Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol eg e oue fo yel gle ele fol eg ug he eleo ehque. The uy log he le of Vo(986), Chuh(989) Gh Ohbo(997). We gve eul fo he geel oe eleo eg of he fo ( L) hh e oe, fo L le h o equl o he ube of geeo of he ely gle ele geelze yl eg. We geelze eul by ele uho. Reul efyg he e of eble fol effe of he eleo eg be o he foo vlble fo he ely fol eg e gve. Sle foul fo lulg he lo of foo ue o ofoug h blo e gve. Effey h ee o he ube of ee obo eee o ee gve e of eo of he ely fol eg oe h h of he eulg eleo eg. Key o he: ol e, geelze yl eg, lo of foo. INTRODUCTION To ou gle ele fol eg hvg fo,,,,, fo oug level, e f ou gle ele ely blo eg, y, fo, fo oug level, ug y of he o eho uh h fo,,,. We he ele l level of he -h fo of elee fo ll ee obo hee fo ou y of he l elee level. If level e elee fo fo e efe o he eulg eg -h oe eleo eg. Ele o o h e oe by Boe (947). He ue fe Eule geoey o ou yel fol eg blo. Khe Svv eee he eho of fe geoee o he ouo of ble ofoue yel fol eg heeby oug he e of eleo. Joh De (975) ooe le eho of ofoug be o geelze yl eg fo AJST, Vol. 4, No. : Deebe, 003 e of geeg ee o geeo hoe h he ofoug e oul ely be eee fo hee geeo. Vo (986) oue ely ohogol ge ele fol eg blo. He ue he eleo ehque, hee he elee fo he f fo hou lo of geely o ob f oe eleo eg. Chuh (989) geelze he o by Vo (986), by oug effe gle ele eg ug he eleo ehque. Sg fo geelze yl eg, he oue effe oe gle ele eleo eg of he fo ( L). Gh Ohbo (998) Coue eleo eg of he fo ( L) hh e oe fo L le h o equl o he ube of geeo of he ely gle ele geelze yl eg. 7
Geele Deleo Deg The obeve of he ee e o gve geel eul ueful evlug he effey of he oue eleo eg be o he lo of foo o he eble effe. To ou he ely gle ele fol eg hh fo ou level fo,,, e ue he eho by Joh De (975). We eve eeo fo lo of foo ue o eleo o he eble effe. Cofoug e o he eleo eg e lo gve. NOTATION We ue he fe effe le oel y µ τ β ε (.) h hee y h h h eoe he obeve yel fo ee obo he h-h blo; τ eoe he effe of ee obo ; β h eoe he effe of he h- h blo ε h e uoele o eo h zeo e ve σ. Le y (y h ) τ τ eoe he v veo of obevo ee effe eevely eh leoghlly oee by. Th ee obo e befoe ohe ee obo... f oly f fo he f u uh h u u e hve u < u fo u The -h o of y τ bove oeo o he -h ee obo he bove gee of he v ee obo. We hll eoe he ee, he blo, he gol of blo ze he ube of blo eevely by N,A,K b. The -h o of he ee N oeo o he -h leoghlly oee ee obo. he q veo of oe of zeo ll be eoe by q 0 q eevely. A geelze eo ll be eoe by... X α hee uh h f ee he eo 0 ohee. A v o veo ll be eoe by hee h... beg S veo of oe f 0, ohee S o veo. The u ve ube eo of he geele X eo α eeee by hle hoe oeog o he eleo eg ll y o ub. LOSS O INORMATION ON ESTIMABLE EECTS We hll ue he oo ( l) D D ( l) (3.) hee D he obe fo ey by eleg he -h o f he -h level elee fo fo he ely eg - o ob euo h he - h olu of he o-h o. We hll lo e D P D (3.) hee o veo fo he olu e of he SI() J (). Ag hee I () - eol ey J () () (). ollog De (978), fo gve o veo, he lo of foo, 0, ue o ofoug h blo, gve by N K N (3.3) Whee N he ee K he gol of blo ze. We oe eleo eg of he fo ( ) h b λ fo λ l blo eh of ze K ( ) S ( l) S geele yl eg eve h K ( ) S λ b λ. Whee he ube of fo, he ube of geeo, he oe of he eleo eg λ Π b h b H C (,g),,,..., gve by Joh De (975). 73 AJST, Vol. 4, No. : Deebe, 003
M. K. GACHII We o gve geel eul o lo of foo fo gve eo effe o hee eg. Le 3.: Lo of foo ue o ofoug h blo o y -fo eo of he fo..., {,,..., } {,,..., } hoe level ee o elee fo ll he fo he eo o ob gve by... λ... hee, f... 0, ohee...... ( ) ( l)... he l blo of g hee... uh h oe o o,, e fo hee fo e hve e,,..., hh e o veo fo SI J.: () () Poof: The o veo gve (.) h beg u veo fo (-l) u veo fo ±,,...,,,...,, y of he olu of he I J fo,,.... Theefoe Bu K () () ( ) ( l) (3.4) (λ \ ( l) )I (λλ ) (3.5) NN............ (3.6) ug (3.) (3.). Hee ug (3.4),(3.5) (3.6) e ob Le 3.. Le 3.: Lo of foo ue o ofoug h blo o y -fo eo of he fo,..., {,,..., } {,,..., } hoe level ee, {,,..., } o elee {,..., } hoe level ee elee o ob gve by λ............ ( ) ( l ) ( l) hee...... gve le 3.,,..., e o veo fo I() J (),..., S S l I J. e o veo fo ( ) ( l) ( l) Poof: The o veo gve (.) h beg u veo fo (-l) u veo fo ±,,...,,,.., I y of he olu of he I () y of he olu of he ( l)i ( J l) (l) fo J fo,,...,,,,.., ().,,...,,,,.. Theefoe ( ) ( l ) ( l) (3.7) Hee ug (3.5),(3.6) (3.7) e ob Le 3. Le 3.3 Lo of foo ue o ofoug h blo o y -fo eo of fo,... fo hh level of ll he fo ee elee fo o ob gve by λ hee........ ( l ) ( l)......,..., e Poof: The o veo... gve le 3. o veo fo ( S l) I ( l) J ( l). gve (.) h beg u veo fo,,...,, (-l) u veo fo,,..., ±,,.., I y of he olu of he ( l)i ( J l) (l) fo,,.... Hee (3.8) ( l ) ( l) Hee ug (3.5),(3.6) (3.8) e ob Le 3.3 AJST, Vol. 4, No. : Deebe, 003 74
Geele Deleo Deg CONOUNDING IN DELETION DESIGNS The follog eul ofoug geele yl eg ue o Joh De (975). The ube of egee of feeo ofoue h blo fo y gve X eo α gve by Y...... Π z (4.) z hee, f, f, f 0 0 0 gve Le 3..... If he ube of egee of feeo (4.) zeo, he he eo uofoue h blo f Π ( ) he eo olly ofoue h blo. Coe y eo beee he fo, y he eo of he fo,... Le (( ) A (4.) hee fo he -h geeo,...,,,...,. g... ; Coe ll he f f ube oe he -h, -h,... -h o of A le h... be he bolue vlue f f, f. Defe follo of he ee ( ) H f, f f 0 HC(h \ {,,..., } {,,... }) f,,..., f f 0, f f > 0 < f < (4.3) I ou e he ee obo he l blo e of he fo ( u 0,,..., ;,,..., ) u g u g... u g The ube of ee he l blo h -h, -h,..., -h fo ll zeo gve by hee... Π HC(,Hf Hf ), f H 0 f g HC(... \ {,,...,g} {,,..., } f g < g... g HC( \ {,,..., } {,,..., }... f >. (4.4) hee g uh h H g 0 H g H g.. 0 g f H 0. Le Y be eoe by Y... hee h h he -h, -h,..., h -h g uy he ee zeo. The be ho h fo he eo of he fo,... ofoue h blo gve by, he ube of egee of feeo ( Y \ {,,..., } {,,..., }) Y......... g g g (4.5) We o gve he follog eul o ofoug eleo eg. Chuh (989) ue ofoug eleo eg he follog uo heoe belo ue X o he. Le α be gve eo. The he fo,... o ly { },,..., be oe o hee uully eluve ehuve e Ω, Ω, Ω 3 follo: Ω o he fo hoe level ee o elee fo o ob, h,,..., o he fo ; Ω he fo { } hoe level ee elee fo o ob hee fo e o he fol e V, h he fo {,,..., } ; Ω 3 o he fo hoe level ee elee fo o ob hee fo e he fol e V, h,,..., ; he fo { } 0,,,...,. We hll e he fol e V V (,,..., ) f... ll ohe ' e zeo, hee {,,..., } {,,..., }, he e hve he follog heoe. 75 AJST, Vol. 4, No. : Deebe, 003
M. K. GACHII Theoe : (Chuh (989)) Le he o veo (,,...,,,,..., ) v le D he v (,,...,, g,,,..., ) hee g ( ) Ω ; h g belog o he oe e of Ω eoe he e u. Whee D D g D... D D hee efe eo 3. The follog eul o eleo eg follo: Theoe If S geele yl eg geee by geeo uh h g e he (-)-h o of ey of oe g, g,..., g e he 3 l (-) o of ey of oe f > eve he ll he effe ll he eo of he f ( ) fo e olly ofoue h blo hle he effe of fo y effe volvg h fo y ube of he f ( ) fo e lly ofoue h blo ove. Poof: o he effe eo of he f fo, e hve fo he effe of fo,,,...,, h HC ( 0, 0,...,0) 0, hh gve HC (,0) hee 0 e >. Y Theefoe ll he effe of he f ( ) fo e olly ofoue h blo ug (4.5). o o fo eo of fo, <,,...,, he A (4.) he zeo hh gve H 0. Theefoe (,) HC Y o ( ) ( ) 0 e >. Hee by (4.5) ll he o fo eo of h fo e olly ofoue h blo. To ob he eul fo he geel e e oee by uo. Aue he -fo eo e olly ofoue h blo fo,,...,, h Y... ( ) 0 We h o ho h he ( ) fo eo e olly ofoue h blo. o (4.5) Y ( Y \ {,,..., } C{,,.. })...... g g g Y Y 0 Y...... ( ) ( ) ( ) ( ) heefoe ll he -fo eo,,,,-- e olly ofoue h blo. o he effe of fo e hve ( 0, 0,...,0,,0,...,0) H C h Thu (, ) H C heefoe Y < ( ) Hee oe egee of feeo fo he effe of fo ofoue h blo. Theefoe he effe of fo lly ofoue. o he o fo eo of he fo,,,...,, he A of he fo 0 AJST, Vol. 4, No. : Deebe, 003 76
Geele Deleo Deg A 0 0...000...0 0 (( )) Hee H 0. 00...00...0 0 H C (,) Theefoe Y ( ) < ( ) follo h he o fo eo of h fo e lly ofoue h blo. I geel fo he -fo eo... fo < <... <,,..., e ge... Theefoe ug (4.5) Y... ( ) 0 ( ) < ( ) ( ) ( ) 3... hee ll eo volvg fo y ube of he f ( ) fo e lly ofoue h blo. I be ho h he egee of feeo ofoue h blo e fo he effe eo of he f ( ) fo fo he effe of fo ll he eo volvg fo y ube of he f ( ) fo ove. Ag he lo of foo o he effe of fo o y o fo eo of he fo,,,...,. ue o ofoug h blo, gve by ( ) The follog eul e o he effey of he l of eleo eg eve fo he geele yl eg bove h ee o eble effe. Theoe 3: The effe of fo ll he eo volvg h fo y ube of he f ( ) fo e lly ofoue hee eleo eg. Poof: I h e V (,,...,, ) e hee {,,..., } {,,..., } fo e e,,...,. Hee Ω {,,..., }, e Ω {,,..., } Ω { φ } 3. Teefoe he oe e Ω gve by P ( Ω ) { φ, }, { }, }..., { }, {, }...,, {,,...,} } follo fo heoe h f D he V... V (,,..,, ) V (,,..,,, e e ) (,,..,)... V (,,..,,,,..,) e Bu ll hee fol e e lly ofoue hee he eul. Ug l gue be ho h ll he ohe effe eo e fully eble hee eg. Thee eleo eg e eve fo geele yl eg geee by geeo uh h g e he ( ) h o of ey of oe e he l o of ey of oe,g,..., he g hee follo h ( ) g 3 lo of foo o fo ue o ofoug h blo gve by e ( )( ) l...,,...,,..., 77 AJST, Vol. 4, No. : Deebe, 003
M. K. GACHII ug le 3. Bu f... ee obo he l blo, he 0,, fo,,..., fo,,..., hee 0,,4,..., 0,,,..., fo, 3,...,. Theefoe fo hee eleo eg.... I be ho h l,, ( S )...... 0 0 ( )( l)... f 0 l,,..., f 0 l, f 0, f 0 The follog eul o lo of foo follo. Theoe 4: Lo of foo o he effe of fo o y o fo eo of he fo,,,..., ue o ofoug h blo, ove l, gve by ( )... 4 ( ) ( ) ( ) ( ) ( )... ( ) ( )( ) ( ) ( ) 0 3 Ele Coe 4 4 geele yl eg geee by 0 0 0, g 0 0 0 g 3 0 0 0. g Thu e hve b 8 blo of ze 3. I 4 follo by Theoe h he effe of fo olly ofoue h blo hle he effe of fo he o fo eo e lly ofoue h blo. All he ohe effe eo e fully eble. Lo of foo o effe of fo o o o fo eo gve by 0.33 ( ) o he eleo eg of he fo ( 4 ),,, 4 4 he effe of fo he o fo eo e lly ofoue ug Theoe 3. All ohe effe eeo e fully eble. The lo of foo o he effe of fo o o he o fo eo o follo; () o he 4 3 3 eleo eg 3 ( 4 ) 3 4 3 3 3 ( 4 ) ( 4 ) 3 3 0 () o he 4 3 eleo eg 0.0 0.3 Noe h fo 4 6 geele yl eg geee by g 0 0 0 0 0, g 0 0 0 0 0 g 3 0 0 0 0 0, g 4 0 0 0 0 0 g 5 0 0 0 0 0. Thu 4 b 8 4. The e effe h ee o eble he 4 4 geele yl eg e ll o eble h eg e he oly oe. Lo of foo o o he o fo eo e he e. Hoeve () o he 4 5 3 eleo eg 0.98 () o he eleo eg 0.5 () o he 4 3 3 3 eleo eg 0.03 (v) o he 4 3 4 eleo eg 0. CONCLUDING REMARKS Thee eleo eg e oee oe effe h he ely geelze yl eg hey ee eve fo h ee o lo of foo. o ele fo l ug Theoe 4 bove ( ) ( ) ( ) < < ( ) o oe eleo eg he effe h ee oeble he ely geelze yl eg e he e effe h e o-eble he eulg eleo eg he e he bove ele. Lly AJST, Vol. 4, No. : Deebe, 003 78
Geele Deleo Deg he eleo eg e oe eool h he ely geele yl eg hey e eve fo beue hey eque lle ube of ee obo hu lle ube of eeel u. REERENCES Boe, R.C.(947): Mhel heoy of yel fol eg. Sy,8,07-66. Chuh,C.K(989) : Couo of effe gle ele eg.pe. De,A.M (978) : The ly of eo gle ele geelze yl Deg. Joul of he Royl Sl Soey,e.B.40,70-89. Gh,K.M (993) : O he ouo of eleo eg.ph.d. he,uvey of Nob. Gh K.M Ohbo J.W (997): Ayel gle ele eg,souh Af Sl Joul,(998) Vol 3,-8. Joh J.A De A.M (975) : Sgle ele fol eee geelze yl Deg: I Syel gee, Joul of he Royl Sl Soey, e.b. 37, 63-7. Vo, D.T(986) : oe eleo eg he ouo of effe ely Ohogol fol eg ll blo, Joul of he Ae Sl Aoo, 8,83-88. 79 AJST, Vol. 4, No. : Deebe, 003