Power Transformations and Unit Mean and Constant Variance Assumptions of the Multiplicative Error Model: The Generalized Gamma Distribution

Transcription

1 Briish Journl of Mhemics & Comuer Science (): 88-06, 0 SCIENCEDOMAIN inernionl Power Trnsformions nd Uni Men nd Consn Vrince Assumions of he Mulilicive Error Model: The Generlized Gmm Disribuion J. Ohkwe * nd D. C. Chikezie Dermen of Mhemics/Comuer Science/Physics, Fculy of Science, Federl Universiy, Ouoke, P.M.B. 6, Yengo, Byels Se, Nigeri. Dermen of Sisics, Fculy of Biologicl nd Physicl Sciences, Abi Se Universiy,P.M.B. 000, Uuru, Abi Se, Nigeri. Auhors conribuions The firs uhor JO conceived he ide of his er, however he heoreicl resuls were joinly esblished by boh uhors. The second uhor DCC mde he firs drf subjec o suervision by he firs uhor. Finlly boh uhors red nd roved he finl mnuscri. Originl Reserch Aricle Received: 5 Augus 0 Acceed: 5 Seember 0 Published: Ocober 0 Absrc Aims: To sudy he imlicions of ower rnsformions nmely; inverse-squre-roo, inverse, inverse-squre nd squre rnsformions on he error comonen of he mulilicive error nd deermine wheher he uni-men nd consn vrince ssumions of he model re eiher reined or violed fer he rnsformion. Mehodology: We sudied he disribuions of he error comonen under he vrious disribuionl forms of he generlized gmm disribuion nmely; Gmm (, b, ), Chi-squre, Exonenil, Weibull, Ryleigh nd Mxwell disribuions. We firs esblished he funcions describing he disribuionl chrcerisics of ineres for he generlized ower rnsformed error comonen nd secondly lied he uni-men condiions of he unrnsformed disribuions o he esblished funcions. Resuls: We esblished he following imorn resuls in modeling using mulilicive error model, where d rnsformion is bsoluely necessry;(i) For he inverse-squre-roo rnsformion, he uni-men nd consn vrince ssumions re roximely minined for ll he disribuions under sudy exce he Chi-squre disribuion where i ws violed. (ii) For he inverse rnsformion, he uni-men ssumions re violed fer he rnsformion exce for he Ryleigh nd Mxwell disribuions. (iii) For he inverse-squre rnsformion, he uni-men ssumion is violed for ll he disribuions under sudy. (iv) For he squre rnsformion, i is only he Mxwell disribuion h minined he uni-men ssumion. (v) *Corresonding uhor: ohkwe.johnson@yhoo.com;

2 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 For ll he sudied rnsformions he vrinces of he rnsformed disribuions were found o be consn bu greer hn hose of he unrnsformed disribuion. Conclusion: The resuls of his sudy hough resriced o he disribuionl forms of he generlized gmm disribuion, however hey rovide useful frmework in modeling for deermining where riculr ower rnsformion is successful for model whose error comonen hs riculr disribuion. Keywords: Error comonen; men; mulilicive error model; ower rnsformion; vrince. Inroducion A mulilicive error model (MEM) is defined by [] s X (), N = µ ξ where X, N is rel-vlued, discree ime sochsic rocess defined on [0, + ), µ, defined condiionlly on Ψ = µ ( θ, Ψ ) is osiive quniy h evolves deerminisiclly ccording o he rmeer vecor, θ. Ψ is he informion vilble for forecsing X, N nd ξ is rndom vrible wih robbiliy densiy funcion defined over [0, + ) suor wih uni men nd unknown consn vrince, ξ ~ V, ( ) +. Th is There is no quesion h he disribuion of ξ in () cn be secified by mens of ny robbiliy densiy funcion (df) hving he chrcerisics in (). Exmles re Gmm, Logφ, φ (which imlies Norml, Weibull, nd mixures of hem []). [] fvor Gmm = φ ); [], in Auoregressive Condiionl Durion (ACD) model frmework considered Weilbull ( + φ ), φ (in his cse, ( φ ) ( φ ) () = + + ). As resul of he bove suggesed secificions, he error comonen ξ would be generlly sudied under he generlized gmm disribuion (GGD) which ccording o [] cn be reresened by = ξ f ( ξ ) bc ( ξ ) ( ξ ) c e =, ξ > 0 c () where ( she rmeer) nd b re rel numbers. c cn in rincile ke ny rel vlue bu normlly we consider he cse where c 0. The reson of using he GGD s he sudy disribuion is becuse he vrious disribuionl forms (The -rmeer gmm, Chi-squre, Exonenil, Weilbull, Ryleigh nd Mxwell disribuions) of he GGD for vrious vlues of, b nd c, hve 89

3 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 he disribuionl chrcerisics given in (). The disribuionl forms of () for vrious vlues of, b nd c re given in Tble. For more deils on he generlized gmm disribuion, see [5]. Tble : Relion of he GGD o oher Disribuions S/N Generlized Gmm Disribuion (GG(, b, c)) b c Gmm, b, b Gmm ( ) Chi- squre n Exonenil α Weibull α 5 Ryleigh 6 Mxwell I is no n oversemen o sy h sisics is bsed on vrious d rnsformions. Bsic sisicl summries such s smle men, vrince, z-scores, hisogrms, ec., re ll rnsformed d. Some more dvnced summries such s rincil comonens, eriodogrms, emiricl chrcerisics funcions, ec., re lso exmles of rnsformed d. According o [6], rnsformions in sisics re uilized for severl resons, bu unifying rgumens re h rnsformed d ; (i) re esier o reor, sore nd nlyze (ii) comly beer wih riculr modeling frmework nd (iii) llow for ddiionl insigh o he henomenon no vilble in he domin of non-rnsformed d. For exmle, vrince sbilizing rnsformions, symmerizing rnsformions, rnsformions o ddiiviy, llce, Fourier, Wvele, Gbor, Wigner-Ville, Hugh, Mellin, rnsforms ll sisfy one or more of oins lised in (i iii). Mny imorn resuls in sisicl nlysis follow from he ssumion h he oulion being smled or invesiged is normlly disribued wih common vrince nd ddiive error srucure. For he mulilicive error model where normliy ssumion is ou of he quesion, he ssumions of ineres re h he error comonen hs uni men nd consn vrince. When he relevn heoreicl ssumions reling o seleced mehod of nlysis re roximely sisfied, he usul rocedures cn be lied in order o mke inferences bou unknown rmeers of ineres. In siuions where he ssumions re seriously violed severl oions re vilble [7]: (i) Ignore he violion of he ssumions nd roceed wih he nlysis s if ll ssumions re sisfied. (ii) Decide wh is he correc ssumion in lce of he one h is violed nd use vlid rocedure h kes ino ccoun he new ssumion. (iii) Design new model h hs imorn secs of he originl model nd sisfies ll he ssumions, e.g. by lying roer rnsformion o he d or filering ou some susec d oin which my be considered oulying. (iv) Use disribuion-free rocedure h is vlid even if vrious ssumions re violed. For more deils on he bove lised oions, see [8]. 90

4 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 Mos reserchers, however, hve oed for (iii) which hs rced much enion s documened by [9] nd [0] mong ohers. In his sudy our ineres would cener on rnsformion s remedy for siuions where he ssumions for rmeric d nlysis re seriously violed. D rnsformions re he licions of mhemicl modificions o he vlues of vrible.however cuion should be exercised in he choice of he ye of rnsformion o be doed so h he fundmenl srucure of he series is no disored nd hereby rendering he inerreion very difficul or imossible. There re wo mjor mehods of d rnsformions nmely Brle nd Box nd Cox mehods of d rnsformion, however for ese of licion we would only consider he Brle s echniques. [] used he simle relion beween men nd sndrd deviion over severl grous for choice of rorie rnsformion. [] hd shown how o ly Brle s rnsformion echnique o ime series d using he Buys-Bllo ble. For deils on Buys-Bllo ble, see []. According o [], he relionshi beween vrince nd men over severl grous is wh is needed for choice of rorie rnsformion. If we ke rndom smles from oulion, he mens nd sndrd deviions of hese smles will be indeenden (nd hus uncorreled) if he oulion hs norml disribuion []. [] showed h Brle s rnsformion for ime series d is o regress he nurl logrihms of he grou sndrd deviions ˆ,,,..., i. i m X,,,..., i. i = m nd ( = ) gins he nurl logrihms of grou mens deermine he sloe, of he relionshi log ˆ = α + log X + error () e i. e i. For non-sesonl d h require rnsformion, we sli he observed ime series X, =,,..., n chronologiclly ino m firly equl differen rs nd comue ( X, i,,..., i. = m ) nd ( ˆ i., i,,..., m) = for he rs. For sesonl d wih he lengh of he eriodic inervls, s, he Buys-Bllo ble nurlly riions he observed d ino m eriods or rows for esy licion. []) lso showed h Brle s rnsformion my lso be regrded s he ower rnsformion X ( ), (5) e { log X, = Summry of rnsformions for vrious vlues of is given in Tble. 9

5 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 Tble : Brle s Trnsformion for some vlues of. S/N Required Trnsformion 0 No rnsformion X log e X X 5 X 6 7 X X Recenly here re vrious sudies on he effecs of rnsformion on he error comonen of he mulilicive error models whose disribuionl chrcerisics is given in (). The overll im of such sudies is o esblish he condiions for successful rnsformion [5]. According o [5], successful rnsformion is chieved when he desirble roeries of d se remin unchnged fer rnsformion. For he MEM where he normliy ssumion of he error comonen is ou of he quesion, herefore in his sudy we shll be ineresed in he uni men nd consn vrince ssumions. For he urely mulilicive ime series model whose error comonen in ddiion o being normlly disribued is clssified under he chrcerisics given in (), [6], [7] nd [8] hd resecively invesiged he effecs of logrihm, squre nd inverse rnsformions on he error comonen, ξ, where ~ N (, ) ξ. [6] discovered h he logrihm rnsform; Y = Log ξ cn be ssumed o be normlly disribued wih men, zero nd he sme vrince, for < 0.. Similrly [7] discovered h he squre rnsform; Y = ξ is successful in he inervl 0 < 0.07, where is he sndrd deviion of he originl error comonen before rnsformion wheres [8] discovered h he inverse rnsform Y = cn be ssumed o be normlly disribued wih men, one nd he sme vrince ξ. rovided 0.0 The licion of ower rnsformion o model () gives X (6) * * * N =, µ ξ 9

6 Briish Journl of Mhemics & Comuer Science (), 88-06, 0, N, N, N * X = X = X, µ = µ * where, ξ * ξ ξ = = nd + ξ ~ V (, ), =,,,,, (7) where ξ is he generlized ower rnsformed error comonen of model (). The mos oulr ower rnsformions re log X, X,/ X, X nd e / X.The resuls of he rnsformions on model () re given in Tble. The logrihm rnsformion convers he mulilicive error model () o he ddiive model while he oher lised rnsformions leve he model sill mulilicive. For he logrihm rnsformion Y = log X = log µ + log ξ = µ + ξ (8) * * e e e In Tble he following noions were doed Y = Trnsformed observed series * µ = The Trnsformed funcion of µ * ξ = Trnsformed error comonen I is cler from Tble, h only he logrihm rnsformion lers he ssumions lced on he error comonen of he mulilicive error model nd s resul ineres in his er would be cenered on he rnsformions h leves model () sill mulilicive. Tble : Trnsformions of he mulilicive Error Model * Y µ ξ Model for Y log e X log e µ log e X µ / X / µ / X µ / X / µ / / X /µ Assumion on ξ Addiive * + ξ ~ ( V 0, ξ Mulilicive * + ξ ( ~ V, ) ξ Mulilicive * + ξ ~ ( V, ξ Mulilicive + ξ ( ~ V, ) ξ Mulilicive * + ξ ( ~ V, ) / ξ Mulilicive + ξ ( ~ V, ) * e 9

7 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 Since model (6) is sill mulilicive error model nd herefore * ξ mus lso be chrcerized wih uni men nd some consn vrince, which my or my no be equl o. Thus, in his er, we wn o sudy he effec of ower rnsformions on non-norml disribued error comonen of mulilicive error model whose disribuionl chrcerisics belong o he Generlized Gmm disribuion. The urose is o deermine if he ssumed fundmenl srucure of he error comonen (uni men nd consn vrince) is minined fer he ower rnsformion nd lso o invesige wh hens o nd in erms of equliy or nonequliy. According o [5], he overll reson for concenring on he error comonen of model (6) is s lne s he nose on he fce: he reson is h he ssumions for model nlysis re lwys lced on he error comonen. In his ferile cdemic minefield, [5] hd sudied he imlicion of squre roo rnsformion on he uni men nd consn vrince ssumions of he error comonen of model () whose disribuionl chrcerisics belong o he Generlized Gmm Disribuion for he vrious forms; Chi-squre, Exonenil, Gmm (, b, ), Weibull, Mxwell nd Ryleigh disribuions. From he resuls of he sudy, he uni men ssumion is roximely minined for ll he given disribuionl forms of he GGD. However here were reducions in he vrinces of he disribuions exce hose of he Gmm (, b, ), for >, Ryleigh nd Mxwell h incresed, hence hey concluded h squre-roo rnsformion is no rorie for mulilicive error model wih Gmm (, b, ) for > or Ryleigh or Mxwell disribued error comonen. Finlly, [5] recommended h squre-roo rnsformion, where licble for mulilicive error model re successful for he sudied disribuions if he vrince of he rnsformed error comonen < 0.5. Ou of he six oulr ower rnsformions nmely logrihm, squre roo, inverse, inverse squre roo, squre nd inverse squre, only he effec of squre roo rnsformion on he error comonen of he mulilicive error model wih regrd o uni men nd consn vrince hd been sudied by [5], leving he ohers ye o be sudied. Wheres he logrihm rnsformion convers he mulilicive error model o n ddiive model, he ohers sill leve i mulilicive herefore in his er we sudy he effec of he ower rnsformions nmely; inverse squre roo, inverse, inverse squre nd squre on he error comonen of he mulilicive error model wih he overll im of invesiging on wh hens o he uni men nd consn vrince ssumions fer he rnsformions. This sudy would be crried ou under he generlized gmm disribuion considering h ll he suggesed disribuions of he error comonen of he mulilicive error model ([]; [];[]) re he vrious forms of he generlized gmm disribuion. This er is orgnized s follows; Secion One conins he inroducion while he disribuionl chrcerisics of he generlized ower rnsformed error re conined in Secion Two. While he resuls of he sudy re in Secion hree, he conclusion, cknowldgemens, uhors conribuions nd references re resecively conined in Secions four, five, six nd seven.. For simliciy, he noion ξ = ξ nd Y = Y would be doed. 9

8 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 DISTRIBUTIONAL CHARACTERISTICS OF THE GENERALIZED POWER TRANSFORMED ERROR COMPONENT Suose he disribuionl chrcerisics of ξ belongs o he generlized gmm disribuion whose robbiliy densiy funcion denoed s f ( ξ ) is (), le where is ower rnsformion, herefore y ξ = (9) ξ = y nd d ξ = d y y herefore bsed on he resul in [], he robbiliy densiy funcion of y would be given by c y ( bc ) bc c y e f y = y bc bc ( y ) c y e =, y > 0 b c (0) (0) is robbiliy densiy funcion (df) nd in wh follows, we hve o show h is inegrl is uniy ( Th is, f By doing he subsiuion 0 y d y = ). We roceed s follows; bc bc c ( y ) f ( y) d y = y e d y () 0 0 c, 0 c w= y < w< () 95

9 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 in (0) we hve he following resuls c c c w d y w w y = ; = ; d y = d w () d w c c Now subsiuing he resuls in () nd () ino () we obin bc bc c c c w w w = 0 c 0 () f y d y e d w Afer series of lgebric evluion in () we hve h 0 bc b f ( y) d y = =, bc Hving shown h (0) is roer df by showing h is inegrl is uniy, we now roceed o obin is generlized k h momen s follows, where k is osiive ineger: By definiion E Y = y f y d y, hence k k 0 bc bc k k c k ( y ) E ( Y ) = y f ( y) d y = y y e d y = 0 0 bc k c ( y ) y e d y (5) bc + 0 By lying he subsiuion in () nd is resuls in () ino (5), we obin bc + k bc c c w 0 k c w w E ( Y ) = e d w b c = bc bc + k k ( ) b + c w w e d w 0 c c 96

10 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 = k b + c k (6) For k = nd, we obin E Y = E Y = b + c b + c (7) (8) Hence he vrince of Y denoed s is given by b+ b c + c = (9) The men nd vrince of he unrnsformed disribuion given in () hd been obined by [5] s nd E ( ξ ) = b + c = Vr ( ξ ) = E ( ξ ) ( E ( ξ )) = b b + + c c (0) () The mens nd he vrinces of he vrious forms re given in [5]. For he uni men condiion nd is imc on he vrince for he vrious forms of he disribuions under sudy s obined by [5], see Tble. 97

11 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 Tble : Condiion for Uni Men nd is imlicion on he Vrince of he secil cses of he originl GGD Disribuion Men Condiion for Uni Vrince Men Gmm (, b, ) b = b Chi- squre n n = α α = Exonenil.0 Weibull α = = α α.0 Ryleigh π = ( π ) π π = 0. Mxwell π π = = 0. π 8 Similrly for he vrious ower rnsformions under sudy nmely, inverse ( = ), inverse squre roo ( = ), inverse squre ( = ) nd squre ( = ), he corresonding mens nd vrinces of he vrious disribuions under sudy re obined nd he resuls re given in Tbles 5-8 while he mens nd vrinces resuling from he licions of he uni men condiions of he unrnsformed disribuions o hose of he rnsformed disribuions re given in Tbles 9-. RESULTS AND DISCUSSION Tbles 5 hrough 8 give he heoreicl exressions for he men nd vrinces of he vrious forms fer inverse squre roo, inverse, inverse squre nd squre rnsformions resecively. I is imorn o noe h he men nd vrince of he exonenil disribuion s well s he vrince of he Ryleigh disribuion re undefined (Tble 8) for squre rnsformion. In Tbles 9 hrough, he mens nd vrinces of he rnsformed disribuions resuling from he licions of he uni-men-condiions for he vrious forms of he unrnsformed GGD re resecively given for he vrious rnsformions: (Tble 9 for inverse-squre-roo; Tble 0 for inverse; Tble for inverse-squre; Tble for squre rnsformions). I is seen in Tble h he mens nd vrinces resuling from he licion of he uni-men-condiions of he unrnsformed disribuion o he rnsformed disribuion re osiively defined for he Gmm (, b, ) where = b >, bu undefined for he Chi-squre, Exonenil nd Weibull disribuions. Furhermore he vrince of he Ryleigh disribuion is no lso defined under his condiion. For he inverse-squre-roo rnsformion (Tble 9), exce he Chi-squre disribuion h hs men.0 o he neres whole number, he mens of ll he oher forms under sudy re roximely uniy o he neres whole number, however, he vrinces incresed. Th is 98

12 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 > for ll he disribuions. Here he uni-men nd consn vrince ssumions re roximely minined for ll he disribuions under sudy exce he Chi-squre disribuion where he uni men ssumion is violed. Tble 5: Men nd Vrince of he secil cses of he originl GGD under inverse squre roo rnsformion = Vr ξ Disribuion Men E( ξ ) Vrince Gmm (, b,) b + Chi- squre n + n Exonenil α Weibull α α Ryleigh 7 Mxwell 9 b + b( b+ )( b+ ) n + n( n+ )( n+ ) 8 n α 8 6 α α α α For he inverse rnsformion (Tble 0), only he mens of he Ryleigh nd Mxwell disribuions re roximely.0, ohers hve men.0 o he neres whole number. Also > for ll he disribuions. Here he uni-men ssumions re violed exce for he Ryleigh nd Mxwell disribuions. Furhermore, for he inverse-squre rnsformion (Tble ), hey were increses in he vrinces, > for ll he disribuions however he mens re ll.0 o he neres 99

13 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 whole number. Under his rnsformion, he uni-men ssumion is violed for ll he disribuions under sudy. Finlly, for he squre rnsformion (Tble ), eiher he men or he vrince or boh re undefined for he vrious disribuions exce he Mxwell disribuion h minined he uni men even hough is vrince incresed fer he rnsformion. Here i is only he Mxwell disribuion h minined he uni-men ssumion. Tble 6: Men nd Vrince of he secil cses of he originl GGD under inverse rnsformion = Disribuion Men E( ξ ) Vrince Vr ( ξ ) b( b+ ) b( b+ ) Gmm (, b, ) ( b+ )( b+ ) b ( b+ ) Chi- squre n( n+ ) n( n+ ) ( n+ )( n+ 6) n( n+ ) Exonenil Weibull α α α ( 6 α ) α Ryleigh Mxwell α α α α 6 Tble 7: Men nd Vrince of he secil cses of he originl GGD under inverse squre rnsformion = Disribuion Men E( ξ ) Vrince Vr ( ξ ) b( b+ )( b+ ) b( b+ )( b+ ) Gmm (, b, ) 6 ( b+ )( b+ )( b+ 5 ) b ( b+ )( b+ ) Chi- squre n( n+ )( n+ ) n( n+ )( n+ ) ( n+ 6)( n+ 8)( n+ 0) n( n+ )( n+ ) Exonenil 6α Weibull α α Ryleigh Mxwell α α ε α α

14 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 Tble 8: Men nd Vrince of he secil cses of he originl GGD under squre rnsformion = Disribuion Men E( ξ ) Vrince Vr ( ξ ) Gmm (, b,) b b ( b ) ( b ) Chi- squre n n ( n ) ( n ) Exonenil Undefined Undefined Weibull, α > α, α > α α Ryleigh Undefined Mxwell ( ( )) CONCLUSION In his sudy, we invesiged he imlicion of ower rnsformions nmely, inverse-squreroo, inverse, inverse-squre nd squre rnsformions on he uni-men nd consn vrince ssumions of he error comonen of he mulilicive error model. The disribuions of he error comonen sudied were he vrious forms of he generlized gmm disribuion nmely Gmm (, b, ), Chi-squre, Exonenil, Weibull, Ryleigh nd Mxwell disribuions. The urose of he sudy is o invesige on wheher he uni-men nd consn vrince ssumions necessry for modeling using he mulilicive error model re eiher violed or reined fer he vrious ower rnsformions. Firsly, he funcions describing disribuionl chrcerisics of ineres for he generlized ower rnsformed error comonen were esblished nd secondly he uni-men condiions of he unrnsformed disribuions were lied o he esblished funcions wih view o sudying heir imcs on he rnsformed disribuion. From he resuls of he sudy, he following were discovered; (i) For he inverse-squre-roo rnsformion (Tble 9), exce he Chi-squre disribuion h hs men.0 o he neres whole number nd he Gmm(,b,) whose men nd vrince deends on he rmeer (=b), he mens of ll he oher forms under sudy re roximely uniy. However, he vrinces incresed. Th is > for ll he disribuions. Here he uni-men nd consn vrince ssumions re roximely minined for ll he disribuions under sudy exce he Chi-squre disribuion where i is violed. 0

15 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 (ii) For he inverse rnsformion (Tble 0), only he mens of he Ryleigh nd Mxwell disribuions re roximely.0, ohers hve men.0 o he neres whole number. Also > for ll he disribuions. Here he uni-men ssumions re violed exce for he Ryleigh nd Mxwell disribuions. (iii) For he inverse-squre rnsformion (Tble ), here were increses in he vrinces fer rnsformion. Th is > for ll he disribuions however he mens re ll.0 o he neres whole number. Under his rnsformion, he uni-men ssumion is violed for ll he disribuions under sudy. (iv) For he squre rnsformion (Tble ), eiher he men or he vrince or boh re undefined under he licion of he uni men condiion for he Chi-squre, Exonenil, Weibull nd Ryleigh disribuions however he Mxwell disribuion minined he uni men ssumion even hough is vrince incresed fer he rnsformion. Here i is only he Mxwell disribuion h minined he uni-men ssumion. Finlly i is imorn o noe h under his rnsformion he men nd vrince of he Gmm (,b,) re osiively defined for >. Tble 9: Alicion of he Uni Men condiion of he originl GGD nd is imlicions on he Men nd Vrince of he secil cses under inverse squre roo rnsformion Disribuion Gmm (, b,) Uni men of he Unrnsforme d Disribuion b Chi- squre n = Exonenil Weibull = ( + ) Men E( ξ ) Vrince Vr ( ξ ) =.6 = ( )( ) ( + ) + + n + ( ) n( n+ )( n+ ) 8 =.5 n α = =. α = =.. α 8 =. 6 α α α α =. = Ryleigh π Mxwell π = 7 = ( π ) 5 7 π =. ( ) 9 = 0.5 = ( ) = 0. 0

16 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 Tble 0: Alicion of he Uni Men condiion of he originl GGD nd is imlicions on he Men nd Vrince of he secil cses under inverse rnsformion = S/n Disribuion Men E( ξ ) Vrince Vr ( ξ ) ( +) ( + ) Gmm (, b, ) ( + )( + ) ( + ) Chi- squre n ( n+ ) ( n+ )( n+ 6 ) n ( n+ ) Exonenil.0 Weibull = 96.0 α = α ( α ) α α =.0 5 Ryleigh =. 6 Mxwell =. 6 =.0 α = 0.0 ( ) α α α =.6 6 = 0.9 Tble : Alicion of he Uni Men condiion of he originl GGD nd is imlicions on he Men nd Vrince of he secil cses under inverse squre rnsformion = Disribuion Men E( ξ ) Vrince Vr ( ξ ) ( + )( + ) ( + )( + ) Gmm (, b, ) 5 ( + )( + )( + 5 ) ( + )( + ) Chi- squre n( n+ )( n + ) n( n+ )( n+ ) ( n+ 6)( n+ 8)( n+ 0) n( n+ )( n+ ) = 5.0 Exonenil 6α = 6.0 = ( α ) α ( ( α )) = 68.0 α Weibull Ryleigh Mxwell = 6.0 α α 6 68α = 68.0 = 6.8 ( ( ) ) 8 =.6 6 = =.9 ( ) 0

17 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 Tble : Alicion of he Uni Men condiion of he originl GGD nd is imlicions on he Men nd Vrince of he secil cses under squre rnsformion = Disribuion Men E( ξ ) Vrince Vr ( ξ ) Gmm (, b,) ( ) ( ) Chi- squre * * Exonenil * * * * Weibull Ryleigh * =.0 Mxwell =. = 0.9 Noe: * mens undefined under he licion of he uni men condiion of he unrnsformed disribuion Finlly, exce for he Gmm (,b,) which under he licion of he uni men condiion hs men nd vrince h deend on where =b, we mke he following recommendions bsed on he resuls of his sudy; (i) Inverse-squre-roo rnsformion is rorie for Exonenil, Weibull, Ryleigh nd Mxwell disribued error comonens. (ii) Inverse rnsformion is rorie for d se whose error comonen belongs o Ryleigh or Mxwell disribuions. (iii) Inverse squre rnsformion is no rorie for d se whose error comonen belong o he six sudied disribuions. (iv) Squre rnsformion is only rorie for d se whose error comonen belongs o Mxwell disribuion. ACKNOWLEDGEMENTS The Auhors wish o exress heir rofound griude o he Referees for heir wonderful commens nd suggesions h heled o imrove he quliy of his er. Comeing Ineress Auhors hve declred h no comeing ineress exis. 0

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19 Briish Journl of Mhemics & Comuer Science (), 88-06, 0 [6] Iwueze IS. Some Imlicions of Truncing he N (, ) Disribuion o he lef Zero. Journl of Alied Sciences. 007;7(): [7] Ohkwe J, Iwuoh O, Ouonye EL. Condiions for Successful Squre Trnsformion in Time Series Modeling. Alied Mhemics, 0;: doi: 0.6/m [8] Nwosu CR, Iwueze IS, Ohkwe J. Condiion for successful inverse rnsformion of he error comonen of he mulilicive ime series model. Asin Journl of Alied Sciences. 0:6(): 5. DOI: 0.9/js Ohkwe nd Chikezie; This is n Oen Access ricle disribued under he erms of he Creive Commons Aribuion License (h://creivecommons.org/licenses/by/.0), which ermis unresriced use, disribuion, nd reroducion in ny medium, rovided he originl work is roerly cied. Peer-review hisory: The eer review hisory for his er cn be ccessed here (Plese coy se he ol link in your browser ddress br) 06