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1 Mathematics 14,, 1-8; doi:1.339/math11 OPEN ACCESS mathematics ISSN Article O the Folded Normal Distributio Michail Tsagris 1, Christia Beeki ad Hossei Hassai 3, * arxiv: v1 [stat.me] 14 Feb 14 1 School of Mathematical Scieces, Uiversity of Nottigham, NG7 RD, UK; mtsagris@yahoo.gr School of Busiess ad Ecoomics, TEI of Ioia Islads, 311 Lefkada, Greece; christiabeeki@gmail.com 3 Statistical Research Cetre, Executive Busiess Cetre, Bouremouth Uiversity, BH8 8EB, UK; * Author to whom correspodece should be addressed; hhassai@bouremouth.ac.uk; Tel.: Received: 1 October 13; i revised form: 6 Jauary 14 / Accepted: 6 Jauary 14 / Published: 14 February 14 Abstract: The characteristic fuctio of the folded ormal distributio ad its momet fuctio are derived. The etropy of the folded ormal distributio ad the Kullback Leibler from the ormal ad half ormal distributios are approximated usig Taylor series. The accuracy of the results are also assessed usig differet criteria. The maximum likelihood estimates ad cofidece itervals for the parameters are obtaied usig the asymptotic theory ad bootstrap method. The coverage of the cofidece itervals is also examied. Keywords: folded ormal distributio; etropy; Kullback Leibler; maximum likelihood estimates 1. Itroductio Maily studied i the 196s, the folded ormal distributio is a special case of the Gaussia distributio occurrig whe the sig of the variable is always positive. I 1961, a method of estimatig the parameters based upo the estimatig equatios of the momets was discussed i [1], where they also gave some examples of its applicatios i the idustrial sector. The folded ormal distributio was used to study the magitude of deviatio of a automobile strut aligmet []. The properties of the multivariate folded ormal distributio with its possible applicatios were studied i [3]. I additio, tables with probabilities for a rage of values of the vector of parameters were provided, ad a applicatio of the model with real data was illustrated. A alterative method usig the secod ad fourth

2 Mathematics 14, 13 momets of the distributio was proposed i [4], whilst [5] performed maximum likelihood estimatio ad calculated the asymptotic iformatio matrix. Thereafter, the sequetial probability ratio test for the ull hypothesis of the locatio parameter beig zero agaist a specific alterative was evaluated i [6] with the idea of illustratig the use of cumulative sum cotrol charts for multiple observatios. I [7], the author dealt with the hypothesis testig of the zero locatio parameter regardless of the variace beig kow or ot. The distributio formed by the ratio of two folded ormal variables was studied ad illustrated with a few applicatios i [8]. The folded ormal distributio has bee applied to may practical problems. For istace, itroduced i [9] is a ecoomic model to determie the process specificatio limits for folded ormally distributed data. Through this paper, we will examie the folded ormal distributio from a differet perspective. I the process, we will cosider the study of some of its properties, amely the characteristic ad momet geeratig fuctios, the Laplace ad Fourier trasformatios ad the mea residual life of this distributio. The etropy of this distributio ad its Kullback Leibler divergece from the ormal ad half ormal distributios will be approximated via the Taylor series. The accuracy of the approximatios are assessed usig umerical examples. Also reviewed here is the maximum likelihood estimates (for a itroductio, see [1]), with examples from simulated data give for illustratio purposes. Simulatio studies will be performed to assess the validity of the estimates with ad without bootstrap calibratio i low sample cases. Numerical optimizatio of the log-likelihood will be carried out usig the simplex method [1].. The Folded Normal The folded ormal distributio with parameters (µ, ) stems from takig the absolute value of a ormal distributio with the same vector of parameters. The desity of Y, with Y N (µ, ) is give by: f (y) = Thus,X = Y, deoted by Y FN (µ, ), has the followig desity: f (x) = 1 π The desity ca be writte i a more attractive form [5]: x +µ ) ( µx ) f (x) = π e ( cosh 1 π e 1 (y µ) (1) [e 1 (x µ) +e 1 (x+µ)] () ad by expadig thecosh via a Taylor series, we ca also write the desity as: x +µ ) ( 1) ( µx ) f (x) = π e ( ()! We ca see that the folded ormal distributio is ot a member of the expoetial family. The cumulative distributio ca be writte as: F (x) = 1 [ ( ) ( )] x µ x+µ erf +erf (5) = (3) (4)

3 Mathematics 14, 14 where erf is the error fuctio: erf (x) = pi x e t dx (6) The mea ad the variace of Equatio () is calculated usig direct calculatio of the itegrals as follows [1]: [ ( µ µ f = π e +µ 1 Φ µ )] (7) f = µ + µ f (8) where Φ(.) is the cumulative distributio fuctio of the stadard ormal distributio. The third ad fourth momets about the origi are calculated i [4]. We develop the calculatio further by providig the characteristic fuctio ad the momet geeratig fuctio of Equatio (). Figure (1) shows the desities of the folded ormal for some parameter values. Figure 1. The black lie is the desity of the N (µ, ) ad the red lie of the FN (µ, ). The parameters i the left figure (a) are µ = ad = 3 ad i the right figure (b) µ = ad = 4. desity desity (a) (b).1. Relatios to Other Distributios The distributio of Z = X/ is a o-cetral χ distributio with oe degree of freedom ad o-cetrality parameter equal to (µ/) [11]. It is clear that whe µ =, a cetral χ 1 is obtaied. The half ormal distributio is a special case of Equatio (), with µ = for which [1] showed that it is the limitig form of the folded (cetral) t distributio as the degrees of freedom of the latter go to ifiity. Both distributios are further developed i the bivariate case i [13].

4 Mathematics 14, 15 The folded ormal distributio ca also be see as the the limit of the folded o-stadardized t distributio as the degrees of freedom go to ifiity. The folded o-stadardized t distributio is the distributio of the absolute value of the o-stadardizedtdistributio withv degrees of freedom: g(x) = Γ( ) [ ] v+1 Γ ( ) 1+ 1 (x µ) v+1 [ ] (x+µ) v+1 (9) v vπ v v.. Mode of the Folded Normal Distributio The mode of the distributio is the value of x for which the desity is maximised. I order to fid this value, we take the first derivative of the desity with respect to x ad set it equal to zero. Ufortuately, there is o closed form. We ca, however, write the derivative i a better way ad ed up with a o-liear equatio. df (x) dx = (x µ) [ x e 1 x e 1 (x µ) (x+µ) (x µ) +e 1 (x+µ) ] µ (1+e µx ) µ e 1 (x+µ) = (1) ] [ e 1 (x µ) e 1 (x+µ) = (11) (1 e µx ) = (1) (µ+x)e µx = µ x (13) x = µ x log µ µ+x (14) We saw from umerical ivestigatio that whe µ <, the maximum is met whe x =. Whe µ, the maximum is met at x >, ad whe µ becomes greater tha 3, the maximum approaches µ. This is of course somethig to be expected, sice, i this case, the folded ormal coverges to the ormal distributio..3. Characteristic Fuctio ad Other Related Fuctios of the Folded Normal Distributio Forms for the higher momets of the distributio whe the momet is a odd ad eve umber is provided i [4]. Here, we derive its characteristic ad, thus, the momet geeratig fuctio. ϕ x (t) = E ( e itx) = = = e itx 1 (x µ) dx+ π e A dx+ π e itx f X (x)dx = We will work ow with the formsaad B. e itx 1 [e 1 (x µ) +e 1 (x+µ)] dx π e itx 1 (x+µ) dx π e B dx (15) π A = itx 1 (x µ) = i tx x +µx µ = x x(i t+µ)+µ (16) = [x (i t+µ)] + 4 t i tµ = (x a) t +iµt (17)

5 Mathematics 14, 16 where a = i t+µ. Thus, the first part of Equatio (15) becomes: e A πdx = e t +iµt e (x α) = e t +iµt [ 1 Φ The secod expoet,b, usig similar calculatios becomes: t dx = e π +iµt [1 P (X )] (18) ( a )] [ = e t +iµt 1 Φ ( µ )] it (19) B = itx 1 (x+µ) = [x (i t µ)] t iµt () ad, thus, the secod part of Equatio (15) becomes: e B ( πdx = t µ )] e [1 Φ iµt it Fially, the characteristic fuctio becomes: [ 1 Φ ϕ x (t) = e t +iµt ( µ +it )]+e t iµt [1 Φ Below, we list some more fuctios that iclude expectatios. ( µ +it )] (1) () 1. The momet geeratig fuctio of Equatio () exists ad is equal to: M x (t) = ϕ x ( it) = e t [1 Φ ( +µt µ ( )]+e t t µ )] [1 Φ µt t (3) We ca see that the characteristic geeratig fuctio ca be differetiated ifiitely may times, sice the first derivative cotais the desity of the ormal distributio, ad thus, it always cotais some expoetial terms. The folded ormal distributio is ot a stable distributio. That is, the distributio of the sum of its radom variables do ot form a folded ormal distributio. We ca see this from the characteristic (or the momet) geeratig fuctio Equatio () or Equatio (3).. The cumulat geeratig fuctio is simply the logarithm of the momet geeratig fuctio: ( ) t { K x (t) = logm x (t) = +µt log 1 Φ ( µ [ ( µ )]} )+e t µt 1 Φ t (4) 3. The Laplace trasformatio ca easily be derived from the momet geeratig fuctio ad is equal to: E ( e tx) = e t [1 Φ ( µt µ ( )]+e +t t µ )] [1 Φ +µt +t (5) 4. The Fourier trasformatio is: ˆf (t) = e πixt f (x)dx = E ( e πixt) (6) However, this is closely related to the characteristic fuctio. We ca see that E(e πixt ) = φ x ( πt). Thus, Equatio (6) becomes: [ ˆf (t) = φ x ( πt) = e 4π t iπµt 1 Φ ( µ )] iπt (7) [ ( + e 4π t +iπµt µ )] 1 Φ iπt (8)

6 Mathematics 14, The mea residual life is give by: E(X t X > t) = E(X X > t) t (9) where t R +. The above coditioal expectatio is give by: E(X X > t) = t xf (x) P (x > t) dx = t [ erf xf (x) dx (3) 1 F (t) ( ) ( )] x µ x+µ +erf. The cotets The deomiator i Equatio (3) is writte as 1 1 withi the itegral i the umerator of Equatio (3) could be replaced by 1 F (t), as well, but we will ot replace it. The calculatio of the umerator is doe i the same way as the calculatio of the mea. Thus: t xf (x)dx = = = Fially, Equatio (3) ca be writte as: 1 x 1 1 t π e (x µ) dx+ x 1 t π e (x+µ) dx (31) [ ( )] e (t µ) t µ +µ 1 Φ + ( ) e (t µ) t µ π µφ (3) π [ ( )] e(t µ) t µ π +µ 1 Φ (33) E(X t X > t) = Etropy ad Kullback Leibler Divergece e(t µ) π [ erf +µ [ 1 Φ ( )] t µ ( ) ( )] t (34) x µ x+µ +erf Whe studyig a distributio, the etropy ad the Kullback Leibler divergece from some other distributios are two measures that have to be calculated. I this case, we tried to approximate both of these quatities usig a Taylor series. Numerical examples are displayed to show the performace of the approximatios Etropy The etropy is defied as the egative expectatio of logf (x). E = E[ logf (x)] = logf (x)f (x)dx { 1 = f (x)log [e 1 (x µ) +e 1 (x+µ)]} dx π = log π f (x)dx f (x)log e (x µ) = log π + x µx+µ f (x) e(x+µ) 1+ e (x µ) dx f (x)log (1+e µx )dx (35)

7 Mathematics 14, 18 Let us ow take the secod term of Equatio (35) ad see what is equal to: 1 x f (x) = µ + by exploitig the kowledge of variace Equatio (8) (36) µ xf (x) = µ µ f sice the first momet is give i Equatio (7) ad (37) µ Fially, the third term of Equatio (35) is equal to: A = f (x) f (x) = µ (38) ( 1) +1 e µx dx (39) =1 by makig use of the Taylor expasio for log(1+x) aroud zero, but istead of x, we have e µx. Thus, we have maaged to break the secod itegral of etropy Equatio (35) dow to smaller pieces of: A = = =1 ( 1) +1 =1 e ax 1 1 π e (x µ) dx ( 1) +1 e ( µ+a ) µ ( [1 Φ µ a )] ( 1) +1 =1 e ( µ a ) µ [1 Φ ( µ a )] ( 1) +1 e ax 1 1 π e (x+µ) dx by iterchagig the order of the summatio ad the itegratio, fillig up the square i the same way to the characteristic fuctio ad witha = µ. The fial form of the etropy is give i Equatio (4): ( )] E log π µ µµ f ( 1) +1 =1 ( 1) +1 =1 e (µ µ) µ [1 Φ ( µ =1 )] + µ e (µ µ) µ [1 Φ µ + µ Figure shows the true value of Equatio (4), whe = 5 ad µ rages from zero to 5, thus for values ofθ = µ from zero to five. The true value was calculated usig umerical itegratio. Rprovides this optio with the commad itegrate. The secod ad third order approximatios (usig the first two ad three terms of the ifiite sums i Equatio (4)), are also displayed for compariso. (4)

8 Mathematics 14, 19 Figure. Etropy values for a rage of values ofθ = µ with = 1 (a) ad = 5 (b). etropy True value Secod order approximatio Third order approximatio etropy True value Secod order approximatio Third order approximatio θ (a) θ (b) We ca see that the secod order approximatio is ot as good as the third order, especially for small values of θ. The Taylor approximatio of Equatio (4) is valid whe the value, a, is close to zero. As with the logarithm approximatio, the expasio is aroud zero; thus, whe we start goig further away from zero, the approximatio loses its) accuracy. The same is true i our case. Whe the values of θ are small, the the value oflog (1+e µx is far from zero. As θ icreases, ad, thus, the expoetial term decreases, the Taylor series approximates true value better. This is why we see a small discrepacy of the approximatios o the left of Figure, which become egligible later o. 3.. Kullback Leibler Divergece from the Normal Distributio The Kullback Leibler divergece [14] of oe distributio from aother i geeral is defied as the expectatio of the logarithm of the ratio of the two distributios with respect to the first oe: [ KL(f g) = E f log f ] = f (x)log f (x) g g(x) dx The divergece of the folded ormal distributio from the ormal distributio is equal to: [e 1 (x µ) +e 1 KL(FN N) = = 1 [e 1 (x µ) +e 1 (x+µ)] log π 1 π 1 [e 1 (x µ) +e 1 (x+µ)] log (1+e µx )dx π 1 1 π e (x µ) (x+µ)] which is the same as the secod itegral of Equatio (35). Thus, we ca approximate this divergece by the same Taylor series: ( )] ( 1) +1 KL(FN N) e (µ µ) µ [1 Φ µ + µ =1 ( )] ( 1) +1 + e (µ µ) µ µ [1 Φ + µ =1 dx

9 Mathematics 14, Figure 3. Kullback Leibler divergece from the ormal for a rage of values ofθ = µ with = 1 (a) ad = 5 (b). Kullback Leibler divergece True value Secod order approximatio Third order approximatio Kullback Leibler divergece True value Secod order approximatio Third order approximatio θ (a) θ (b) Figure 3 presets two cases of the Kullback Leibler divergece, for illustratio purposes, whe the first two ad three terms of the ifiite sum have bee used. I the first graph, the stadard deviatio is equal to oe, ad i the secod case, it is equal to five. The divergece seems idepedet of the variace. The chage occurs as a result of the value of θ. It becomes clear that whe the value of the mea to the stadard deviatio icreases, the folded ormal coverges to the ormal distributio Kullback Leibler Divergece from the Half Normal Distributio As metioed i Sectio.1, the half ormal distributio is a special case of the folded ormal distributio with µ =. The Kullback Leilber divergece of the folded ormal from the half ormal distributio is equal to: = KL(FN ( µ, ) FN ( µ =, ) ) = = log = log+ 1 [e 1 (x µ) +e 1 (x+µ)] log π f ( x;µ, ) dx+ ( µx µ f ( x;µ, ) ( log ) f ( x;µ, ) dx+ = log+ µµ f µ +KL(FN N) 1 π [e 1 (x µ) +e 1 1 π e x e µ +µx +e µ µx (x+µ)] )dx dx f ( x;µ, ) log (1+e µx )dx where f (x;µ, ) stads for the folded ormal Equatio () ad µ f is the expected value give i Equatio (7). Figure 4 shows the approximatios to the true value whe = 1 ad = 5. This time, we used the third ad fifth order approximatios, but eve the, for small values of θ, the approximatios were ot satisfactory.

10 Mathematics 14, 1 Figure 4. Kullback Leibler divergece from the half ormal for a rage of values of θ = µ with = 1 (a) ad = 5 (b). Kullback Leibler divergece True value Secod order approximatio Third order approximatio Kullback Leibler divergece True value Third order approximatio Firfth order approximatio θ (a) θ (b) The previous result caot lead to a iequality regardig the Kullback Leibler divergeces from the two other distributios. Whe µ >, the the divergece from the half ormal will be greater tha the divergece from the ormal, ad whe µ <, the opposite is true. However, this is ot strict, sice it ca be the case for either iequality that the relatioship betwee the divergeces is ot true. Istead, we ca use it as a rule of thumb i geeral. 4. Parameter Estimatio We will show two ways of estimatig the parameters. The first oe ca be foud i [1], but we review it ad add some more details. Both of them are essetially the maximum likelihood estimatio procedure, but i the first case, we perform maximizatio, whereas i the secod case, we seek the root of a equatio. The log-likelihood of Equatio () ca be writte i the followig way: l = logπ + l = logπ + l = logπ ] log [e (x i µ) +e (x i +µ) ( log [e (x i µ) (x i µ) + 1+e (x i +µ) ( ) log 1+e µx i e (x i µ) )] (41)

11 Mathematics 14, where is the sample size of thex i values. The partial derivatives of Equatio (41) are: l µ = l = + l = + (x i µ) (x i µ) + µ 4 4 (x i µ) + µ 4 4 x i e µx i 1+e µx i = (x i µ) x i e µx i 1+e µx i x i 1+e µx i x i 1+e µx i, ad By equatig the first derivative of the log-likelihood to zero, we obtai a ice relatioship: x i 1+e µx i = (x i µ) (4) Note that Equatio (4) has three solutios, oe at zero ad two more with the opposite sig. The example i Sectio 4.1 will show graphically the three solutios. By substitutig Equatio (4), to the derivative of the log-likelihood w.r.t ad equatig to zero, we get the followig expressio for the variace: = (x i µ) + µ (x i µ) = (x i µ ) = x i µ (43) The above relatioships Equatios (4) ad (43) ca be used to obtai maximum likelihood estimates i a efficiet recursive way. We start with a iitial value for ad fid the positive root of Equatio (4). The, we isert this value of µ i Equatio (43) ad get a updated value of. The procedure is beig repeated util the chage i the log-likelihood value is egligible. Aother easier ad more efficiet way is to perform a search algorithm. Let us write Equatio (4) i a more elegat way. ( ) x i 1+e µx i x i 1+e µx i where is defied i Equatio (43). 1+e µx i ( ) x i 1 e µx i +µ = 1+e µx i +µ = It becomes clear that the optimizatio the log-likelihood Equatio (41) with respect to the two parameters has tured ito a root search of a fuctio with oe parameter oly. We tried to perform maximizatio via the E-M algorithm, treatig the sig as the missig iformatio, but it did ot prove very good i this case A Example with Simulated Data We geerated 1 radom values from the FN(,9) i order to illustrate the maximum likelihood estimatio procedure. The estimated parameter values were equal to (ˆµ =.183,ˆ = 8.65). The correspodig 95% cofidece itervals for µ ad were (.78,3.585) ad (.,14.18)

12 Mathematics 14, 3 respectively. Equatio (41), estimates ofµ. Figure 5 shows graphically the existece of the three extrema of the log-likelihood oe miimum (always at zero) ad two maxima at the maximum likelihood Figure 5. The left graph (a) shows the three solutios of the log-likelihood. The right three-dimesioal figure (b) shows the values of the log-likelihood for a rage of mea ad variace values. values of equatio (4) log likelihood values mea variace 5. mea values (a) (b) 4.. Simulatio Studies Simulatio studies were implemeted to examie the accuracy of the estimates usig umerical optimizatio based o the simplex method [1]. Numerical optimizatio was performed i [15], usig the optim fuctio. The term accuracy refers to iterval estimatio rather tha poit estimatio, sice the iterest was o costructig cofidece itervals for the parameters. The umber of simulatios was set equal to R = 1,. The sample sizes raged from to 1 for a rage of values of the parameter vector. The R-package VGAM[16] offers algorithms for obtaiig maximum likelihood estimates of the folded ormal, but we have ot used it here. For every simulatio, we calculated 95% cofidece itervals usig the ormal approximatio, where the variace was estimated from the iverse of the observed iformatio matrix. The maximum likelihood estimates are asymptotically ormal with variace equal to the iverse of the Fisher s iformatio. The sample estimate of this iformatio is give by the secod derivative (Hessia matrix) of the log-likelihood with respect to the parameter. This is a asymptotic cofidece iterval. Bootstrap cofidece itervals were also calculated usig the percetile method [17]. For every simulatio, we produced the bootstrap distributio of the data with B = 1 bootstrap repetitios. Thus, we calculated the.5% lower ad upper quatiles for each of the parameters. I additio, we calculated the correlatios for every pair of the parameters. Tables 1 to 4 preset the coverage of the95% cofidece itervals for the two parameters at differet pairs of sample size ad mea. The rows correspod to the sample size, whereas the colums correspod to the ratio θ = µ, with = 5 fixed.

13 Mathematics 14, 4 Table 1. Estimated coverage probability of the 95% cofidece itervals for the mea parameter, µ, usig the observed iformatio matrix. Values of θ Sample size What ca be see from Tables 1 ad is that whist the sample size is importat, the value of θ, the mea to stadard deviatio ratio, is more importat. As this ratio icrease the coverage probability icreases, as well, ad reaches the desired omial 95%. This is also true for the bootstrap cofidece itervals, but the coverage is i geeral higher ad icreases faster as the sample size icreases i cotrast to the asymptotic cofidece iterval. What is more is that whe the value of θ is less tha oe, the bootstrap cofidece iterval is to be preferred. Whe the value of θ becomes equal to or more tha oe, the both the bootstrap ad the asymptotic cofidece itervals produce similar coverages. The results regardig the variace are preseted i Tables 3 ad 4. Whe the value ofθis small, both ways of obtaiig cofidece itervals for this parameter are rather coservative. The bootstrap itervals ted to perform better, but ot up to the expectatios. Eve whe the value of θ is large, if the sample sizes are ot large eough, the omial coverage of95% is ot attaied. Table. Estimated coverage probability of the bootstrap 95% cofidece itervals for the mea parameter, µ, usig the percetile method. Values of θ Sample size

14 Mathematics 14, 5 Table 3. Estimated coverage probability of the 95% cofidece itervals for the variace parameter,, usig the observed iformatio matrix. Values of θ Sample size The correlatio betwee the two parameters was also estimated for every simulatio from the observed iformatio matrix. The results are displayed i Table 5. The correlatio betwee the two parameters is always egative irrespective of the sample size or the value of θ, except for the case whe θ = 4. I this case, the correlatio becomes zero as expected. As the value of θ grows larger, the probability of the ormal distributio, which lies o the egative axis, becomes smaller util it becomes egligible. I this case, the distributio equals the classical ormal distributio for which the two parameters are kow to be orthogoal. Table 4. Estimated coverage probability of the bootstrap 95% cofidece itervals for the variace parameter,, usig the percetile method. Values of θ Sample size

15 Mathematics 14, 6 Table 5. Estimated correlatios betwee the two parameters obtaied from the observed iformatio matrix. Values of θ Sample size Table 6 shows the probability of a ormal radom variable beig less tha zero whe = 5 ad the same values ofθ as i the simulatio studies. Table 6. Probability of a ormal variable havig egative values. Values of θ Whe the ratio of mea to stadard deviatio is small, the area of the ormal distributio i the egative side is large, ad as the value of this ratio icreases, the probability decreases util it becomes zero. I this case, the folded ormal is the ormal distributio, sice there are o egative values to fold o to the positive side. This of course is i accordace with all the previous observatios ad results we saw. 5. Applicatio to Body Mass Idex Data We fitted the folded ormal distributio o real data. These are observatios of the the body mass idex of 7 New Zealad adults, accessible via the R package VGAM [16]. These measuremets are a radom sample from the Fletcher Challege/Aucklad Heart ad Health survey coducted i the early 199s [18]. Figure 6 cotais a histogram of the data alog with the parametric (folded ormal) ad the o-parametric (kerel) desity estimatio. It should be oted that the fitted folded ormal here coverges i distributio to the ormal.

16 Mathematics 14, 7 Figure 6. The histogram o the left shows the body mass idices of 7 New Zealad adults. The gree lie is the fitted folded ormal ad the blue lie is the kerel desity. The perspective plot o the right shows the log-likelihood of the body mass idex data as a fuctio of the mea ad the variace. Desity log likelihood values mea variace Body mass idices The estimated parameters (usig the optim commad i R) were ˆµ = 6.685(.175) ad ˆ = 1.34(1.14), with their stadard error appearig iside the paretheses. Sice the sample size is very large, there is o eed to estimate their stadard errors ad, cosequetly, 95% cofidece itervals, eve though their ratio is oly1.51. Their estimated correlatio coefficiet was very close to zero ( 1 4 ), ad the estimated probability of the folded ormal with these parameters below zero is equal to zero. 6. Discussio We derived the characteristic fuctio of this distributio ad, thus, its momet fuctio. The cumulat geeratig fuctio is simply the logarithm of the momet geeratig fuctio, ad therefore, it is easy to calculate. The importace of these two fuctios is that they allow us to calculate all the momets of the distributio. I additio, we calculated the Laplace ad Fourier trasformatios ad the mea residual life. The etropy of the folded ormal distributio ad the Kullback Leibler divergece of this distributio from the ormal ad half ormal distributios were approximated usig the Taylor series. The results were umerically evaluated agaist the true values ad were as expected. We reviewed the maximum likelihood estimates ad simplified their calculatio ad saw some properties of them. Cofidece itervals for the parameters were obtaied usig the asymptotic theory ad the bootstrap methodology uder the umbrella of simulatio studies.

17 Mathematics 14, 8 The coverage of the cofidece itervals for the two parameters was lower tha the desired omial i the small sample cases ad whe the mea to stadard deviatio ratio was lower tha oe. A alterative way to correct the uder-coverage of the mea parameter is to use a alterative parametrizatio. The parameters θ = µ ad are calculated i [5]. If we use θ ad µ, the the coverage of the iterval estimatio ofµis corrected, but the correspodig coverage of the cofidece iterval for is still low. The correlatio betwee the two parameters was always egative ad decreasig as the value of θ was icreasig, as expected, util the two parameters become idepedet. A applicatio of the folded ormal distributio to real data was exhibited, providig evidece that it ca be used to model o-egative data adequately. Coflicts of Iterest The authors declare o coflict of iterest. Refereces 1. Leoe, F.C.; Nelso, L.S.; Nottigham, R.B. The folded ormal distributio. Techometrics 1961, 3, Li, H.C. The measuremet of a process capability for folded ormal process data. It. J. Adv. Mauf. Techol. 4, 4, Chakraborty, A.K.; Chatterjee, M. O multivariate folded ormal distributio. Sakhya 13, 75, Eladt, R.C. The folded ormal distributio: Two methods of estimatig parameters from momets. Techometrics 1961, 3, Johso, N.L. The folded ormal distributio: Accuracy of estimatio by maximum likelihood. Techometrics 196, 4, Johso, N.L. Cumulative sum cotrol charts for the folded ormal distributio. Techometrics 1963, 5, Sudberg, R. O estimatio ad testig for the folded ormal distributio. Commu. Stat.-Theory Methods 1974, 3, Kim, H.J. O the ratio of two folded ormal distributios. Commu. Stat.-Theory Methods 6, 35, Liao, M.Y. Ecoomic tolerace desig for folded ormal data. It. J. Prod. Res. 1, 48, Nelder, J.A.; Mead, R. A simplex method for fuctio miimizatio. Comput. J. 1965, 7, Johso, N.L; Kotz, S.; Balakrisha, N. Cotiuous Uivariate Distributios; Joh Wiley & Sos, Ic.: New York, NY, USA, Psarakis, S.; Paaretos, J. The folded t distributio. Commu. Stat.-Theory Methods 199, 19, Psarakis, S.; Paaretos, J. O some bivariate extesios of the folded ormal ad the folded t distributios. J. Appl. Stat. Sci., 1,

18 Mathematics 14, Kullback, S. Iformatio Theory ad Statistics; Dover Publicatios: New York, NY, USA, R Developmet Core Team. R: A Laguage ad Eviromet for Statistical Computig, 1. Available olie: (accessed o 1 December 13). 16. Yee, T.W. The VGAM package for categorical data aalysis. J. Stat. Softw. 1, 3, Efro, B.; Tibshirai, R. A Itroductio to the Bootstrap; Chapma ad Hall/CRC: New York, NY, USA, MacMaho, S.; Norto, R.; Jackso, R.; Mackie, M.J.; Cheg, A.; Vader Hoor, S.; Mile, A.; McCulloch, A. Fletcher challege-uiversity of Aucklad heart ad health study: Desig ad baselie fidigs. N. Zeal. Med. J. 1995, 18, c 14 by the authors; licesee MDPI, Basel, Switzerlad. This article is a ope access article distributed uder the terms ad coditios of the Creative Commos Attributio licese (