On The Gamma-Half Normal Distribution and Its Applications

Transcription

1 Joural of Moder Applied Statistical Methods Volume Issue Article O The Gamma-Half Normal Distributio ad Its Applicatios Ayma Alzaatreh Austi Peay State Uiversity, Clarksville, TN Kriste Kight Austi Peay State Uiversity, Clarksville, TN Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Alzaatreh, Ayma ad Kight, Kriste (3) "O The Gamma-Half Normal Distributio ad Its Applicatios," Joural of Moder Applied Statistical Methods: Vol. : Iss., Article 5. DOI:.37/jmasm/ Available at: This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.

2 Joural of Moder Applied Statistical Methods Copyright 3 JMASM, Ic. May 3, Vol., No., /3/$95. O The Gamma-Half Normal Distributio ad Its Applicatios Ayma Alzaatreh Kriste Kight Austi Peay State Uiversity, Clarksville, TN A ew distributio, the gamma-half ormal distributio, is proposed ad studied. Various structural properties of the gamma-half ormal distributio are derived. The shape of the distributio may be uimodal or bimodal. Results for momets, limit behavior, mea deviatios ad Shao etropy are provided. To estimate the model parameters, the method of maximum likelihood estimatio is proposed. Three real-life data sets are used to illustrate the applicability of the gamma-half ormal distributio. Key words: T-X families; gamma-x family; uimodal; bimodal; Shao etropy. Itroductio I recet years, advacemets i techology ad sciece have resulted i a wealth of iformatio, which is expadig the level of kowledge across may disciplies. This iformatio is gathered ad aalyzed by statisticias, who hold the resposibility of accurately assessig the data ad makig ifereces about the populatio of iterest. Without this precise evaluatio of data, each field remais limited to its curret state of kowledge. I the last decade, it has bee discovered that may well-kow distributios used to model data sets do ot offer eough flexibility to provide a adequate fit. For this reaso, ew methods are beig proposed ad used to derive geeralizatios of wellkow distributios. With these distributios, strog applicatios have bee made to real-life scearios. Alzaatreh, et al. (3b) proposed the T- X families of distributios. These families of distributios were used to geerate a ew class of distributios which offer more flexibility i modelig a variety of data sets. Several members of the T-X families have bee studied Ayma Alzaatreh is a Assistat Professor i the Departmet of Mathematics & Statistics. him at: alzaatreha@apsu.edu. Kriste Kight is a studet i the Departmet of Mathematics ad Statistics. her at: kkight3@my.apsu.edu. i the literature (e.g., Alzaatreh, et al. (3a); Alzaatreh, et al. (3b); Alzaatreh, et al. (a); Alzaatreh, et al. (b); Lee, et al. (3)). Oe well-kow distributio is the halformal distributio, which has bee used i variety of applicatios. Previous work by Blad ad Altma (999) used the half-ormal distributio to study the relatioship betwee measuremet error ad magitude. Blad (5) exteded the work of Blad ad Altma (999) by usig the distributio to estimate the stadard deviatio as a fuctio so that measuremet error could be cotrolled. I his work, various exercise tests were aalyzed ad it was determied that variability of performace does declie with practice (Blad, 5). Maufacturig idustries have utilized the halformal distributio to model lifetime processes uder fatigue. These idustries ofte produce goods with a log lifetime eed for customers, makig the cost of the resources eeded to aalyze the product failure times very high. To save time ad moey the half ormal distributio is used i this reliability aalysis to study the probabilistic aspects of the product failure times (Castro, et al., ). Due to the fact that the half-ormal distributio has oly oe shape, various geeralizatios of the distributio have bee derived. These geeralizatios iclude the geeralized half-ormal distributio (Cooray, et al., 8), the beta-geeralized half-ormal (Pescrim, et al., ) ad the Kumaraswamy 3

3 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS geeralized half-ormal (Cordeiro, et al., ). Several of the correspodig applicatios iclude the stress-rupture life of kevlar 49/epoxy strads placed uder sustaied pressure (Cooray, et al., 8), failure times of mechaical compoets ad flood data (Cordeiro, et al., ). I this article the gamma ad half ormal distributios are combied to propose a ew geeralizatio of the half-ormal distributio, amely, the gamma half-ormal distributio. Let Fx ( ) be the cumulative distributio fuctio (CDF) of ay radom variable X ad rt () be the probability desity fuctio (PDF) of a radom variable T defied o [, ). The CDF of the T-X family of distributios defied by Alzaatreh, et al. (3b) is give by ( F x ) log ( ) { } Gx ( ) = rtdt ( ) = R log( Fx ( ) (.) Whe X is a cotiuous radom variable, the probability desity fuctio of the T-X family is f( x) gx ( ) = r log Fx ( ) Fx ( ) = hxr ( ) Hx ( ), ( ( )) ( ) (.) where hx () ad H ( x ) are the hazard ad the cumulative hazard fuctios of the radom variable X associated with f ( x ). If a radom variable T follows the gamma distributio with parameters ad, ( ) t/ rt () = Γ( ) t e, t, the the defiitio i (.) leads to the gamma-x family with the PDF gx ( ) = ( ( )) ( ) f( x) log F( x) F( x). Γ( ) (.3) Whe =, the gamma-x family i (.3) reduces to the gamma-geerated distributio itroduced by Zografos ad Balakrisha (9). Whe = ad / =, the gamma-x family reduces to the distributio of the first order statistics of the radom variable X. If X is the half ormal radom variable with the desity fuctio x / f( x) = e, x>, the (.3) gives gx ( ) = π e πγ( ),, > ; x > x x x ( log( Φ( ))) ( Φ( )), (.4) where Φ is the CDF of the stadard ormal distributio. A radom variable X with the PDF g(x) i (.4) is said to follow the gamma-half ormal distributio with parameters, ad. From (.), the CDF of the gamma- half ormal distributio is obtaied as Gx ( ) = γ{, log( Φ( x/ ))}/ Γ ( ), t (.5) u where γ (, t) = u e du is the icomplete gamma fuctio. A series represetatio of Gx ( ) i (.5) ca be obtaied by usig the series expasio of the icomplete gamma fuctio from Nadarajah ad Pal (8) as k + k ( ) x γ (, x) =. (.6) k = k!( + k) From (.6), the CDF of the gamma half-ormal distributio ca be writte as k + k ( ) [ log( Φ( x / ))] Gx ( ) =. Γ + k ( ) k = k!( + k) (.7) The hazard fuctio associated with the gammahalf ormal distributio is 4

4 AYMAN ALZAATREH & KRISTEN KNIGHT gx ( ) hx ( ) = Gx ( ) x / e ( log( Φ( x/ ))) ( Φ( x/ )) = π Γ γ Φ x x >. (.8) [ ( ) {, log( ( / ))}] Some Properties of the Gamma-Half Normal Distributio Lemma gives the relatio betwee the gamma-half ormal distributio ad the gamma distributio. Lemma (Trasformatio) If a radom variable Y follows the gamma distributio with parameters ad, the the radom variable Y X = Φ (.5 e ) follows the gamma-half ormal distributio with parameters, ad. Lemma (Trasformatio) Proof The results follow by usig the trasformatio techique. The limitig behaviors of the gammahalf ormal PDF ad the hazard fuctio are give i Lemma. Lemma The limit of the gamma-half ormal desity fuctio as x is ad the limit of the gamma-half ormal hazard fuctio as x is. Also, the limit of the gamma-half + ormal ad hazard fuctio as x is give by, > lim gx ( ) = lim hx ( ) =, + + = x x π, <. (.), Lemma Proof Sice the radom variable X is defied o (, ), this implies lim gx ( ) =. Usig x L Hôpital s rule it ca be show that lim hx ( ) =. Now, hx ( )[ Gx ( )] = gx ( ) x implies that lim gx ( ) = lim hx ( ). Results i + + x x (.) follow immediately from defiitio (.4). The modes of the gamma-half ormal distributio ca be obtaied by takig the derivative of gx ( ). The derivative with respect to x of (.4) ca be simplified to g'( x) = e π Γ( ) where x x x ( log( Φ( ))) [ Φ( )] k( x), kx ( ) = x x x log( Φ ( )) + ( ) hz ( ) x x + ( ) hz ( )log( Φ( ) (.) Settig (.) to, the critical values of gx ( ) are x = ad the solutio of the equatio kx= ( ). The solutio of kx= ( ) is equivalet to the equatio x= hz ( x/ ) log( Φ ( x / )), (.3) where hz ( x/ ) = φ( x/ )/( Φ ( x/ )). Corollary If ad, the gamma-half ormal distributio is uimodal ad the mode is at x =. 5

5 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS Corollary Proof If <, the Lemma implies that x = is a modal poit. Whe < ad, it follows from (.3) that x <, ad hece equatio (.3) has o solutio, thus, x = is a uique modal poit. The proof is complete by otig that whe = ad, the PDF of gamma-half ormal i (.4) is a strictly decreasig fuctio. Figures -3 show various graphs of gx ( ) ad hx ( ). These figures idicate that the gamma-half ormal PDF may take o a variety of shapes for differet values of, ad. The shapes rage from reversed-j shape, bimodal, right-skewed ad approximately symmetric. As decreases, the right tail of the gamma-half ormal distributio becomes loger. Bimodality appears whe is less tha. Figure 3 idicates that the gamma-half ormal hazard fuctio is either a bathtub shape or icreasig failure rate shape. Whe <, ad for certai values of, the gamma-half ormal distributio becomes bimodal. It is difficult to fid aalytically the regio where the distributio is bimodal. However, a umerical solutio is obtaied to determie the umber of roots of the derivative of the gamma-half ormal distributio. Figure 4 shows the boudary regio of ad where the gamma-half ormal distributio is bimodal. Lemma 3 If Q( λ), < λ < deotes the quatile fuctio for the gamma-half ormal distributio, the ( ) Q( λ) = Φ.5exp{ γ (, λγ ( ))}. (.4) Figure : The Gamma-Half Normal PDF for Various Values of, ad g(x) alpha =, beta =, theta= alpha =., beta =, theta=.95 alpha =.8, beta =3, theta=.9 alpha =.9, beta =4, theta=.45 alpha=, beta=3, theta= x 6

6 AYMAN ALZAATREH & KRISTEN KNIGHT Figure : The Gamma-Half Normal PDF for Various Values of, ad g(x) alpha =, beta =, theta= alpha =, beta =.5, theta=.55 alpha =, beta =, theta=.9 alpha =5, beta =, theta=. aplha=, beta=6, theta= x Figure 3: The Gamma-Half Normal Hazard Fuctio for Various Values of, ad g(x) alpha=.5, beta=.5, theta=. alpha=.7, beta=, theta= alpha=.9, beta=.7, theta=. alpha=4, beta=.9, theta= alpha=8, beta=, theta= x 7

7 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS Lemma 3 Proof The proof follows by takig the iverse fuctio of (.5). The etropy of a radom variable X is a measure of variatio of ucertaity (Réyi, 96). Shao s etropy (Shao, 948), for a radom variable X with PDF g(x) is defied as E{ log ( g( X) )}. Sice 948 may applicatios have bee used with Shao s etropy i differet areas, icludig egieerig, physics, biology, ecoomics ad iformatio theory. Accordig to Alzaatreh, et al. (3b), the Shao etropy of the gamma-x family of distributios is give by { ( ( ))} T ηx = E log f F e + ( ), + log + log Γ ( ) + ( ) ψ( ) (.5) where ψ is the digamma fuctio ad T is the gamma radom variable with parameters ad. Theorem 3 The Shao etropy for the gammahalf ormal distributio is give by η = log + log π + log X ( ) ( ) log + σ + μ log Γ ( ) + ( ) ψ( ), (.6) where μ ad σ are the mea ad variace of the gamma-half ormal, respectively. Figure 4: Bimodal Regio for the Gamma-Half Normal Desity Fuctio where = 8

8 AYMAN ALZAATREH & KRISTEN KNIGHT Theorem 3 Proof First it is ecessary to fid E{ log f ( F ( ))}.5 e T, where f ( x ) ad Fx ( ) are the PDF ad CDF of the halformal distributio, respectively. From the CDF of the half-ormal distributio it follows that x + F ( x) = Φ ( ) ad hece, { log ( ( ))} T E f F e = log + log π + log, T + ( ) E( Φ (.5 e )) where T follows the gamma distributio. By Lemma, T Φ (.5 e ) follows the gamma-half ormal with parameters, ad T. Hece E( Φ (.5 e )) = σ + μ where σ ad μ are the variace ad the mea for the gamma-half ormal distributio. The result i (.5) follows from equatio (.4). Momets ad Mea Deviatios th The r momets for the gamma-half ormal distributio i (.4) ca be writte as r EX ( ) = π Γ( ) x x x r ( log( Φ( ))) ( Φ( )) dx. x e (3.) Usig the substitutio u = log( Φ( x/ )), (3.) reduces to EX ( ) Γ( ) e u e du (3.) r r r u r u/ ( ) = ( (.5 )). Φ Because o closed form is foud for (3.), umerical itegratio ca be used to th calculate the r momets. Table provides the mode, mea, media ad variace of the gamma-half ormal distributio for various values of ad whe =. Equatios (.) ad (3.) are used for these calculatios. For fixed ad, the mode, mea, media ad variace are icreasig fuctios of. Also, for fixed ad, the mode, mea, media ad variace are icreasig fuctios of. Figure 5 displays the skewess ad kurtosis graphs of the gamma-half ormal distributio for differet values of ad with =. For fixed, the skewess ad kurtosis are decreasig fuctios of ; for fixed, the skewess ad kurtosis are decreasig fuctios of. Lemma 5 If the media is deoted by M, the the mea deviatio from the mea, D( μ ), ad the mea deviatio from the media, D( M ), for the gamma-x distributio are give by D ad ( μ) = μγ{, log( Φ( μ)}/ Γ( ) I μ where DM ( ) = μ I M, (3.3) k+ Im = a( k, i) γδ (, i( m)), Γ ( ) k= i= i ck k + k + ( ) aki (, ) = ( π /) k+ i ( + i ) ad δi ( m) = ( i+ / )log( Φ( m/ )). Lemma 5 Proof If gx ( ) ad Gx ( ) are the PDF ad the CDF of the gamma-half ormal distributio, the the mea deviatios from the mea ad the media ca be writte as ad D( μ) = μg( μ) xg( x) dx (3.4) DM ( ) = μ xgxdx ( ). (3.5) M μ 9

9 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS Cosider the itegral: I = m = m π Γ( ) xg( x) dx m xe x x x ( log( Φ( ))) ( Φ( )) (/ ) (3.6) By substitutig u = log( Φ( x/ )) ad because erf ( x/ ) = Φ( x/ ), equatio (3.6) ca be writte as m u log( Φ( )) u Im = erf ( e ) u e du Γ( ) (3.7) Usig the series represetatio for u erf ( e ) (see Wolfram website), results i ck k+ k+ erf ( x) = ( π / ) x, k = k + (3.8) k cc m k m where ck = m= ( m+ )(m+ ) ad c =. Usig (3.8), equatio (3.7) reduces to I m ck π = ( ) Γ ( ) k + k= k+ m u log( Φ( )) u k+ ( e ) u e du. Usig the series expasio of (3.9) reduces to dx (3.9) ( u k e ) +, The results follow by substitutig I m i D( μ ) ad D( M ) i (3.3). Order Statistics The desity fuctio of the r th order statistic, X r :, for a radom sample of size draw from (.3), is f x = gx Gx Br (, r+ ) Gx (4.) r r ( ) ( )( ( )) ( ( )). Usig the biomial expasio, (4.) ca be writte as f r : gx ( ) ( x) = Br (, r+ ) r r j r+ j ( ) ( Gx ( )). j= j From (.7), equatio (4.) ca be writte as (4.) r j gx ( ) ( ) r fr : ( x) = r+ j Br (, r+ ) j= Γ( ) j k k r+ j + ( ) ( log( Φ( x / ))) k + k = k!( k+ ) r gx ( ) =... Br (, r+ ) j= k= k= sk + j ( ) r sk + ( r+ j ) r+ j k P ( ) r j k j + = Γ x sk + ( r+ j ) ( log Φ( )). I m k+ = a( k, i) γ ( δ, i), Γ ( ) k= i= where i c ( ) (, ) k k+ k + aki = ( π /) k+ i ( + i) ad δi ( m/ ) = ( i+ / )log( Φ( m/ )).

10 AYMAN ALZAATREH & KRISTEN KNIGHT Table : Mode, Mea, Media ad Variace for Some Values of ad with = Mode Mea Media Variace , , ,

11 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS Usig (.4), f r : where r ( x) =... Br (, r+ ) s kr+ j j= k k Γ ( s + ( r+ j) ) k sk + ( r+ j) r+ j Pk Γ( ) r gx ( s + ( ),, ), k r + j j (4.3) r j+ r j+ k = k ad i k i i i= i= P = k ( k + ). The result i (4.3) shows that the PDF of th the r order statistics of the gamma half-ormal distributio ca be expressed i terms of ifiite sums of the gamma half-ormal PDFs. Usig the same techique as Ristic ad Balakrisha (), the asymptotic distributio of the sample miimum X: ca be obtaied by utilizig Theorem i Arold, et al. (99), which G( ε x) γ states that if lim = x, the the ε G () G( ε ) asymptotic distributio of X : will be of Weibull type with shape parameter γ. Figure 5: The Gamma-Half Normal Skewess ad Kurtosis Graphs for Various Values of ad whe = Skewess beta =.5 beta =.8 beta = beta =3 beta=7 Kurtosis 3 4 beta =.5 beta =.8 beta = beta =3 beta= Alpha Alpha

12 AYMAN ALZAATREH & KRISTEN KNIGHT For the gamma half-ormal distributio, () G = ad G( εx) g( εx) lim = x lim G( ε) g( ε) log( Φ ( εx / )) = x lim+ ε log( ( ε / )) Φ φ( εx / ) = x xlim x. + = ε φ( ε / ) (4.4) + + ε ε Hece, the asymptotic distributio X r : is of Weibull type with shape parameter. The asymptotic distributio of the sample maximum X : ca be viewed as G ( x ), where G ( ) ( ) x = G x ad G is the CDF of X :. Parameter Estimatio Let a radom sample of size be take from the gamma-half ormal distributio. The log-likelihood fuctio for the gamma-half ormal distributio i (.4) is give by log L(, ) = log log logπ log Γ( ) log x i= i= i= + ( ) log( log( Φ( x / ))) + ( ) log( Φ( xi / ). i i (5.) The derivatives of (5.) with respect to, ad respectively, are give by log L = ψ ( ) log + log( log( Φ( x / ))), i= i (5.) log L = log( ( x / )), Φ i i= (5.3) log L = + 3 xi i= xh i z( xi / ) + ( ) i= log( Φ ( xi / )) ( ) xh i z( xi / ) i= (5.4) Settig (5.), (5.3) ad (5.4) to zero ad solvig them simultaeously results i ˆ, ˆ ad ˆ. The iitial values for the parameters, ad ca be obtaied by assumig the radom sample, xi, i =,, is take from the half-ormal distributio with parameter. By equatig the populatio mea to the sample mea of xi, i =,, ad solvig for, the iitial value ˆ = π /x. Assumig y log( ( / ˆ i = Φ xi )), i =,, are take from the gamma distributio with parameters ad (see Lemma ). By equatig the populatio mea ad the populatio variace of gamma distributio (with parameters ad ) to the correspodig sample mea ad sample variace of yi, i=,, ad solvig for ad, the iitial values are = y s ad = sy / y / y, where y ad s y are the sample mea ad the sample variace for y, y,, y. 3

13 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS Whe = =, the gamma-half ormal distributio reduces to the half-ormal distributio; thus, the likelihood ratio test ca be used to determie whether the gamma-half ormal distributio or the half-ormal distributio is the best model for fittig a give data set. The likelihood ratio test ca be used for testig the hypothesis H : = = agaist Ha : or, which is based o λ = L ˆ ˆ ˆ ( )/ L a (,, ), where L ad L a are the likelihood fuctios for the half-ormal ad the gamma-half ormal distributios, respectively. The quatity logλ follows the Chi-square distributio with degrees of freedom asymptotically. Applicatio Three data sets were applied to the gamma-half ormal distributio, ad compared with the half-ormal, geeralized half-ormal, beta geeralized half-ormal ad iverse Gaussia distributios. The first two data sets (see Tables ad 4), were aalyzed by Raqab, et al. (8). This data represets the tesile stregth data measured i GPa for sigle-carbo fibers that were tested at gauge legths of mm ad mm. The third data set (see Table 6) was aalyzed by Cheg, et al. (98) ad represets the flood level for the Susquehaa River at Harrisburg, PA. The maximum likelihood estimates, KS (Kolmogorov-Smirov) test-statistics ad p-values for the fitted distributios are reported i Tables 3, 5 ad 7. The data i Tables ad 4 are fitted to the gamma-half ormal, half-ormal, geeralized half-ormal ad beta geeralizedhalf ormal distributios. The half-ormal distributio did ot produce a adequate model for the data. However, the geeralized halformal, beta geeralized ad gamma-half ormal each provide a good fit for the two data sets. Amog the three geeralizatios of the half-ormal distributio, the gamma-half ormal provides the best fit for the first data set, ad geeralized half-ormal provides the best fit for the secod. Whe graphig the first data set, a approximately symmetric distributio is obtaied. The distributio of the secod data set, however, is a right-skewed shape. This suggests that the gamma-half ormal distributio is able to model data of both approximately symmetric ad right-skewed shapes. Figures 6 ad 7 display the empirical ad fitted cumulative distributio fuctios; these figures support the results i Tables 3 ad 5, respectively. Table : Sigle Carbo Fibers at mm The third data set (see Table 6) was aalyzed by Cheg, et al. (98) ad fitted to the iverse Gaussia distributio. These results, as well as the comparisos made to the halformal, beta geeralized half-ormal ad gamma-half ormal distributios, are reported i Table 7. The geeralized half-ormal distributio was diverget for the third data set. I view of these results, the gamma half-ormal ad iverse Gaussia distributios give a moderate fit to the data. The half-ormal distributio does ot give a adequate fit to the data, while the geeralized half-ormal provided the best fit. I viewig the distributio of the third data set, aother right-skewed distributio is observed. This cofirms the fact that the gamma-half ormal distributio ca be used to fit data of a right-skewed shape. Figure 8 displays the empirical ad fitted cumulative distributio fuctios. 4

14 AYMAN ALZAATREH & KRISTEN KNIGHT Table 3: Parameter Estimates for Sigle Carbo Fibers at mm Distributio Half-Normal Geeralized Half-Normal Beta Geeralized Half-Normal Gamma-Half Normal Parameter Estimates ˆ =.533 â =.83 ˆq =.5879 ˆ μ =. a ˆ =.374 b ˆ =.369 ˆ =.86 ˆ =.9766 â =.8794 ˆb = 3.75 ˆ =.3934 KS P-value Coclusio The gamma-half ormal distributio, a ew geeralizatio of the half-ormal distributio, was derived usig the method proposed by Alzaatreh, et al. (3b). Various properties of the distributio were studied icludig the momets, mea deviatios from the mea ad media, hazard fuctio, modality ad Shao etropy. The maximum likelihood method was proposed for the estimatio of the gamma-half ormal parameters. I order to demostrate the applicability of the gamma-half ormal distributio it was fitted to three real data sets ad compared with the half-ormal, geeralizedhalf ormal, iverse Gaussia ad beta geeralized half-ormal distributios. Results show that the gamma-half ormal distributio provides a adequate fit for each data set. Because the distributio was fitted to data sets with right-skewed ad approximately symmetric shapes, this idicates that the gamma-half ormal distributio offers flexibility that exteds beyod the half-ormal distributio. Although the gamma-half ormal distributio ca be bimodal, it was difficult to fid data i the literature with the specific form of bimodality. The maximum likelihood fuctios may be further studied uder differet types of cesorig for future applicatios of the gamma-half ormal distributio. Refereces Alzaatreh, A., Famoye, F., & Lee, C. (3a).Weibull-Pareto distributio ad its applicatios. Commuicatios i Statistics: Theory & Methods, 4(9), Alzaatreh, A., Lee, C., & Famoye, F. (3b). A ew method for geeratig families of cotiuous distributios. To Appear Metro: Iteratioal Joural of Statistics. Alzaatreh, A., Famoye, F., & Lee, C. (a). Gamma-Pareto distributio ad its applicatios. Joural of Moder Applied Statistical Methods, (), Alzaatreh, A., Lee, C., & Famoye, F. (b). O the discrete aalogues of cotiuous distributios. Statistical Methodology, 9, Arold, B. C., Balakrish, N., & Nagaraja, H. N. (99). A first course i order statistics. New York, NY: Wiley. Blad, J. M., & Altma, D. G. (999). Measurig agreemet i method compariso studies. Statistical Methods i Medical Research, 8, Blad, J. M. (5). The half-ormal distributio method for measuremet error: two case studies. Upublished lecture available o 5

15 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS Castro, L., Gomez, H., & Valezuela, M. (). Epsilo half-ormal model: properties ad iferece. Computatioal Statistics & Data Aalysis, 56(), Cheg, R. C. H., & Ami, N. A. K., (98). Maximum likelihood estimatio of parameters i the iverse Gaussia distributio, with ukow origi. Techometrics, 3, Cordeiro, G., Pescim, R., & Ortega, E. (). The Kumaraswamy geeralized halformal distributio for skewed positive data. Joural of Data Sciece,, Cooray, K., & Aada, M. (8). A Geeralizatio of the Half-Normal distributio with applicatios to lifetime data. Commuicatios i Statistics-Theory ad Methods, 37, Lee, C., Famoye, F. & Alzaatreh, A. (3). Methods for geeratig families of cotiuous distributio. To Appear i WIREs Computatioal Statistics. Nadarajah, S., & Pal, M. (8). Explicit expressios for momets of gamma order statistics. Bulleti of the Brazilia Mathematical Society, 39(), Pescrim, R., et al. (). The beta geeralized half- ormal distributio. Computatioal Statistics ad Data Aalysis, 54, Raqab, M., Madi, M., & Debasis, K. (8). Estimatio of P(Y<X) for the 3- parameter geeralized expoetial distributio. Commuicatios i Statistics-Theory ad Methods, 37(8), Reyi, A. (96). O measures of etropy ad iformatio, I: Proceedigs of the Fourth Berkeley Symposium o Mathematical Statistics ad Probability, I, Berkeley, CA: Uiversity of Califoria Press. Ristic, M., & Balakrisha, N. (). The gamma-expoetiated expoetial distributio. Joural of Statistical Computatio ad Simulatio, 8(8), 9-6. Shao, C. E. (948). A mathematical theory of commuicatio. Bell System Techical Joural, 7, Wolfram. GammaBetaErf/IverseErf/6///4/, retrieved o December 4,. Zografos, K., & Balakrisha, N. (9). O families of beta ad geeralized gamma-geerated distributios ad associated iferece. Statistical Methodology, 6,

16 AYMAN ALZAATREH & KRISTEN KNIGHT Figure 6: CDF for Fitted Distributios for Gauge Legth of mm Data CDF Empirical Half Normal Geeralized Half Normal Beta Geeralized Half-Normal Gamma-Half Normal Stregth data measured i GPA Table 4: Sigle Carbo Fibers at mm Distributio Parameter Estimates Table 5: Parameter Estimates for Sigle Carbo Fibers at mm Half-Normal ˆ =.49 Geeralized Half-Normal â =.5347 ˆq =.4798 ˆ μ =.699 Beta Geeralized Half-Normal a ˆ =.9544 b ˆ =.5 ˆ =.954 ˆ =.47 Gamma-Half Normal â =.46 ˆb =.46 ˆ =.459 KS P-value

17 ON THE GAMMA-HALF NORMAL DISTRIBUTION AND ITS APPLICATIONS Figure 7: CDF for Fitted Distributios for Gauge Legths of mm Data CDF Empirical Half Normal Geeralized Half Normal Beta Geeralized Half-Normal Gamma-Half Normal Stregth data measured i GPA 8

18 AYMAN ALZAATREH & KRISTEN KNIGHT Table 6: Maximum Flood Level for Susquehaa River Table 7: Parameter estimates for maximum flood levels Distributio Half-Normal Geeralized Half-Normal Beta Geeralized Half-Normal Gamma-Half Normal Parameter Estimates a ˆ = ˆ 76. ˆq =.444 Diverget b = ˆ =.9 ˆ =.483 â =.78 ˆb =.45 ˆλ =.94 KS P-value Figure 8: CDF for Fitted Distributios for Maximum Flood Level CDF Empirical Half Normal Beta Geeralized Half-Normal Iverse Gaussia Gamma-Half Normal Maximum flood level 9