A FAMILY OF GOODNESS-OF-FIT TESTS FOR THE CAUCHY DISTRIBUTION RODZINA TESTÓW ZGODNOŚCI Z ROZKŁADEM CAUCHY EGO

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1 JAN PUDEŁKO A FAMILY OF GOODNESS-OF-FIT TESTS FO THE CAUCHY DISTIBUTION ODZINA TESTÓW ZGODNOŚCI Z OZKŁADEM CAUCHY EGO Abrac A w family of good-of-fi for h Cauchy diribuio i propod i h papr. Evry mmbr of hi family i affi ivaria ad coi agai ay o Cauchy diribuio. ul of h Mo Carlo imulaio prformd o vrify fii ampl bhaviour of h w ar prd. Kyword: Cauchy diribuio, good-of-fi, mpirical characriic fucio Srzczi W arykul zapropoowao ową rodzię ów zgodości z rozkładm Cauchy go. Każdy z rodziy afiiczi izmiiczy i zgody przciwko każd alraywi i będąac rozkładm Cauchy go. Zaprzowao akż wyiki ymulaci umryczych przprowadzoych w clu zbadaia zachowaia owych ów dla kończoych prób. Słowa kluczow: ozkład Cauchy go, zgodości, mpirycza fukca charakryycza Dr Ja Pudłko, Iyu Mamayki, Wydział Fizyki, Mamayki i Iformayki Soowa, Polichika Krakowka.

2 . Iroducio L X, X,... b a quc of i.i.d. radom variabl wih diribuio fucio F. W coidr h problm of ig h hypohi: H : F F agai H : F F whr F i h family of h Cauchy diribuio, i.. wih F F: F( x) F xm / σ, m, σ, Th locaio paramr m i h mdia ad h cal paramr σ rpr half of h irquaril rag i hi ca. I rc yar hr wr vral papr dvod o hi problm. Gürlr ad Hz [5] ad Maui ad Takmura [9] coidrd h aiic of h form whr ϕ i h mpirical characriic fucio { ( ( ) ) ( ) ( )} F ( x) / + π arca x. () D ϕ ϕ wd, of h adardizd wih uiabl imaor daa () () () ϕ () xp( iy ) ( ) ( ) Y X mˆ / σ ˆ,,...,, φ () xp( ) i h horical characriic fucio of h adard Cauchy diribuio ad w() i h wigh fucio. Th wigh w() xp( λ ) coidrd i [5, 9] rul wih imply ad clod form of h aiic, amly D, λ λ λ 4 k, λ + Y Y ( + λ) + Y ( k) λ Gürlr ad Hz [5] howd ha wih h ampl mdia ad half of h irquaril rag a h imaor of locaio ad cal, rpcivly, h bad o D,λ i coi agai ach alraiv diribuio havig a uiqu mdia ad uiqu uppr ad lowr quaril. I hi papr w propo aohr aiic of h form (). Sic h mo impora propri of a diribuio ar drmid by h bhaviour of h characriic fucio i a ighbourhood of zro, pcially i h ca of havily aild diribuio lik h Cauchy o, o hould u h wigh puig mor ma aroud zro. For hi rao w will u uboudd i zro wigh fucio. Wih h wigh w () xp( )/,

3 whr λ > ad (; ), h aiic agai ha clod form, amly 3 Sic h family of h Cauchy diribuio i clod wih rpc o h affi raformaio o i ird i affi ivaria. To obai a affi ivaria aiic of h form () i i ough o adardiz h ampl wih quivaria imaor i (), i.. imaor mˆ mˆ (,...,X ) ad σ ˆ (X,...,X ) uch ha for vry a > ad b w hav ad ( )/ Y D, λ, Γ( ) ( ( + λ) ( ( + λ) + Y ) co ( ) arca + λ ( )/ + ( + ( YYk) ) YYk λ co ( )arca. k, λ mˆ ( ax + b,, ax + b) amˆ ( X,, X ) + b ( ax + b,, ax + b) a ( X,, X ). Th prviou auhor coidrd h ampl mdia ad half of h irquaril rag [5], h maximum liklihood imaor (MLE) ad h EISE imaor [9]. Sic h u of EISE do o improv h powr of h complicaig h calculaio a h am im w do o coidr h imaor i hi papr. Th papr i orgaizd a follow. Scio coai a rviw of propri of imaor propod i [] by Pudłko. I Scio 3 ad 4 hr ar mai rul of h papr,.i. horm cocrig h wak covrgc of D,λ, wh h ampl com from h Cauchy diribuio, i limi diribuio ad h coicy of corrpodig agai ach o-cauchy alraiv diribuio. Scio 5 pr h rul of h umrical imulaio prformd o vrify h fii ampl bhaviour of h w.. Eimaor of h paramr of h Cauchy diribuio Th choic of h paramr ud o adardiz h daa i () i vry impora o h prformac of h. I hi papr bid MLE ad ordr imaor (ampl mdia ad half of h quaril rag) w will u imaor propod by Pudłko i []. Th imaor ar dfid a argum (mˆ θˆ,α, ) miimizig h diac Eimaor dfid a argum miimizig δ α ( θ) ϕ ( ) ϕθ ( ) d, + α () () () ϕ ϕ θ wd wr propod idpdly by [6] ad [] bu h auhor coidrd boudd wigh fucio w.

4 4 whr φ i h mpirical characriic fucio, φ θ () im σ i h characriic fucio of h Cauchy diribuio wih h mdia m ad h irquaril rag σ. I [] wa howd ha θˆ,α may b quivally dfid by whr Γ i h Gamma fucio ad Z arca((x m)/σ): A i wa howd i [] h family of h abov imaor ca b coiuouly clod by akig for α h ML imaor Eimaor θˆ,α ar affi quivaria, rogly coi, aympoically ormally diribud wih h covariac marix whr B i h Ba fucio ad I i h idiy marix ad hav h followig Bahadur rpraio mˆ, α l ( X ) + op(), (3) (, α σ) l( X ) + op(), wih θˆ α α α, α : argmi Γ( α) σ co Z co( αz ), θ Θ θˆ, argmax logσ log ( σ + ( X m) ). θ Θ σ Σ( θ ), I ( α) ˆ ( 3α) B( α, α) α α l ( x) co zi(( α) z), α α α α l ( x) co zc ( o(( α) z) ). α whr (5) 3. Aympoic bhaviour of h propod aiic Th followig uful rpraio of D,λ, ca b obaid by raighforward algbra ( 4) D Zˆ () Zˆ, λ, W will coidr Ẑ () a a radom lm i h Frch pac C() of coiuou fucio o h ral li dowd wih h mric d, () ( co( X ) + i( X ) (co( mˆ) + i( mˆ)) ).

5 ( 6 ρ ( f, g) ) ρ( f, g) ρ ( f, g), + 5 whr ρ (f; g) up f() g(). Now w formula h followig horm o h covrgc of h proc Ẑ. Thorm. L X X,... b a quc of idpd, idically, Cauchy diribud radom variabl. Th hr xi a crd Gauia proc Z i C() uch ha whr d do wak covrgc. If mˆ ad ar h ampl mdia ad half of h irquaril rag, rpcivly, ha h covariac krl of Z i for all,, whr For h imaor θˆ,α w hav I paricular, for h maximum liklihood imaor w hav Zˆ d Z i C( ), π π π K (,) ( J ( ) + J ( )) ( J ( ) + J ( )), J () i( x) x dx J co( x), ( ) dx. + + x 4 ( α) ( 3α) B( α, α) Γ( α, ) Γ ( α, ) ( 7) Kα (,) + + α Γ( α) Γ( ) if > α if <. () 8 K (,) + ( + ). ( ) Proof. I h ca of h MLE, h ampl mdia ad half of h irquaril rag hi horm wa provd i [5, 9] rpcivly. Hr w prov h ca of imaor θˆ,α. L S. By C(S) w do h pac of ral-valud coiuou fucio o S wih h uprmum orm. Uig h Thorm of Cörgő ad h oaio hri (Scio 3. of [3]) w will how ha Z () i wakly covrg i C(S) o h zro ma Gauia proc wih h covariac krl K α (, ). Aumpio (i), (ii) ad (vi) do o dpd o h choic of h imaor ad wr vrifid i [5]. Aumpio (iv) i a coquc of h Bahadur rpraio of h imaor θˆ,α prd i h prviou cio. I ordr o vrify h Aumpio (v) w ima

6 6 up ( l( x, θ) + Dl x ( x, θ) ) xu α α α α up max( co zi( z( α)), co zco( z( α)) xu α + max( co α α α zi( z( α)), co zco( z( α)) ) α α α ( + ) + <. α ) Hc, Ẑ covrg wakly i C(S) o h zro ma Gauia proc wih h covariac krl of h form whr By h dirc calculaio w hav T T K (,) + H(, θ ) E(( l X )( l X ) ) H (, θ ) H (, θ ), k( xlxdf,)( ) ( x) H(, θ ), k( xlxdf, )( ) ( x), H (, θ) kxd (,) θ F( x, θ). T H (, θ ) (, ). L u ow calcula x compo of K(, ). ElX l X T T (( )( ) ) E( Λ ψ( X, θ )( Λ ψ( X, θ )) ) Λ CΛ Σ( θ ) I, ( α) ( 3α) B( α, α) hu, T T H(, θ ) ElX ( ( ) l( X ) ) H(, θ ) ( α) ( + ). ( 3α) B( α, α) α α dx kxl (, ) ( xdf ) ( x) (co( ) i( ))co i( ( )) π( α) x + x z z α + x 3α π( α) π/ co α kxl (, ) ( xdf ) ( x) π( α) α π( α) (co( x) + i( x))co zi( z( α))i( a z) dz, dx ( co( x) + i( x)) + x α dx zi( z( α)) + x

7 3α 4 π/ π/ α co( a z) dz co zco( z ( α) )co( a z) dz π( α) π( α) 3α / π α co zco( z( ( ) ( ) α))co( a z) dz, α π α 7 (comp. [4] formula 3.73.). Hc, 3α π( α) + π/ 3α π( α) H (, θ ), kxlxdf (, )( ) ( x) co 3 α I h abov igral hr i miu wh > ad plu i aohr ca. I h ca of > uig h formula ad 9.4 of [4] w hav α π( α) π/ whr W i h Whiakr fucio, ad Γ(, ) do h icompl Gamma fucio. I h cod ca ( < ) by formula of Gradhy, yzhik [4] w obai Thu K α (, ) i of h form (7). Sic h covrgc of Ẑ i C(S) wa howd for ay compac S ; Ẑ covrg o Z alo i h Frch pac C() wih h mric ρ (comp. [8], p. 6). Now w pr h horm o h covrgc of h aiic D,λ,. Thorm. Udr h aumpio of Thorm w hav () 9 D zi( z( α))i( a zdz ) ( α) π/ α co zco( z( α))co( a z) dz ( ) α co zco z( α) a z dz H (, θ ), kxlxdf (, )( ) ( x) ( α )/ W α, λ, α + α ( α )/,( α)/ ( ) Γ( α) ( α) Γ( α, ) Γ( α) ( α) Γ( α, ), Γ( α) H (, θ ), kxlxdf (, )( ) ( x). σλ ˆ Zˆ () d D, : λ Z () d d. ( α).

8 8 Proof. Sic K (,) d < ad K α (,) d <, by h Tolli Thorm w hav Thu, D λ, i fii wih probabiliy. By h followig Taylor xpaio whr θ θ θˆ θ wih probabiliy, Ẑ ha h form whr Z i h followig proc ED Z λ, () d <. By raighforward calculaio i i ay o how ha h proc Z ha zro ma fucio ad h am covariac krl a h proc Z ad ha Z covrg wakly o Z i C(S). Sic hi covrgc ak plac for ay compac S, Z covrg wakly do Z i h Frch pac C(). Furhr i hi proof h followig covrgc will b dd F( x, θˆ ) F( x, θ ) θˆ θ, F( x, θ ), θ Zˆ () kxd (, ) F ( ( x) F( x, θˆ )) kxd (, ) F ( ( x) F( x, θ )) + kxd (, ) Fx ( (, θ ) F( x, θˆ )) kxd (, ) F ( ( x) F( x, θ )) ( θˆ θ ), kxd (, ) θf( x, θ) kxd (, ) F ( ( x) F( x, θ )) l( X ), H (, θ ) + l( X ), H (, θ) ( θˆ θ), H(, θ ) Z () + ( θˆ θ ), H(, θ ) H(, θ ) + l( X ) ( θˆ θ), H (, θ) Z () + ( θˆ θ ), H(, θ ) H(, θ ) + o ( ), Hθ (, ), P Z (): kxd (,) F ( ( x ) F ( x )) l X H ( ), (, ) θ ( ) co( X ) + i( X ) l ( X ) + l ( X ).

9 P ( ) (Z ˆ () Z()) d, ( ) ( Zˆ ( ) Z ( )) ad I ordr o obai () w calcula λ ( ) Z () d Z () d P P d. 9 ( ) ( Zˆ ( ) Z ( )) d ( ( θˆ θ ), H(, θ ) H(, θ ) + o ( ), H(, θ )) τ τ i i, whr τ m ˆ ad τ σ ˆ. H i ar boudd ad coiuou o h S Θ, whr S i ay compac ad Θ i clour of crai ighborhood of θ, h quc τ ad τ ar igh ad H i (, θ ) H i (; θ ) covrg o wih probabiliy for i, ; hu, i h coquc, w obai (). Covrgc () ca b obaid aalogouly. Uig h Taylor xpaio whr δ ad h Schwarz iqualiy w ima P d ( Hi (, θ ) Hi(, θ)( H (, θ ) H (, θ ) d + τiop ( ) ( Hi(, θ ) Hi(, ) H (, ) d θ θ i, op () i, + Hi (, θ) H (, θ) d, δ ( ), Z () ( 3) λσ ˆ d Z () d Z () / 4 ( Z d) λ () A i wa howd i Gürlr i Hz [5] h quc 4 ( Z () d) δ / d λ ( δ ) d /.

10 i igh. Sic wih probabiliy, h la igral i (3) covrg wih probabiliy o Γ(3 )/λ 3 ad i h coquc w obai (). Covrgc ( 4) Z () λ d d Z () λ d ca b provd aalogouly o Hz ad Wagr ([7], proof of.7, pp. -) By () ad () w hav Zˆ () / ( Zˆ ( ) Z( )) d λ Z () d / P d, / hu Zˆ () ad i coquc w hav Fially, applyig (), (4), (5) ad h Sluky Lmma w obai λσ Zˆ () d Z d Z () () hc / d λ Z () / d + o P (), ( 5) Zˆ () d Z () d op (). + D + Z () d Z () λ + Z () d Z () λ d d d, d Zˆ () d Z () d D, λ, λ,. d Th covariac krl of h proc Z drmi a igral opraor o h pac L () λ( )/ + K: L ( ) f K (,) f() d L ( ). ( ) / α Thorm (iii) of Bucu ([]) guara ha h krl of hi opraor ha h rpraio a aboluly ad uiformly covrg ri λ( )/ + ( 6) K (,) α ηφ() (), ( ) / φ

11 whr η ar igvalu of h opraor K ordrd oicraigly (... ), ad ϕ ar h corrpodig igfucio. L u dfi h followig ochaic proc ( 7) Y () ηφ () N, whr N, N, i a quc of idpd radom variabl diribud accordig o h adard ormal low. Sic h ri (7) i covrg i ma, Y i crd Gauia proc wih h covariac fucio (6), hr i h covariac fucio of h proc Takig io accou orhoormaliy of h igfucio w obai D whr L do qualiy of probabiliy law. Hc h limi diribuio of aiic D,λ, i h am a h diribuio of η N. λ, / Z (). / Z() d Y () d ηφ () N d η N, 4. Coicy I ordr o obai coi good-of-fi for h Cauchy diribuio h followig procdur ca b applid: fir w ima h paramr ad h w compu h aiic ad compar i valu wih criical valu for fixd igicac lvl. I [] i wa howd ha imaor cao b compud if, ad oly if, #{k : X θˆ,α k m}/ α for om m. For Cauchy diribud ampl h probabiliy of uch v i qual o. Thu, i ha ca h hypohi H hould b rcd. Th followig horm guara coicy of h bad o h aiic D,λ, agai ay o Cauchy alraiv. L u r ha hi horm do o impo ay rricio o h alraiv diribuio. Thorm.3. of Gürlr ad Hz [5] ca b provd aalogouly. I hi way o ca obai coicy of coidrd by Gürlr ad Hz [5] agai ay o Cauchy alraiv wihou aumpio o uiqu of h mdia ad irquaril rag. Thorm 3. L X, X,... b a quc of idpd, idically diribud radom variabl wih commo characriic fucio φ, mˆ ad b ay arlir coidrd imaor. Tha λ im/ ( 8) limif,, if ( / ) D σ λ ( m, σ) ϕ σ d Θ wih probabiliy.

12 L u oic ha righ-hadid of (8) i qual o if, ad oly if, φ i characriic fucio of h Cauchy diribuio. Proof. For poiiv coa T ad K w will do Uig ubiuio / ad applyig h Mikowki iqualiy w ima a follow Thu, w obai D T, λ, T T / im ix σ T / i( X mˆ )/ σ ˆ ˆ d imˆ ix σ ˆ, K, K σ, K : [ T/ σ, T/ σ] [ K, K]. imˆ ix ϕ(), K mˆ mˆ ϕ() λ d d d / d. / (9) D, λ, ϕ () ϕ(), K, K σ ˆ ˆ m ϕ() d d / /. Th fir igral i h la iqualiy ca b imad i h followig way, K ϕ () ϕ() up ϕ ( ) ϕ( ) σ ˆ, K T up ϕ ( ) ϕ( ). σ ˆ, K d mi( T/ σ K ˆ, ) d

13 Sic h mpirical characriic fucio covrg uiformly o ay compac wih probabiliy o characriic fucio (comp. Cöorgő [], Thorm..) h fir igral o h righ-hadiz of (9) covrg o wih probabiliy. Thrfor w obai im limif,, if ( ) D σ σ λ ( m, σ) ϕ Θ σ, K ( m, σ) Θ[ TT, ] [ σk, σk] im/ σ if ϕ( / σ) λσ d. d 3 Lig K ad T o ifiiy compl h proof. 5. Simulaio rul I hi cio w pr rul of h umrical imulaio prformd o vrify h fii ampl bhaviour of h w. Sic h rul of imulaio prd i Gürlr ad Hz [5] how h advaag of h bad o h aiic D,λ,, wih h ampl mdia ad half of h irquaril rag a imaor ovr ohr, w compar h bhaviour of h w wih h coidrd by Gürlr ad Hz. Tabl pr criical valu for four dir imaor ad diffr valu of paramr ad λ imad from ampl (for ad 5) draw from h adard Cauchy diribuio. Tabl ad 3 pr imad powr of coidrd for om alraiv for ad 5; rpcivly. I h abl N(, ) do adard ormal diribuio, do Sud diribuio wih dgr of frdom, Log(, ) do logiic diribuio, U(, ) do uiform diribuio ovr h irval (, ), La(, ) do h adard Laplac diribuio, χ do chi-quar diribuio wih dgr of frdom, G ad B do Gamma ad Ba diribuio, rpcivly. From hi abl w draw a cocluio ha uig > i h aiic D,λ, do o iflu igicaly o h, whil applyig h w imaor coidrably improv θˆ,α h powr of h. Criical valu o h igificac lvl α. Tabl 5 θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 λ λ λ λ λ λ

14 4 Eimad powr for o h igificac lvl α. Tabl Alraiv diribuio λ λ.5 λ 5 θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 N(, ) Log(, ) U(, ) La(, ) χ G(, ) B(.5,.5) λ λ λ θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 N(, ) Log(, ) U(, ) La(, ) χ G(, ) B(.5,.5)

15 5 Eimad powr for 5 o h igificac lvl α. Tabl 3 Alraiv diribuio λ λ.5 λ 5 θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 N(, ) Log(, ) U(, ) La(, ) χ G(, ) B(.5,.5) λ λ λ θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 θˆ M θˆ ML θˆ.6 θˆ.8 N(, ) Log(, ) U(, ) La(, ) χ G(, ) B(.5,.5) 99

16 6 frc [] Bucu J., Poiiv igral opraor i uboudd domai, J. Mah. Aal. Appl. 96, 4, [] Cörgő S., Mulivaria mpirical characriic fucio, Z. Wahrch. Vrw. Gabi 55, 98, 3-9. [3] Cörgő S., Krl raformd mpirical proc, J. Mulivaria Aal. 3, 983, 5-7. [4] Gradhy I.S., yzhik I.M., Tabl of igral, ri, ad produc, Acadmic Pr, Nw York Lodo Toroo 98. [5] Gürlr N., Hz N., Good-of- for h Cauchy diribuio bad o h mpirical characriic fucio, A. I. Sai. Mah. 5,, [6] Hahco C.., Th igrad quard rror imaio of paramr, Biomrika 64, 977, [7] Hz N.,Wagr T., A w approach o BHEP for mulivaria ormaliy, J. Mulivaria Aal. 6, 997, -3. [8] Karaza I., Shrv S., Browia moio ad ochaic calculu, Sprigr, Nw York 988. [9] Maui M., Takmura A., Empirical characriic fucio approach o goodof- for h Cauchy diribuio wih paramr imad by MLE or EISE, A. I. Sai. Mah. 57, 5, [] Pudłko J., Good-of- bad o mpirical characriic fucio (i polih), Ph.D. Thi, Jagilloia Uivriy, Cracow 7. [] Thoro J.C., Paulo A.S., Aympoic diribuio of characriic fuciobad imaor for h abl law, Sahya 39, 977,

1973 AP Calculus BC: Section I

1973 AP Calculus BC: Section I 97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f