Characteristic Function for the Truncated Triangular Distribution., Myron Katzoff and Rahul A. Parsa

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1 Secion on Survey Reserch Mehos JSM 009 Chrcerisic Funcion for he Trunce Tringulr Disriuion Jy J. Kim 1 1, Myron Kzoff n Rhul A. Prs 1 Nionl Cener for Helh Sisics, 11Toleo Ro, Hysville, MD. 078 College of Business n Pulic Aminisrion, Drke Universiy, Des Moines, IA 501 Asrc. Two ypes of proceures for msking micro cn e consiere: ing inepenen noise n muliplying y inepenen noise. The ringulr isriuion is one of he opions for incorporing muliplicive noise. The ringulr isriuion runce roun 1 is eer opion hn he unrunce ringulr isriuion n is in use. The isriuionl form of he runce ringulr isriuion ws evelope n is now ville. Invesigion of he sisicl properies of mske y using he runce ringulr isriuion requires knowlege of he momens of he runce ringulr isriuion. In his pper, we firs evelop he chrcerisic funcion for he runce ringulr isriuion; hen, he momens re erive from he chrcerisic funcion. We presen generl formul for he momens. This will help users wih recovering he esime momens of he originl. Key wors: msking, muliplicive noise, momen 1. Inroucion Since roun 1980, he U.S. Energy Informion Aminisrion (EIA) hs een using muliplicive noise for msking he numer of heing n cooling ys in n re, ec., in heir pulic use micro file from he Resienil Energy Consumpion Survey. EIA uses noise which follows he runce norml isriuion [Hwng, (1)]. Evns, e l [] propose he use of muliplicive noise o msk economic. They consiere noise which follows isriuions such s norml n runce norml isriuions. Kim n Winkler [] consiere muliplicive noise which follows he runce norml isriuion. The U.S. Bureu of he Census uses runce ringulr isriuion for msking he Commoiy Flow Survey (Aou []). Kim [5] evelope he proiliy ensiy funcion (pf) of he runce ringulr isriuion n showe h he esime from he mske y he isriuion is unise if he ringulr isriuion is symmeric ou 1 n runce symmericlly ou 1. Muliplicive noise hs he following form: Disclimer: The finings n conclusions in his pper re hose of he uhors n o no necessrily represen he views of he Nionl Cener for Helh Sisics, Ceners for Disese Conrol n Prevenion. 70

2 Secion on Survey Reserch Mehos JSM 009 y xe, i1,,..., n, i i i h where y i is he mske vrile for he i uni such s person, househol, eslishmen, ec., x i is he corresponing un-mske vrile n e i (>0) is he noise. If pulic use micro file is mske y muliplicive noise which follows he runce ringulr isriuion, hen users nee o know he momens of he runce ringulr isriuion in orer o esime he momens of he originl. The es wy of eriving momens of isriuion is eveloping he chrcerisic funcion of he isriuion. Thus, in his pper, we evelop he chrcerisic funcion of he runce ringulr isriuion. Using he chrcerisic funcion, we erive he momens of he isriuion. Finlly, we generlize he formul for ny momen..1 Tringulr Disriuion Trunce Tringulr Disriuion A runce ringulr isriuion is moifie form of ringulr isriuion, n hus we firs consier ringulr isriuion which is shown in Figure 1. The ringulr isriuion is very useful. I cn e use for pproximing he norml, gmm n e isriuions. The ringulr isriuion is nlyiclly esier o hnle hn he norml isriuion. f(e) m e Figure 1. Tringulr Disriuion 70

3 Secion on Survey Reserch Mehos JSM 009 The ringulr isriuion of rnom vrile e s shown ove hs he following form. 1 e, f() e m 1 e, m e m. (1) m e Here m is he moe n is unique.. Trunce Tringulr Disriuion When ringulr isriuion-se noise is use, one mus voi using he numer close o one (1) for noise, ecuse muliplying y numer very close o 1 oes no chnge he originl vlue h much, n hus he originl vlue oes no ge ny proecion. In iion, he proiliy ensiy for e is he grees, when e is ner 1. This suggess h he lrges numer of unis o no ge proecion. Hence, i hs een suggese [()] o runce he mi-secion, or he secion ner 1 of he ringulr isriuion. The runce ringulr isriuion hs he following shpe. f(e) m c e Figure. Trunce Tringulr Disriuion Suppose he isriuion is runce n c, c, s shown in Figure. In his cse, he pf hs he following form [Kim, (5)]: 705

4 Secion on Survey Reserch Mehos JSM 009 f() e ( m) m c m ( m ) m c m ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( e), e ( e), ce. () Le r enoe he egree (or re) of runcion. If we complee he ringle of Figure wih se o form ringulr isriuion, he egree of runcion is he moun of proiliy uner h ensiy funcion from o c. Since he inervl of runcion enclosing he moe of h ringulr isriuion nee no hve enpoins plce symmericlly ou he mein m of he ringulr isriuion wih se, m is now relly convenien symolic rifc for reling he evelopmen of secion.1 o he cse when we runce he ringulr isriuion. As we show nex, m is n unknown h is he roo of quric equion. We hve h r = 1 p, where 1 1 p e e m c m 1 ( ) ( c) m m e e () In he semen of he runce ringulr isriuion funcion, m is no orinrily inclue (Figure clls his poin o he enion of he reer). From equion () i follows h m A B B A C, where A p( ), B c p C ( ) ( c) p( ). ( ) ( ) ( ) n In he expression for m, we hve oh + n -. Selecion of m vlue will epen on wheher m is in he runce region or no. Noe h he isseminion gency shoul le he users know he egree of runcion (r). Chrcerisic Funcion The ensiy funcion of he runce ringulr isriuion is given y equion (). By leing in h equion 706

5 Secion on Survey Reserch Mehos JSM 009 k m c m ) ( ) ( ) ( ) (, () he runce ringulr isriuion cn e expresse s k( m)( x), x f( x) k ( m )( x), c x. (5) Noe h he ove proiliy ensiy funcion hs wo prs. The firs pr represens he sie o he lef of he region of runcion n he secon pr he sie o he righ of he region of runcion. The chrcerisic funcion of he runce ringulr isriuion ue o he lef sie of he isriuion is, k ( ix ix ix m ) ( x ) e x k ( m ) xe x k ( m ) e x. (6) Using inegrion y prs, ignoring k ( m) for he ime eing, we le x u n ix ix e v e x. Then x u n v. Thus he firs erm in equion (6) wihou i k ( m) ecomes 1 ix x ix ix xe x e e x i i x ix e i 1 i e ix i i 1 i i e e e e. (7) i i i The secon erm of equion (6) ecomes i i ix ix i i e x e e e. (8) Puing equions (7) n (8) ino equion (6), we hve 707

6 Secion on Survey Reserch Mehos JSM 009 k( m) k( m) k( m) k( m) e e e e e e i i i i i i i i i i k ( m) k ( m) i i i i i i e e e e. (9) Similrly, he chrcerisic funcion of he runce ringulr isriuion ue o he righ sie of he isriuion is, ix ix ix c c km ( ) ( xe ) x km ( ) e x km ( ) xe x c. k( m) x k( m) e k( m) e e c c i i i ix ix ix c k( m ) k( m ) k( m ) i i i i ic i ic i ic e e e ce e e km ( ) km ( ) i i ic ic i ic ce e e e Thus he chrcerisic funcion is. (10) e e e e ce e e e () k( m) k( m ) i i i i i i ic ic i ic (11).1 Firs Momen The firs erivive of () is, '( ) k( m) i ie i e i ie i e i e i e i ie 708

7 Secion on Survey Reserch Mehos JSM 009 km ( ) ic ice ic c e ic ie ic ce i e i e i ie (1) To ge he firs momen, we normlly replce y 0 in he ove. However, y 0 oing so, we ge or, where is non-zero consn. Thus, we mus pply 0 0 L Hopil s rule firs, someimes more hn once for he sme erm. The resuls re s follows. e e e e ik( m) e e e i i i i i i i ic i i ic ic ic ce ic e e ce ikm ( ) ce ce ce e e e ce ce e ik( m) ik( m ) 6 6 i i i ic ic i By seing = 0 in he ove n iviing y i, we ge he firs momen,. k ( m) km ( ) 6 6 '(0) ( ) (c c ). (1) For verificion, he firs momen of he runce ringulr isriuion cn e erive y srighforwr inegrion s follows. E ( x ) k ( m ) x ( x ) x k ( m ) x ( x ) x c k ( m) km ( ) c c. (1) 6 6 By replcing k in equion (1) wih he expression in equion (), Ex ( ) m ( ) m c ( c) ( )( m) ( c)( m) 709

8 Secion on Survey Reserch Mehos JSM 009 In cse = c n m = m, he ove simplifies furher.. Secon Momen By king he secon erivive of () in equion (10), we ge i i i i i i 6ie e i e ie e i e ''( ) k( m) i i i i ic ic 6e 6e ie e 6ice c e km ( ) ic ic ic ic i ic i i ic e ie ce ic e 6e 6e ie e. (15) Applying L Hopil s rule repeely, we ge ie ie ie ie ie k ( m) ie ie i i i i i i i ie ie ie ice ice km ( ) ice i i i ic ic ic i i i ic ic ie ie ic ic ie ice ice ice ice. Seing = 0 in he ove n iviing y 0, i, we ge he secon momen roun k ( m) km ( ) (0) 1 1 c c ''. (16). Thir Momen By king he hir erivive of () in equion (11) n pulling he sme erms ogeher, we ge 710

9 Secion on Survey Reserch Mehos JSM 009 '''( ) k( m) i ie i 1i e i i e i e i 6ie i 18ie 6e i e e e e 6 e i e i i i i i i i 5 5 km ( ) ic ice ic 1c e ic ic e ic c e ic 6ie i 18ie ic ic ic i ic i 6ce ic e c e e e 6 e i e 5 5. (17) Applying L Hopil s rule repeely, seing = 0 in '''( ) n iviing he comine erms y i, we ge he hir momen roun 0 s follows: k ( m) km ( ) '''(0) 5 c 5c. (18) From equions (1), (16) n (18), one cn infer h he fourh momen roun 0 is k ( m) km ( ) 6 5 ''''(0) 5 6 5c 6c 6, (19) 0 0 n h he fifh momen roun 0 is, k ( m) km ( ) 7 6 '''''(0) 6 7 6c 7c 7. (0). h p Momen Theorem. The h p momen of he runce ringulr isriuion is p k ( ) (0) ( 1) ( ) ( p1)( p) m p p1 p p p 711

10 Secion on Survey Reserch Mehos JSM 009 km ( ) ( p1)( p) p p1 p ( p 1) c ( p ) c. (1) Proof. The resul cn e prove y mhemicl inucion. When p = 1, k ( m) km ( ) c c '(0) ( ) ( ). The ove formul is he sme s wh we oserve in equions (1) n (1) for p = 1. Suppose i is rue for p = h. Th is, h k ( ) (0) ( 1) ( ) ( h1)( h) Then when p = h + 1, m h h1 h h h km ( ) ( h1)( h) h h1 h ( h 1) c ( h ) c. k ( m) (0) [( h1) 1] [( h1) ] [( h1) 1][( h1) ] h 1 ( h 1) ( h 1) 1 ( h 1) km ( ) ( h1) ( h1) 1 ( h1) [( h1) 1] c [( h1) ] c. [( h1) 1][( h1) ] The ove shows he heorem hols when p = h Concluing Remrks If pulic use micro file is mske y muliplicive noise which follows he runce ringulr isriuion, hen users nee o know he momens of he runce ringulr isriuion in orer o esime he momens of he originl 71

11 Secion on Survey Reserch Mehos JSM 009. The es wy of eriving momens of isriuion is eveloping he chrcerisic funcion of he isriuion. Thus, in his pper, we evelope he chrcerisic funcion of he runce ringulr isriuion. Using he chrcerisic funcion, we erive he momens of he isriuion. Finlly, we generlize he formul for he momens. 5. References 1. Hwng, J.T. (1986) Muliplicive Errors-in-Vriles Moels wih Applicions o Recen D Relese y he U.S. Deprmen of Energy, Journl of he Americn Sisicl Associion, Vol. 81, No. 95, Evns, T., Zyz, L., n Sln, J. (1998) Using Noise for Disclosure Limiion of Eslishmen Tulr D, Journl of Officil Sisics, Vol. 1, No., pp Kim, J..J. n Winkler, W. E. (001) Muliplicive Noise for Msking Coninuous D, Proceeings of he Survey Mehos Reserch Secion, Americn Sisicl Associion, CD Rom.. Personl communicion in 007 wih Aou, J., Cornell Universiy. 5. Kim, J..J. (007) Applicion of Trunce Tringulr n Trpezoil Disriuions for Developing Muliplicive Noise, Proceeings of he Survey Mehos Reserch Secion, Americn Sisicl Associion, CD Rom. 71