α <1 and β ≤1, bimodal. It is difficult to find analytically the it follows from (2.3) that x < 0 , and hence region where the distribution is bimodal. equation (2.3) has no solution, thus, x = 0 is a However, a numerical solution is obtained to unique modal point. The proof is complete by determine the number of roots of the derivative noting that when α =1 and β ≤1, the PDF of of the gamma-half normal distribution. Figure 4 shows the boundary region of α and β where gamma-half normal in (1.4) is a strictly decreasing function. the gamma-half normal distribution is bimodal. Figures 1-3 show various graphs of gx() and hx(). These figures indicate that the Lemma 3 If Q(),0λλ<< 1 denotes the quantile gamma-half normal PDF may take on a variety of shapes for different values of α,and.β θ function for the gamma-half normal distribution, then The shapes range from reversed-J shape, bimodal, right-skewed and approximately λθ=− Φ−−11 − βγαλα Γ symmetric. As β decreases, the right tail of the Q( )() 0.5exp{ ( , ( ))} . gamma-half normal distribution becomes longer. (2.4) Bimodality appears when α is less than 1. Figure 3 indicates that the gamma-half normal hazard function is either a bathtub shape or increasing failure rate shape.
Figure 1: The Gamma-Half Normal PDF for Various Values of α, β and θ
alpha =1, beta =1, theta=1 alpha =0.2, beta =1, theta=0.95 alpha =0.8, beta =3, theta=0.9 alpha =0.9, beta =4, theta=0.45 alpha=1, beta=3, theta=1
g(x)
0.0 0.2 0.4 0.6 0.8 1.0
012345 x
106
AYMAN ALZAATREH & KRISTEN KNIGHT
Figure 2: The Gamma-Half Normal PDF for Various Values of α, β and θ
alpha =2, beta =1, theta=2 alpha =2, beta =0.5, theta=1.55 alpha =2, beta =2, theta=0.9 alpha =5, beta =2, theta=1.2 aplha=2, beta=6, theta=1
g(x)
0.0 0.2 0.4 0.6 0.8 1.0
02468 x
Figure 3: The Gamma-Half Normal Hazard Function for Various Values of α, β and θ
alpha=0.5, beta=0.5, theta=2.2 alpha=0.7, beta=2, theta=1 alpha=0.9, beta=1.7, theta=1.2 alpha=4, beta=0.9, theta=1 alpha=8, beta=1, theta=1
g(x)
0.0 0.5 1.0 1.5 2.0 0123456 x
107
ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS
Lemma 3 Proof where ψ is the digamma function and T is the The proof follows by taking the inverse gamma random variable with parameters α and function of (1.5). β . The entropy of a random variable X is a measure of variation of uncertainty (Rényi, Theorem 3 1961). Shannon’s entropy (Shannon, 1948), for a The Shannon entropy for the gamma- random variable X with PDF g(x) is defined as half normal distribution is given by − EgX{}log() ( ) . Since 1948 many applications have been used with Shannon’s ηπθ=− + + X log 2 log log entropy in different areas, including engineering, 1 physics, biology, economics and information +++−+σμ22 α β β 2 ( ) (1 ) log theory. 2θ According to Alzaatreh, et al. (2013b), the +Γ+− log (ααψα ) (1 ) ( ), Shannon entropy of the gamma-X family of (2.6) distributions is given by
where μ and σ are the mean and variance of ηαβ=−−−1 −T + − X EfFe{}log()() 1 (1 ) the gamma-half normal, respectively. , ++Γ+−βααψα log log ( ) (1 ) ( ) (2.5)
Figure 4: Bimodal Region for the Gamma-Half Normal Density Function where θ = 1
108
AYMAN ALZAATREH & KRISTEN KNIGHT
Theorem 3 Proof Table 1 provides the mode, mean, First it is necessary to find median and variance of the gamma-half normal −− α β −−EfF{log()1 () 1 0.5 eT } , where f ()x distribution for various values of and when θ =1. Equations (2.2) and (3.2) are used and Fx() are the PDF and CDF of the half- for these calculations. For fixed α and θ , the normal distribution, respectively. From the CDF mode, mean, median and variance are increasing of the half-normal distribution it follows that functions of β . Also, for fixed β and θ , the −−x +1 Fx11()=Φθ ( ) and hence, mode, mean, median and variance are increasing 2 functions of α. Figure 5 displays the skewness and kurtosis graphs of the gamma-half normal −−EfFe{log()−−1 () 1 T } distribution for different values of α and β with θ =1. For fixed α , the skewness and =− log 2 + logπθ + log , kurtosis are decreasing functions of β ; for fixed
1 −− β +Φ− ( )Ee (θ 12 (1 0.5T )) , the skewness and kurtosis are decreasing 2θ 2 functions of α . where T follows the gamma distribution. By Lemma 5 −− Lemma 1, θ Φ−1(1 0.5e T ) follows the If the median is denoted by M, then the mean deviation from the mean, D()μ , and the gamma-half normal with parameters α , β and −− mean deviation from the median, D()M , for θ . Hence Ee((10.5))θσΦ−122T =+μ the gamma-X distribution are given by 2 where σ and μ are the variance and the mean −1 for the gamma-half normal distribution. The DI(μμγαβ )=Φ−Γ− 2 { , log(2 ( μ )}/ ( α ) 2 μ result in (2.5) follows from equation (2.4).
and Moments and Mean Deviations DM()=−μ 2 I , (3.3) The r th moments for the gamma-half M normal distribution in (1.4) can be written as 2θ ∞+21k r where Iakim= (,)(γαδ , ( )), EX()= miΓ α ()ki==00
2 − x 1 i −1 +− ∞ −− c + 21(1)k 2 2 xxα − k 21k r 2θ 1 β = π x edx(−Φ log(2 ( ))) (2 Φ ( )) . aki(,) ( /2) α α θ ++β πθΓ() α β θθ 21kii (1) δ =− +β Φ − θ (3.1) and i (mi ) ( 1/ )log(2 ( m / )).
Using the substitution ux=−log(2 Φ ( − /θ )) , Lemma 5 Proof (3.1) reduces to If gx() and Gx() are the PDF and the CDF of the gamma-half normal distribution, − rrθ ∞ then the mean deviations from the mean and the ruru=Φ(1) −−11/αβ −− EX()α ( (0.5)) e u e du . median can be written as Γ αβ 0 () (3.2) μ DG()μμμ=− 2 () 2 xgxdx () (3.4) 0 Because no closed form is found for and (3.2), numerical integration can be used to M th DM()=−μ 2 xgxdx (). (3.5) calculate the r moments. 0
109
ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS
Consider the integral: μ The results follow by substituting Im in D() m Ixgxdx= () and D()M in (3.3). m 0
2 − x −− 2 m 2 xxαβ−− Order Statistics = 2θ −Φ1(1/)1 Φ xe ( log(2 ( ))) (2 ( )) dx th α 0 πθΓ() α β θθ The density function of the r order (3.6) statistic, X rn: , for a random sample of size n drawn from (1.3), is By substituting ux=−log(2 Φ ( − /θ )) and 1 −− because erf (xx / 2θθ )=Φ 2 ( / ) − 1, f() x=− gx ()(())(1 Gxrnr1 Gx ()). −+ equation (3.6) can be written as Brn(, r 1) (4.1) −m −u θ −Φlog(2 ( )) =−2 θ −−−11u α β Using the binomial expansion, (4.1) can be Ieuedum α erf (1 ) Γ()αβ 0 written as (3.7) gx() fx()= Using the series representation for rn: −+ −− Brn(, r 1) erf1 (1− e u ) (see Wolfram website), results in (4.2) nr− − jrjnr +−1 (1)− (Gx ()) . ∞ = j −++c j 0 12121= k π kk erf (x ) ( / 2) x , = 21k + k 0 From (1.7), equation (4.2) can be written as (3.8)
k −1 cc where c = mk−−1 m gx()nr− (1)− j nr− k ++ fx()= m=0 (1)(21)mm rn: −+ Γα rj+−1 Brn(, r 1)j=0 ( ) j and c = 1. 0 rj+−1 ∞ (−− 1)kk ( log(2 Φ− (x /θ )))α + Using (3.8), equation (3.7) reduces to × βαk +α + k =0 kk!( ) ∞ 2θ c π + nr−∞∞ I = k ()21k gx() m α = .... Γ+()αβ = 2k 1 2 k 0 −+ === (3.9) Brn(, r 1)jk00012 k −m −u −Φlog(2 ( )) ∞ sj+ θ −+−uk21α 1β − k − ()1− euedu . 1 (1) nr 0 ++−α +− srjk (1) rj1 = βαP Γ() j krj+−1 0 k −+uk21 − Using the series expansion of (1− e ) , x srj++−(1)α (−Φ log 2 ( )) k . (3.9) reduces to θ
2θ ∞+21k Iaki= (,)(γ αδ , ), miΓ α ()ki==00 where + i c + 21k − = k π 21k (1) aki(,) ( /2) α 21ki++i (1)β δθ=− +β Φ − θ and i (mi / ) ( 1/ )log(2 ( m / )).
110
AYMAN ALZAATREH & KRISTEN KNIGHT
Table 1: Mode, Mean, Median and Variance for Some Values of α and β with θ =1
α β Mode Mean Median Variance
0.5 0 0.2482 0.1351 0.0887
1 0 0.4347 0.2578 0.23895
0.5 4 0 1.1785 0.8370 1.3385
7 0 1.6911 1.2718 2.4929
9 0 1.9761 1.5177 3.2649
0.5 0 0.4255 0.3292 0.1355
1 0 0.7298 0.5969 0.3442
0.9 4 0, 1.0373 1.8914 1.6854 1.7215 7 0, 1.7998 2.6687 2.4242 3.0935
9 0, 2.1818 3.0968 2.8303 3.9973
0.5 0 0.4674 0.3757 0.1449
1 0 0.7979 0.6745 0.3634
1 4 1.3744 2.0485 1.8627 1.7774
7 2.1441 2.8811 2.6601 3.1744
9 2.5481 3.3388 3.0973 4.0932 0.5 1.3439 1.4376 1.4069 0.2516
1 2.1644 2.2701 2.2349 0.5257
4 4 4.9410 5.1202 5.0607 2.0628 7 6.7013 6.9392 6.8604 3.5516
9 7.6885 7.9315 7.8419 4.5361
0.5 2.0642 2.1197 2.1012 0.2677
1 3.1775 3.2457 3.2229 0.5355
7 4 6.9108 7.0418 6.9982 2.0578
7 8.0469 9.4615 9.4032 3.5509
9 10.5824 10.7830 10.7164 4.5419
111
ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS
Using (1.4), The result in (4.3) shows that the PDF of the rth order statistics of the gamma half-normal 1 nr−∞∞ distribution can be expressed in terms of infinite fx( )= ... rn: −+ sums of the gamma half-normal PDFs. Using the Brn(, r 1)jkk=0 12 same technique as Ristic and Balakrishnan ∞ Γ++α 1 ((srjk )) (2012), the asymptotic distribution of the sample ++α +− srjk () rj1 βαP Γ() minimum X1:n can be obtained by utilizing krj+−1 k Theorem 8.3.6 in Arnold, et al. (1992), which nr− ×++gx(| s ( r j ),,),αβθ Gx()ε γ k = j states that if lim− x , then the ε →G 1 (0) G()ε (4.3) asymptotic distribution of X will be of 1:n rj−+1 rj−+1 Weibull type with shape parameter γ. = =+α where skki and Pkkkii∏ (). i=1 i=1
Figure 5: The Gamma-Half Normal Skewness and Kurtosis Graphs for Various Values of α and β when θ = 1
beta =0.5 beta =0.5 beta =0.8 beta =0.8 beta =1 beta =1 beta =3 beta =3 beta=7 beta=7
Kurtosis Skewness
0 100 200 300 400
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0123456 0123456
Alpha Alpha
112
AYMAN ALZAATREH & KRISTEN KNIGHT
For the gamma half-normal distribution, ∂ log L − = G 1(0)= 0 and ∂α n −−ψα β + −Φ− θ Gx()εε gx () nn( ) log log( log(2 ( xi / ))), = i=1 lim++x lim εε→→00Gg()εε () (5.2) α −1 Φ−εθ = log(2 (x / )) ∂ α n x lim+ logLn 1 ε →0 Φ−εθ =− −log(2 Φ ( −x /θ )), log(2 ( / )) ∂βββ2 i i=1 α −1 φε(/)− x θ α (5.3) ==x xxlim . + ε →0 φεθ− (/) ∂ log L = (4.4) ∂θ
n n 1 2 Hence, the asymptotic distribution X rn: is of −+ x θθ3 i Weibull type with shape parameter α . The i=1 asymptotic distribution of the sample maximum n xh(/) x θ +− (α 1) iz i X nn: can be viewed as Gxn (), where Φ− θ i=1 log(2 (xi / )) Gx()=− 1 G ( − x ) and G is the CDF of X . n n 1 1 1:n −−1 θ ( 1) xhiz ( x i / ) β = Parameter Estimation i 1 Let a random sample of size n be taken (5.4) from the gamma-half normal distribution. The log-likelihood function for the gamma-half Setting (5.2), (5.3) and (5.4) to zero and solving normal distribution in (1.4) is given by them simultaneously results in αˆ , βˆ and θˆ . The initial values for the parameters α , logL (αβ , ) = β and θ can be obtained by assuming the nn random sample, x ,1,,in= is taken from log 2−−n logθπ log i 22 the half-normal distribution with parameter θ . 1 n By equating the population mean to the sample −Γ−αα β − 2 nn log ( ) log 2 xi = θ 2θ i=1 mean of xi ,1,,in and solving for , the n θπˆ = +−αθ − Φ− initial value /2x . Assuming ( 1) log( log(2 (xi / ))) = =− Φ −θˆ = i 1 yiilog(2 (xi / )), 1, , n are taken 1 n from the gamma distribution with parameters α +− ( 1) log(2 Φ− (x /θ ). β i β i=1 and (see Lemma 1). By equating the (5.1) population mean and the population variance of gamma distribution (with parameters α and The derivatives of (5.1) with respect to α , β β ) to the corresponding sample mean and θ = and respectively, are given by sample variance of yii ,1,, n and solving α β α = 22 for and , the initial values are 0 ys/ y β = 2 2 and 0 syy / , where y and sy are the
sample mean and the sample variance for y1 ,
y2 , …, yn .
113
ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS
α ==β that the gamma-half normal distribution is able When 1, the gamma-half normal distribution reduces to the half-normal to model data of both approximately symmetric distribution; thus, the likelihood ratio test can be and right-skewed shapes. Figures 6 and 7 used to determine whether the gamma-half display the empirical and fitted cumulative normal distribution or the half-normal distribution functions; these figures support the distribution is the best model for fitting a given results in Tables 3 and 5, respectively. data set. The likelihood ratio test can be used for testing the hypothesis H :1α ==β against Table 2: Single Carbon Fibers at 20 mm 0 α ≠≠β Ha :1 or 1, which is based on 0.312 0.314 0.479 0.552 0.700 0.803 0.861 0.865 λθ= LL()/ (, αˆ βˆˆ ,)θ , where L and L are 0 a 0 a 0.944 0.958 0.966 0.997 the likelihood functions for the half-normal and 1.006 1.021 1.027 1.055 the gamma-half normal distributions, 1.063 1.098 1.140 1.179 respectively. The quantity −2logλ follows the 1.224 1.240 1.253 1.270 Chi-square distribution with 2 degrees of 1.272 1.274 1.301 1.301 freedom asymptotically. 1.359 1.382 1.382 1.426 1.434 1.435 1.478 1.490 Application 1.511 1.514 1.535 1.554 Three data sets were applied to the 1.566 1.570 1.586 1.629 gamma-half normal distribution, and compared 1.633 1.642 1.648 1.684 with the half-normal, generalized half-normal, 1.697 1.726 1.770 1.773 beta generalized half-normal and inverse 1.800 1.809 1.818 1.821 Gaussian distributions. The first two data sets 1.848 1.880 1.954 2.012 (see Tables 2 and 4), were analyzed by Raqab, et 2.067 2.084 2.090 2.096 al. (2008). This data represents the tensile 2.128 2.233 2.433 2.585 strength data measured in GPa for single-carbon 2.585 fibers that were tested at gauge lengths of 20 mm and 10 mm. The third data set (see Table 6) was analyzed by Cheng, et al. (1981) and The third data set (see Table 6) was represents the flood level for the Susquehanna analyzed by Cheng, et al. (1981) and fitted to the River at Harrisburg, PA. The maximum inverse Gaussian distribution. These results, as likelihood estimates, KS (Kolmogorov-Smirnov) well as the comparisons made to the half- test-statistics and p-values for the fitted normal, beta generalized half-normal and distributions are reported in Tables 3, 5 and 7. gamma-half normal distributions, are reported in The data in Tables 2 and 4 are fitted to Table 7. The generalized half-normal the gamma-half normal, half-normal, distribution was divergent for the third data set. generalized half-normal and beta generalized- In view of these results, the gamma half-normal half normal distributions. The half-normal and inverse Gaussian distributions give a distribution did not produce an adequate model moderate fit to the data. The half-normal for the data. However, the generalized half- distribution does not give an adequate fit to the normal, beta generalized and gamma-half data, while the generalized half-normal provided normal each provide a good fit for the two data the best fit. In viewing the distribution of the sets. Among the three generalizations of the third data set, another right-skewed distribution half-normal distribution, the gamma-half normal is observed. This confirms the fact that the provides the best fit for the first data set, and gamma-half normal distribution can be used to generalized half-normal provides the best fit for fit data of a right-skewed shape. Figure 8 the second. When graphing the first data set, an displays the empirical and fitted cumulative approximately symmetric distribution is distribution functions. obtained. The distribution of the second data set, however, is a right-skewed shape. This suggests
114
AYMAN ALZAATREH & KRISTEN KNIGHT
Table 3: Parameter Estimates for Single Carbon Fibers at 20 mm
Beta Generalized Gamma-Half Distribution Half-Normal Generalized Half-Normal Normal Half-Normal
aˆ =1.3742 aˆ = 2.2823 aˆ = 2.8794 ˆ Parameter b = 0.2369 ˆ θˆ = 1.5323 qˆ= 1.5879 b = 3.1725 Estimates αˆ =1.8610 μˆ = 0.1220 θˆ = 0.3934 θˆ = 0.9766
KS 0.3317 0.0548 0.0863 0.0425
P-value 0.0000 0.9857 0.9985 0.9996
Conclusion References The gamma-half normal distribution, a new Alzaatreh, A., Famoye, F., & Lee, C. generalization of the half-normal distribution, (2013a).Weibull-Pareto distribution and its was derived using the method proposed by applications. Communications in Statistics: Alzaatreh, et al. (2013b). Various properties of Theory & Methods, 42(9), 1673-1691. the distribution were studied including the Alzaatreh, A., Lee, C., & Famoye, F. moments, mean deviations from the mean and (2013b). A new method for generating families median, hazard function, modality and Shannon of continuous distributions. To Appear Metron: entropy. The maximum likelihood method was International Journal of Statistics. proposed for the estimation of the gamma-half Alzaatreh, A., Famoye, F., & Lee, C. normal parameters. In order to demonstrate the (2012a). Gamma-Pareto distribution and its applicability of the gamma-half normal applications. Journal of Modern Applied distribution it was fitted to three real data sets Statistical Methods, 11(1), 78-94. and compared with the half-normal, generalized- Alzaatreh, A., Lee, C., & Famoye, F. half normal, inverse Gaussian and beta (2012b). On the discrete analogues of generalized half-normal distributions. continuous distributions. Statistical Results show that the gamma-half Methodology, 9, 589-603. normal distribution provides an adequate fit for Arnold, B. C., Balakrish, N., & each data set. Because the distribution was fitted Nagaraja, H. N. (1992). A first course in order to data sets with right-skewed and statistics. New York, NY: Wiley. approximately symmetric shapes, this indicates Bland, J. M., & Altman, D. G. (1999). that the gamma-half normal distribution offers Measuring agreement in method comparison flexibility that extends beyond the half-normal studies. Statistical Methods in Medical distribution. Although the gamma-half normal Research, 8, 135-160. distribution can be bimodal, it was difficult to Bland, J. M. (2005). The half-normal find data in the literature with the specific form distribution method for measurement error: two of bimodality. The maximum likelihood case studies. Unpublished lecture available on functions may be further studied under different http://wwwusers.york.ac.uk/~mb55/talks/halfnor types of censoring for future applications of the .pdf. gamma-half normal distribution.
115
ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS
Castro, L., Gomez, H., & Valenzuela, Pescrim, R., et al. (2010). The beta M. (2012). Epsilon half-normal model: generalized half- normal distribution. properties and inference. Computational Computational Statistics and Data Analysis, 54, Statistics & Data Analysis, 56(12), 4338-4347. 945-957. Cheng, R. C. H., & Amin, N. A. K., Raqab, M., Madi, M., & Debasis, K. (1981). Maximum likelihood estimation of (2008). Estimation of P(Y
116
AYMAN ALZAATREH & KRISTEN KNIGHT
Figure 6: CDF for Fitted Distributions for Gauge Length of 20mm Data
Empirical Half Normal Generalized Half Normal Beta Generalized Half-Normal Gamma-Half Normal
CDF
0.0 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 2.5 Strength data measured in GPA
Table 4: Single Carbon Fibers at 10 mm
0.101 0.332 0.403 0.428 0.457 0.550 0.561 0.596 0.597 0.645 0.654 0.674 0.718 0.722 0.725 0.732 0.775 0.814 0.816 0.818 0.824 0.859 0.875 0.938 0.940 1.056 1.117 1.128 1.137 1.137 1.177 1.196 1.230 1.325 1.339 1.345 1.420 1.423 1.435 1.443 1.464 1.472 1.494 1.532 1.546 1.577 1.608 1.635 1.693 1.701 1.737 1.754 1.762 1.828 2.052 2.071 2.086 2.171 2.224 2.227 2.425 2.595 3.220
Table 5: Parameter Estimates for Single Carbon Fibers at 10mm
Beta Generalized Gamma-Half Distribution Half-Normal Generalized Half-Normal Normal Half-Normal aˆ =1.9544 aˆ = 1.5347 aˆ = 2.4260 Parameter bˆ = 0.1522 ˆ θˆ = 1.4019 qˆ= 1.4798 b = 0.4216 Estimates αˆ =1.0954 μˆ = 0.0699 θˆ = 1.4529 θˆ = 0.4722
KS 0.2099 0.0606 0.0863 0.0678
P-value 0.0078 0.9748 0.7361 0.9342
117
ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS
Figure 7: CDF for Fitted Distributions for Gauge Lengths of 10mm Data
CDF
Empirical Half Normal Generalized Half Normal Beta Generalized Half-Normal Gamma-Half Normal
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Strength data measured in GPA
118
AYMAN ALZAATREH & KRISTEN KNIGHT
Table 6: Maximum Flood Level for Susquehanna River .654 .613 .402 .379
.269 .740 .416 .338 .315 .449 .297 .423 .379 .324 .418 .412 .494 .392 .484 .265
Table 7: Parameter estimates for maximum flood levels
Beta Generalized Gamma-Half Distribution Half-Normal Generalized Half-Normal Normal Half-Normal
aˆ = 244.48 aˆ = 0.1780 Parameter bˆ = 176.11 ˆ ˆ= 0.4404 Divergent b =0.245 Estimates q αˆ = 0.1901 λˆ = 0.9140 θˆ = 1.2483
KS 0.4026 0.1507 0.1026
P-value 0.0031 0.7540 0.5638
Figure 8: CDF for Fitted Distributions for Maximum Flood Level
CDF
Empirical Half Normal Beta Generalized Half-Normal Inverse Gaussian Gamma-Half Normal 0.0 0.2 0.4 0.6 0.8 1.0 0.3 0.4 0.5 0.6 0.7 Maximum flood level
119