An Epsilon Half Normal Slash Distribution and Its Applications to Nonnegative Measurements

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1 Ope Joural of Optmzato, 3,, -8 Publshed Ole March 3 ( A Epslo Half Normal Slash Dstrbuto ad Its Applcatos to Noegatve Measuremets Wehao Gu *, Pe-Hua Che, Haya Wu 3 Departmet of Mathematcs ad Statstcs, Uversty of Mesota Duluth, Duluth, USA Departmet of Maagemet Scece, Natoal Chao Tug Uversty, HsChu, Chese Tape 3 Departmet of EPLS, Florda State Uversty, Tallahassee, USA Emal: * wgu@d.um.edu, paulache@g.ctu.edu.tw, hw7d@fsu.edu Receved Jauary 5, 3; revsed February 4, 3; accepted February 8, 3 ABSTRACT We troduce a ew class of the slash dstrbuto usg the epslo half ormal dstrbuto. The ewly defed model exteds the slashed half ormal dstrbuto ad has more urtoss tha the ordary half ormal dstrbuto. We study the characterzato ad propertes cludg momets ad some measures based o momets of ths dstrbuto. A smulato s coducted to vestgate asymptotcally the bas propertes of the estmators for the parameters. We llustrate ts use o a real data set by usg maxmum lelhood estmato. Keywords: Epslo Half Normal Dstrbuto; Slash Dstrbuto; Kurtoss; Sewess; Maxmum Lelhood Estmato. Itroducto The epslo half ormal dstrbuto, proposed by Castro et al. [], s wdely used for oegatve data modelg, for stace, to cosder the lfetme process uder fatgue. We say that a radom varable X has a epslo half ormal dstrbuto wth parameters ad, deoted by EHN,, f ts desty fucto s gve by, for x, x x f x () where. deotes the stadard ormal desty fucto. Whe, the epslo half ormal dstrbuto reduces to the half ormal dstrbuto vestgated [-4]. Castro et al. [] provded mathematcal propertes of the epslo half ormal dstrbuto ad dscussed some feretal aspects related to the maxmum lelhood estmato. O the other had, a radom varable S has a stadard slash dstrbuto SL wth parameter, troduced [5], f S ca be represeted as Z S () U where Z N, ad U U, are depedet. It geeralzes ormalty ad has bee much studed the * Correspodg author. statstcal lterature. For the lmt case, SL yelds the stadard ormal dstrbuto Z. Let, the caocal slash dstrbuto follows, see [6]. It s well ow that the stadard slash desty has heaver tals tha those of the ormal dstrbuto ad has larger urtoss. It has bee very popular robust statstcal aalyss ad studed by some authors. The geeral propertes of ths caocal slash dstrbuto were studed [5,7]. Kafadar [8] vestgated the maxmum lelhood estmates of the locato ad scale parameters. Gómez et al. [9] replaced stadard ormal radom varable Z by a ellptcal dstrbuto ad defed a ew famly of slash dstrbutos. They studed ts geeral propertes of the resultg famles, cludg ther momets. Gec [] proposed the uvarate slash by a scale mxtured expoetal power dstrbuto ad vestgated asymptotcally the bas propertes of the estmators. Wag et al. [] troduced the multvarate sew verso of ths dstrbuto ad examed ts propertes ad fereces. They substtuted the stadard ormal radom varable Z by a sew ormal dstrbuto studed [] to defe a sew exteso of the slash dstrbuto. Olmos et al. [3] troduced the slashed half ormal dstrbuto by a scale mxtured half ormal dstrbuto ad showed that the resultg dstrbuto has more urtoss tha the ordary half ormal dstrbuto. Sce the Copyrght 3 ScRes.

2 W. H. GUI ET AL. epslo half ormal dstrbuto EHN, s a exteso of the half ormal dstrbuto, t s aturally to defe a slash dstrbuto based o t whch sewess ad thc taled stuatos may exst. It leads to a ew model o oegatve measuremets wth more flexble asymmetry ad urtoss parameters. The paper s orgazed as follows: Secto, we troduce the ew slash dstrbuto ad study ts relevat propertes, cludg the stochastc represetato etc. I Secto 3 we dscuss the ferece, momets ad maxmum lelhood estmato for the parameters. Smulato studes are performed to vestgate the behavors of estmators Secto 4. I Secto 5, we gve a real llustratve applcato ad report the results. Secto 6 cocludes our wor.. Epslo Half Normal Slash Dstrbuto.. Stochastc Represetato Defto. A radom varable Y has a epslo half ormal slash dstrbuto f t ca be represeted as the rato X Y (3) U where X EHN, defed () ad U U, are depedet,,,. We deote t as Y EHNS,,. Proposto. Let Y EHNS,,. The, the desty fucto of Y s gve by, for y, yt yt fy y t dt where,,. Proof. From (), the jot probablty desty fucto of X ad U s gve by, for x, u, x x gx, u Usg the trasformato: x y u, u u, the jot probablty desty fucto of Y ad U s gve by, for y, u, h y, u yu yu u The margal desty fucto of Y s gve by yu yu fy y d u u (4) After chagg the varable to fucto wll be obtaed as stated. t u, the dedty Remar.3 If, the desty fucto (4) reduces y t fy y e t dt to π whch s the desty fucto for the slashed half ormal dstrbuto studed by [3]. As, y y lm fy y The lmt case of the epslo half ormal slash dstrbuto s the epslo half ormal dstrbuto. For, the caocal case follows. Fgure shows some plots of the desty fucto of the epslo half ormal slash dstrbuto wth varous parameters. The cumulatve dstrbuto fucto of the epslo half ormal slash dstrbuto Y EHNS,, s gve as follows. For y, F y y Y Y f u du yt yt t dt (5) where. s the cumulatve dstrbuto fucto for the stadard ormal radom varable. Proposto.4 Let YUu EHNu, ad U U,, the Y EHNS,,. Fgure. The desty fucto of EHNS σε,, wth var- ous parameters. Copyrght 3 ScRes.

3 W. H. GUI ET AL. 3 Proof. f y Y YU U f y u f u du y y du u u u Remar.5 Proposto.4 shows that the epslo half ormal slash dstrbuto ca be represeted as a scale mxture of a epslo half ormal dstrbuto ad uform dstrbuto. The result provdes aother way besdes the defto (3) to geerate radom umbers from the epslo half ormal slash dstrbuto EHNS,,... Momets ad Measures Based o Momets for t, I ths secto, we derve the momet geeratg fucto,the -th momet ad some measures based o the momets. where, the the momet geeratg fucto of X s gve by Proposto.6 Let X EHN, Y ty e M t U t M t e t X t e t where ad. Proof. See []. Proposto.7 Let Y EHNS,,, the the momet geeratg fucto of Y s gve by, for t, wt MY t = w e wt (6) (7) wt e wt dw (8), ad. Proof. From Proposto.4 ad usg propertes of the codtoal expectato, we have U t U t e e U t U t u t u t e e u t u tdu Mag the trasformato w u ad the result follows. th Proposto.8 Let X EHN,, the the o-cetral momets are gve by X π (9) for,,. where ad. Proof. See []. Proposto.9 Let Y EHNS,,, where, ad. For,, ad th, the o-cetral momet of Y s gve by Y () π Proof. From the stochastc represetato defed (3) ad the results (9), the clam follows a straghtforward maer. Y XU X U X The followg results are mmedate coseueces of (). Corollary. Let Y EHNS,,, where, ad. The mea ad varace of Y are gve by ad for, Y, () π 3 π Var Y = π For the stadardzed sewess Copyrght 3 ScRes.

4 4 W. H. GUI ET AL. ad urtoss coeffcets where Remar. As we have the followg results. Corollary. Let Y EHNS,,, where, ad. The sewess ad urtoss coeffcets of Y are gve by, 3 BB CC AA for 3 () CC 3 for 4 (3) π AA 4 4 BB 5 3 π π e 4 π 3 π CC, the sewess coeffcet coverges. ad the urtoss coeffcet coverges as well π 7 3 π π 48 3 π π 4 3 π e whch are the correspodg sewess ad urtoss coeffcets for the epslo half ormal dstrbuto EHN,. Fgure shows the sewess ad urtoss coeffcets wth varous parameters for the EHNS,, model. The sewess ad urtoss coeffcets decrease as creases. The parameter does ot affect the two coeffcets. 3. Maxmum Lelhood Iferece I ths secto, we cosder the maxmum lelhood estmato about the parameters of the EHNS model defed (4). For y, yt yt fy y θ t where θ,,. dt Suppose y, y,, y s a radom sample of sze from the epslo half ormal slash dstrbuto EHNS,,. The the log-lelhood fucto ca be wrtte as θ log Y l f y = log log yt yt log t dt (4) The estmates of the parameters maxmze the lelhood fucto. By tag the partal dervatves of the log-lelhood fucto wth respect to,, respectvely ad eualzg the obtaed expressos to zero, we obta the followg maxmum lelhood estmatg euatos. Copyrght 3 ScRes.

5 W. H. GUI ET AL. 5 (a) (b) Fgure. The plot for the sewess ad urtoss coeffcet wth varous parameters. a) Sewess coeffcet; b) Kurtoss coeffcet. l l yt yt yt yt 3 3 t yt yt t dt yt yt yt yt 3 3 t yt yt t dt l yt yt t log td t t dt yt yt dt dt The maxmum lelhood estmatg euatos above are ot a smple form. I geeral, there are o mplct expresso for the estmates. The estmates ca be obtaed through some umercal procedures such as Newto-Raphso method. May programs provde routes to solve such maxmum lelhood estmatg euatos. I ths paper, all the computatos are performed usg software R. The MLE estmators are computed by the optm fucto whch uses L-BFGS-B method. I the followg secto, a smulato s coducted to llustrate the behavor of the MLE. For asymptotc ferece of θ,,, the Fsher I θ plays a ey role. It s well formato matrx ow that ts verse s the asymptotc varace matrx of the maxmum lelhood estmators. For the case of a sgle observato, we tae the secod order dervatves of the log-lelhood fucto (4) ad the Fsher formato matrx s defed as I θ, j y log f Y (5) j for,, 3 ad j,, 3. Proposto 3. Let Y, Y,, Y s a radom sample of sze from the dstrbuto EHNS θ, where Copyrght 3 ScRes.

6 6 W. H. GUI ET AL. θ,, ad ˆθ s the maxmum lelhood estmator of θ, we have d θθ N, I θ ˆ (6) 3 Proof. It follows drectly by the large sample theory for maxmum lelhood estmators ad the Fsher formato matrx gve above. 4. Smulato Study 4.. Data Geerato I ths secto, we preset how to geerate the radom umbers from the epslo half ormal slash dstrbuto EHNS,,. Proposto 4. Let Z, V be two depedet radom varables, where Z N, ad V s such that PV, PV the X V Z EHN,, where ad. Proof. For x, P X x P V Z x P Z x P Z x x x The desty fucto of X s x x f x for x whch proves the result. Usg the defto (3) ad the results Propos- to 4., we ca geerate varates from the epslo half ormal slash dstrbuto EHNS,, wth the followg algorthm. Algorthm 4. Usg the defto (3) to geerate data Geerate Z N,, S U, ad U U, Let V f S. Otherwse V Set X VZ X Set Y U 4.. Behavor of MLE I ths secto, we perform a smulato study to llus trate the behavor of the MLE estmators for parameters, ad. It s ow that as the sample sze creases, the dstrbuto of the MLE teds to the ormal dstrbuto wth mea,, ad covarace matrx eual to the verse of the Fsher formato matrx. However, the log-lelhood fucto gve (4) s a complex expresso. It s ot geerally possble to derve the Fsher formato matrx. Thus, the theoretcal propertes (asymptotcally ormal, ubased etc) of the MLE estmators are ot easly derved. We study the propertes of the estmators umercally. We frst geerate 5 samples of sze 5 ad from the EHNS,, dstrbuto for fxed parameters. The estmators are computed by the optm fucto whch uses L-BFGS-B method software R. The emprcal meas ad stadard devatos(sd) of the estmators are preseted Table. It ca be see from Table that the parameters are well estmated ad the estmates are asymptotcally u- Table. Emprcal meas ad SD for the MLE estmators of, ad. 5 ˆ SD ˆSD ˆ SD ˆ SD ˆSD ˆSD.3.9 (.688).66 (.858).7 (.434).585 (.568).33 (.994).36 (.637) (.579).37 (.384) (.747).33 (.46).83 (.58) 3.83 (.773) (.68).48 (.38).454 (.37).9 (.5656).466 (.39).993 (.34) (.5556).476 (.78) 3.898(.87).833 (.4435).4964 (.397) (.6898) (.359).33 (.36).57(.3) 4.84 (.38).349 (.474).79 (.63) (.8434).9 (.594) (.895) 4.7 (.559).333 (.39) 3.68 (.339) (.7).4597 (.36).644 (.86) 4.57 (.8947).5 (.578).55 (.549) (.356).49 (.389) 3.35 (.678) (.836).4434 (.99) 3.8 (.54) Copyrght 3 ScRes.

7 W. H. GUI ET AL. 7 Ta ble. Summ ary for the lfe of fatgu e fracture. Sample Sze Mea Stadard Devato b b Table 3. Maxmum lelhood parameter estmates(wth (SD)) of the HN, EHN ad EHNS models for the stressrupture data. Model ˆ ˆ ˆ logl AIC BIC HN (.64) EHN (.74) (.46) EHNS (.665) (.56) (.783) (a) based. The emprcal mea suare errors decrease as sample sze creases as expected. 5. Real Data Illustrato I ths secto, we cosder the stress-rupture data set, the lfe of fatgue fracture of Kevlar 49/epoxy that are subject to the pressure at the 9% level. The data set has bee prevously studed [,3,3,4]. Table summarzes descrptve statstcs of the data set where b ad b are sample asymmetry ad urtoss coeffcets, respectvely. Ths data set dcates o egatve asymmetry. We ft the data set wth the half ormal, the epslo half ormal ad the epslo half ormal slash dstrbutos usg maxmum lelhood method. The results are reported Table 3. The usual Aae formato crtero (AIC) ad Bayesa formato crtero (BIC) to measure of the goodess of ft are also computed. AIC logl ad BIC log log L. where s the umber of parameters the dstrbuto ad L s the maxmzed value of the lelhood fucto. The results dcate that EHNS model fts best. Fgures 3(a) ad (b) dsplay the ftted models usg the MLE estmates. 6. Cocludg Remars I ths artcle, we have studed the epslo half ormal slash dstrbuto EHNS,,. It s defed to be the uotet of two depedet radom varables, a epslo half ormal radom varable ad a power of the uform dstrbuto. Ths oegatve dstrbuto exteds the (b) Fgure 3. Models ftted for the lfe of fatgue fracture data set. a) Hstogram ad ftted curves; b) Emprcal ad ftted CDF. 7. Acowledgemets epslo half ormal, the half ormal dstrbuto etc. Probablstc ad feretal propertes are derved. A smulato s coducted ad demostrates the good per- formace of the maxmum lelhood estmators. We apply the model to a real dataset ad the results demostrate that the proposed model s very useful ad flexble for o egatve data. The authors would le to tha the aoymous revew- Copyrght 3 ScRes.

8 8 W. H. GUI ET AL. ers ad the edtor for ther valuable commets ad suggestos to mprove the ualty of the paper. REFERENCES [] L. Castro, H. Gómez ad M. Valezuela, Epslo Half-Normal Model: Propertes ad Iferece, Computatoal Statstcs & Data Aalyss, Vo l. 56, No.,, pp do:.6/j.csda..3. [] A. Pewsey, Improved Lelhood Based Iferece for the Geeral Half-Normal Dstrbuto, Commucatos Statstcs Theory ad Methods, Vol. 33, No., 4, pp do:.8/sta-837 [3] N. Olmos, H. Varela, H. Gómez ad H. Bolfare, A Exteso of the Half-Normal Dstrbuto, Statstcal Papers, Vol. 53, No. 4,, pp. -. [4] M. Wper, F. Gró ad A. Pewsey, Objectve Bayesa Iferece for the Half-Normal ad Half-t Dstrbutos, Commucatos Statstcs Theory ad Methods, Vol. 37, No., 8, pp do:.8/ [5] W. Rogers ad J. Tuey, Uderstadg Some Log- Taled Symmetrcal Dstrbutos, Statstca Neerladca, Vol. 6, No. 3, 97, pp. -6. do:./j tb9.x [6] N. Johso, S. Kotz ad N. Balarsha, Cotuous Uvarate Dstrbutos, Wely, Hoboe, 995. [7] F. Mosteller ad J. Tuey, Data Aalyss ad Regresso: A Secod Course Statstcs, Addso-Wesley Pub. Co., Bosto, 977. [8] K. Kafadar, A Bweght Approach to the Oe-Sample Problem, Joural of the Amerca Statstcal Assocato, Vol. 77, No. 378, 98, pp do:.8/ [9] H. Gómez, F. Qutaa ad F. Torres, A New Famly of Slash-Dstrbutos wth Ellptcal Cotours, Statstcs & Probablty Letters, Vol. 77, No. 7, 7, pp do:.6/j.spl.6..6 [] A. Gec, A Geeralzato of the Uvarate Slash by a Scale-Mxtured Expoetal Power Dstrbuto, Commucatos Statstcs Smulato ad Computato, Vol. 36, No. 5, 7, pp do:.8/ [] J. Wag ad M. Geto, The Multvarate Sew-Slash Dstrbuto, Joural of Statstcal Plag ad Iferece, Vol. 36, No., 6, pp. 9-. do:.6/j.jsp [] A. Azzal, A Class of Dstrbutos Whch Icludes the Normal Oes, Scadava Joural of Statstcs, Vol., No., 985, pp [3] D. Adrews ad A. Herzberg, Data: A Collecto of Problems from May Felds for the Studet ad Research Worer, Vol. 8, Sprger-Verlag, New Yor, 985. [4] R. Barlow, R. Tolad ad T. Freema, A Bayesa Aalyss of the Stress-Rupture Lfe of Kevlar/Epoxy Sphercal Pressure Vessels, I: C. Clarott ad D. Ldley, Eds., Accelerated Lfe Testg ad Experts Opos Relablty, Elsever Scece Ltd., Amsterdam, 988, pp Copyrght 3 ScRes.