GENERALIZED MULTIVARIATE EXPONENTIAL TYPE (GMET) ESTIMATOR USING MULTI-AUXILIARY INFORMATION UNDER TWO-PHASE SAMPLING

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1 Pak. J. Statst. 08 Vol. (), 9-6 GENERALIZED MULTIVARIATE EXPONENTIAL TYPE (GMET) ESTIMATOR USING MULTI-AUXILIARY INFORMATION UNDER TWO-PHASE SAMPLING Ayesha Ayaz, Zahoo Ahmad, Aam Sanaullah and Muhammad Hanf Depatment of Statstcs, Natonal College of Busness Admnstaton and Economcs, Lahoe, Pakstan Emal: ORIC, Lahoe Gason Unvesty, Lahoe, Pakstan Emal: Depatment of Statstcs, COMSATS Insttute of Infomaton Technology Lahoe, Pakstan. Emal: ABSTRACT In ths pape, a genealzed multvaate exponental type estmato s poposed n two-phase samplng usng mult-auxlay vaables when populaton nfomaton on some auxlay vaables s not avalable. The vaance covaance matx of the poposed estmato s deved by usng smple andom samplng wthout eplacement scheme. Some specal cases of poposed estmato ae also deduced. Fnally, empcal as well as smulaton study s conducted to assess the pefomance of poposed estmato. KEY WORDS AND PHRASES Multvaate exponental estmato; mult-auxlay vaables; two-phase samplng; vaance covaance matx, patal nfomaton case.. INTRODUCTION Ove the yeas, authos of suvey samplng ae nteested to constuct estmatos to mpove pecson of estmate usng auxlay nfomaton. Cochan (90) poposed classcal ato method of estmaton fo estmatng the populaton mean by consdeng the postve lnea elaton between study and auxlay vaable. The contbuton of Cochan (977) n ato method of estmaton unde two-phase samplng attacts a lot of attenton fo upcomng suvey statstcans. Chand (97) poposed a chan ato estmato based on nfomaton of two auxlay vaables fo the estmaton of populaton mean n two-phase samplng. Svastava et al. (990) extended the Chand (97) estmato to the genealzed chan ato estmato fo the estmaton of populaton mean n two-phase samplng. Bahl and Tuteja (99) was the fst who suggested the exponentaltype ato and poduct estmatos fo estmatng the populaton mean n sngle-phase samplng desgn and theeafte Sngh and Vshwakama (007) extended ths wok n two-phase samplng. Futhe contbuton s done by Samuddn and Hanf (007), Sngh and Solank (0) and Sanaullah et al. (0, 0) etc. 08 Pakstan Jounal of Statstcs 9

2 0 Genealzed Multvaate Exponental Type (GMET) Estmato Olkn (98) was the fst to suggest an extenson of classcal ato estmato to the multvaate ato estmato takng seveal auxlay vaables to ncease pecson of the estmate. John (969) poposed two multvaate genealzatons of ato and poduct estmatos. Tpath and Khattee (989) estmated means of seveal study vaables usng mult-auxlay vaables n smple andom samplng. Futhe, Tpath (989) extended the esults to the stuaton of two phase samplng. Ahmed (00) poposed chan based geneal estmatos usng mult-auxlay nfomaton unde multphase samplng and studed the popetes of poposed estmatos and suggested the optmum sample szes usng a modfed cost. Ahmad et al. (00a) poposed genealzed multvaate ato estmatos usng mult auxlay vaables unde patal nfomaton case fo mult-phase samplng. Futhe contbuton s by Hanf et al. (009), Ahmad et al. (009, 00b, 0 and 0), Butt et al. (0) and Ngesa et al. (0).Afte studyng the avalable wok, t was obseved that lmted lteatue s avalable on genealzed multvaate exponental estmatos fo the estmaton of populaton mean. In ths pape, ou am s to constuct genealzed multvaate exponental type estmato fo estmatng the populaton mean vecto usng mult-auxlay vaables n two-phase samplng. In Secton, the poposed genealzed multvaate exponental type estmato s pesented along wth povdng the expesson of vaance covaance matx up to fst ode of appoxmaton. Some specal cases of ou poposed estmato ae also gven n ths secton. In Secton, an empcal study s conducted to check the supeoty of poposed estmato. In Secton, a smulaton study has been conducted to demonstate the pefomance of ou poposed estmato. Dscusson, concluson and suggestons fo futue eseach wok ae pesented n Secton, 6 and 7 espectvely.. GENERALIZED MULTIVARIATE EXPONENTIAL ESTIMATOR IN TWO PHASE SAMPLING Suppose a fst phase sample of sze n s selected wth SRSWOR fom a fnte populaton of sze N and x denote the obsevatons of th auxlay vaable collected at fst phase sample whee,..., q. Then a sample of sze n s selected fom fst phase sample wth SRSWOR and y() j denote the obsevatons of jth study vaable collected at second phase sample whee j,..., p and x denote the obsevatons of th auxlay vaable collected at second phase sample whee,..., q. Let and f N n Nn and x() () f N n Nn. Suppose e y Y, e x X y() j () j j e x X ae samplng eos. It s assumed that = E e = 0 whee Y j and x vaables. x() () y j E e = E e x X ae espectve populaton means of study and auxlay Smauddn and Hanf (007) was the fst who use the temnology of full nfomaton, patal nfomaton and no nfomaton cases whle suggestng the estmatos

3 Ayesha Ayaz et al. fo populaton mean n connecton wth the avalablty of auxlay nfomaton. They gave the tem to an estmato as full nfomaton case estmato n whch the nfomaton on all auxlay vaables s known, a patal nfomaton case estmato n whch nfomaton on some auxlay vaables s known and the no nfomaton case n whch nfomaton on auxlay vaables s not known. We suggested the followng genealzed multvaate exponental estmato consdeng patal nfomaton case. We constucted ths estmato n such a way that we can deduce ts specal cases that ae full nfomaton case as well as no nfomaton case and these specal cases ae dscussed below. Theefoe, ths estmato s thee n one. Suppose we have q auxlay vaables fom whch the nfomaton on fst auxlay vaables n known fo populaton and fo emanng s q s not then the multvaate estmato can be constucted as wth t whee, t p t j p x x X x x x. () () () () sq () () j y() j exp j j j x() x() X x () x() x() j and j covaance matx of t p. ae unknown constant to be detemned by mnmzng the vaance To poceed fo vaance covaance matx fo fst ode of appoxmaton, we stat wth unvaate case usng mult-auxlay vaables as o o t x x X x x x. () () () sq () () j y() j exp j j j x() x() X x () x() x() t Y e exp j j y j j e x x () () e X e e x x () () x sq x x j j () () () exp exp, e e e () X e X e e x() x() x() e e () () e e () () t j Yj ey exp j X X j x x x x e e () () e e () () jex e () x sq j x x x x () exp exp. X X X X ()

4 Genealzed Multvaate Exponental Type (GMET) Estmato o o Usng Bnomal expanson and gnong second and hghe ode tems, we have e e () () e e () () j x x x x t j Yj ey exp... j X X e e () () e e () () jex ex sq j x x x x X X X X () () () exp... exp.... Ignong second and hghe ode tems, we have t Y e e e () () jex sq e e () () j x x j x x. () X X X () j j y exp j Appoxmatng the exponental functon up to ode one, we have t Y e e e () () jex sq e e () () j x x j x x. (6) X X X () j j y j jyj jyj sq jyj () () () () () j t Y e e e e e e. (7) j j y x x x x x X X X sq * ** () () () () () j t Y e a e e b e a e e. (8) j j y j x x j x j x x Fo multvaate case of j,,..., p, we can wte the above equaton n matx fom as whee t p = Y + d y + Ad - Bd + Ads, (9) Y [ Y ], d [ e ] d j p e e s x x () (), d ex e () x () *, A [ aj ] p, B [ bj ] p y y j p s, d e, x() and A [ a ]. Fom (9), the vaance covaance matx of t p can be wtten as Σ t E (t Y)(t Y) p p p ** j ps

5 Ayesha Ayaz et al. o (0) d y + A d - Bd + A d s d y + A d - Bd + A d s E. y s s y s Σt E d d + d A - d B + d A + A d d + d A - d B + d A p y y s y s We can have -Bd d + d A - d B + d A +A d d + d A - d B + d A. () dyds Σ y x, d d Σ x, E d d f Σ, E d d f f Σ, E d d f Σ, y y y y y x y y x p p p E f f E f f p s d d Σ and d d Σ. E d d f f Σ, E d d f f Σ, E d d f Σ, x s x x x s E f f E f f s x x s s x s s Then Σ f Σ f f Σ A f Σ B f f Σ A f f f f f f f f f f f f f f A Σ B f f A Σ A t y y x y x y x p p p p p s y x x x x x y x x x x x y x x x A Σ A Σ A A Σ B p ( f f )A Σ A f BΣ f f BΣ A f BΣ B s p BΣ A A Σ A Σ A s p s s xx s Fo optmum values of unknown matces A, B and A, dffeentatng () wth espect to these matces one by one and equatng to zeo and then solvng the esultng nomal equatons we can have y, and px x x x s ypxs s x xs x ypx x ypxs ypx x x xs s s x s x xs x x xs A Σ Σ Σ Σ G Σ Σ B Σ Σ A Σ Σ Σ Σ G, whee G Σ Σ Σ Σ Usng nomal equatons used to get optmum values of unknown matces A, B and A, fom (), we can wte x s () () Σt E d d + d A -d B + d A p y y s f Σ f f Σ A f Σ B f f Σ A () yp ypx ypx ypxs Now substtutng the optmum values of unknown matces A, B and A, we get afte smplfcaton

6 Genealzed Multvaate Exponental Type (GMET) Estmato Σ Σ Σ Σ Σ Σ Σ y Σ pxs ypx x x x G s s y px s x xs x ypx Σ f Σ Σ Σ Σ f f t y y x x y x p p p p. () Sngh and Majh (0) poposed chan ato estmato based on nfomaton of two auxlay vaables as, () () exp x x X t y x x x wth mean squae eo () () () f y x yx y x x yx y x (6) MSE t Y f C C C C f C C C. (7) We ae dealng wth multvaate estmatos, then (6) can be modfed to multvaate case fo p study vaables. The multvaate veson of (6) can be wtten as whee, t t g j p t () exp j x x sq X y, fo j,..., p. (8) x x x () j j j of The matx of optmum values of j X j Yj whee j p s the x A Σ Σ and of j Xj Yj whee j ypx s th j element of th j element ypxs xs C Σ Σ. Notce that the mnmum MSE of (8) fo unvaate case usng two auxlay vaables wll be dentcal to the MSE of (6) gven n (7). The expesson of mnmum vaance covaance matx up to fst ode of appoxmaton can be wtten as p p p f p p p s s p s Σ f Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ. (9) g y y x x y x y x x y x y x x y x The poposed estmato t p can be compaed wth modfed estmato t g. Fo compason empcal and smulaton study s gven n the followng sectons... Specal Cases As the poposed estmato gven n () s geneal n natue and many specal membes of ths geneal class can be deduced. Some mpotant multvaate membes usng mult auxlay vaables ae gven n the followng Table. Fom ths lst futhe unvaate membes can be deduced fo dffeent numbe of auxlay vaables. The optmum values and MSE/vaance covaance matces of any membe can easly be deduced fom () and ().

7 Ayesha Ayaz et al. S# Expesson of t j t p t p t p y y x () () () exp q j j x() x() () () exp q j j X x() X x (MEE fo NIC) x (MEE fo FIC) Optmum Matces Σ Σ ypxq ypxq Σ Σ xq Vaance Covaance Matces fσ y f f p y x x y x Σ Σ Σ p q q p q xq Σ Σ Σ f y Σ p ypxq x q ypxq x() x() j x() x f () y() j exp Σ Σ y px x, sq X x() j Σ y Σ pxs xs f X x () (MEE fo PIC) MEE: Multvaate Exponental Estmato; NIC: No Infomaton Case; FIC: Full Infomaton Case; PIC: Patal Infomaton Case y p ypx x ypx Σ Σ Σ Σ Σ y Σ Σ px x ypx Σ y Σ Σ pxs x s ypxs It s obvous that the geneal class wll always be effcent than ts membes, howeve, by educng the numbe of exponental functon based on auxlay vaables, the bas can possbly be educed. So the specal cases ae not ncluded n the empcal and smulaton study.. EMPIRICAL STUDY Fo empcal study, we use the data of 998 census epots of fve dstcts of Punjab, a povnce of Pakstan, Jhang, Fasalabad, Gujat, Kasu and Salkot. The detal of populatons and vaables descpton s gven n Table A and A espectvely of Appendx-A. We consde thee vaables of nteests denoted by Y s and fve auxlay vaables denoted by X s fo computng the detemnants of vaance covaance matces of poposed multvaate exponental estmato and modfed Sngh and Majh (0) estmato. The vaance covaance matces of study and auxlay vaables used to compute the vaance covaance matces ae gven n Table A fo all fve populatons. The vaance covaance matces of t p and t g along wth pecent elatve effcency (PRE) based on the detemnants and tace ae gven n Table A.. SIMULATION STUDY In ths secton, a smulaton study based on atfcal data of sze 000 s conducted to assess the pefomance of poposed estmato gven n () ove modfed Sngh and Majh (0) estmato gven n (0). Fve auxlay vaables ae geneated usng nomal and lognomal dstbuton so that the pefomance can be assessed fo both symmetc and skewed data. Futhe thee esponse vaables ae geneated usng lnea egesson model wth dffeent egesson coeffcents. Fo symmetc data the auxlay vaables ae

8 6 Genealzed Multvaate Exponental Type (GMET) Estmato geneated by; x N,. ; x N, ; x N8, ; x N7, x N6, and fo skewed data; x LogN, ; x LogN, ; x LogN x LogN x LogN.,.7 ;, ;,. The thee esponse vaables ae geneated by model y coeffcents fo.,.,.,.,.7. y j ae all, fo j x j j y ae. The egesson 0., 0., 0., 0., 0. and y j ae The szes of fst phase and second phase samples ae assumed to be espectvely n 00 and n 00. The numbe of eplcatons fo smulaton study s S 000. We geneated fve auxlay vaables and assumed fo fst thee vaables ( x, x, x ) and s fo last two vaables ( x, x ). The sample values of these vaables ae equed to compute the estmatos t p and t g. We computed the vecto of poposed estmatos by takng the log of () as log(t ) log(y ) A x Bx A x, (0) p () s whee A, B and Aae adjusted vesons of matces of optmum values gven n (), the need of adjustment can easly be seen fom (7) and, whee x x k () k x() k x() k x() k x x x, whee x X k k x() k Xk x() k x x, whee xs x() k x() k x() k x() k x ( x, x, x ) T x (,, ) T x (, ) T s s s k, and. () Smlaly, the modfed estmato gven (8) can be computed by log(t ) log(y ) Ax C log(x ), () g () s whee x (,, ) T x x x ; fo x x k () k x() k x() k x() k fo xs Xk x k Xk x k k () (), x ( x, x ) T, s s s and A and C ae also adjusted veson of the ones that ae gven below (8).The values of optmum matces ae computed fom populaton as well as fom fst-phase sample because pactcally populaton paametes ae usually not avalable. These matces ae used to compute these matces of optmum values. Fnally, we need to take ant-log of these vectos of estmatos gven n (0) and () to obtan the coect vecto of estmatos. Afte computng the vecto t p and t g, wth thee elements each, we obtaned the mean and vaance-covaance matx by,

9 Ayesha Ayaz et al. 7 whee y t k and Σy cov( tk, tl ) t k ( ) S tk S and cov( t, t ) ( t t ) t t S. () k l k k l l S The pefomance of unvaate estmato usng mult-auxlay vaable can be obseved fom the dagonal of vaance covaance matx of a multvaate estmato. The vaance covaance matces of study and auxlay vaables that ae necessay to fnd the matces of optmum values based on populaton symmetc and skewed data ae gven n Table B and B espectvely of Appendx B. These matces ae also computed fom fst phase sample fo each eplcaton and fo one of the eplcaton based on symmetc and skewed data ae gven n Table B and B espectvely. Table B and B6 contans the vecto of paametes and bas of both estmatos t p and t g fo symmetc and skewed data usng both cases of fndng matces of optmum values. The Table B7 contans the matces of optmum values computed fom populaton and fst phase sample data that ae used to compute t p and t g fo symmetc and skewed data. Fnally, the vaance covaance matces based on 000 smulaton of t p and t g along wth the PRE based on the detemnants and tace ae gven n Table B8.. DISCUSSION Consdeng the esults of empcal study, fom Table A, t can be seen that poposed multvaate estmato s effcent than modfed Sngh and Majh (0) estmato. Fom the esults of smulaton study the poposed multvaate estmato s effcent than modfed Sngh and Majh (0) estmato fo symmetc as well as fo skewed data. The pefomance s even bette fo skewed data. If we want to compae the pefomance of unvaate estmatos fo each study vaable sepaately usng multauxlay vaables, then t can be assessed fom the dagonals of vaance covaance matces gven n Table A of empcal study and fom Table B8 of smulaton study. Fom both tables, each unvaate estmato deduced fom poposed multvaate estmato s effcent than ts countepat of modfed Sngh and Majh (0) estmato. Compang the pefomance of poposed and modfed Sngh and Majh (0) estmato based on bas computed n smulaton study gven n Table B and B6, the bas s less fo poposed estmato fo symmetc data fo fst study vaable wheeas the pefomance of modfed Sngh and Majh (0) s bette fo othe two study vaables fo symmetc data as well as fo skewed data when optmum values ae computed fom populaton data. But when optmum values ae computed fom fst phase sample data that s actually possble pactcally, the poposed estmato has less bas than modfed Sngh and Majh (0) estmato fo both symmetc and skewed fo all study vaables. 6. CONCLUSIONS In ths pape, a genealzed multvaate exponental estmato s developed fo twophase samplng usng mult-auxlay vaables. As the poposed class s geneal n natue, some specal cases ae deduced along wth the expessons of mean squae

10 8 Genealzed Multvaate Exponental Type (GMET) Estmato eos. The poposed estmato has potental of utlzng nfomaton of any numbe of auxlay vaables and ethe ths auxlay nfomaton on all auxlay vaables s avalable fo populaton, patally avalable o even not avalable. Because the estmato has such specal cases those can handle all thee stuatons. On the bass of empcal and smulaton studes the pefomance of poposed estmato s fa bette than ts competto n tem of bas and mean squae eos fo both multvaate and unvaate cases. The pefomance s even bette fo hghly skewed data that s usually the type of busness and economc data. Hence the estmato has a lage scope of applcaton. Ths pape also flls the gap n the lteatue fo pacttones those want to estmate populaton mean vecto of coelated esponse vaables ethe wth symmetc o skewed natue but the set of study vaables needs to depend on the same set of auxlay vaables as usually n the multvaate egesson case. Fo ndependent esponse vaables the unvaate veson can be used fo eal lfe applcatons. Futhe the estmated vecto along wth estmated vaance covaance matx can be used fo hypothess testng and confdence egon fo the vecto of populaton mean usng lage sample popetes. 7. FUTURE RESEARCH WORK Cuently we estmated the vecto of mean usng same set of auxlay vaables fo each study vaables. But pactcally t s possble that each study vaable can depend on dffeent set of auxlay vaables. Then how multvaate estmatos can be constucted and how the vaance covaance matx can be deved, etc. REFERENCES. Ahmed, M.S. (00). Geneal Chan estmatos unde mult-phase samplng. J. Appled Statst. Sc., (), -0.. Ahmad, Z., Hanf, M. and Ahmad, M. (009). Genealzed Regesson Cum-Rato Estmatos fo Two-Phase Samplng Usng Mult-Auxlay Vaables. Pak. J. Statst., (), Ahmad, Z., Hanf, M. and Ahmad, M. (00a). Genealzed Mult-Phase Multvaate Rato Estmatos fo Patal Infomaton Case Usng Mult-Auxlay Vaables. Communcaton of the Koean Statstcal Socety, 7(), Ahmad, Z. and Hanf, M. (00b). Genealzed multvaate egesson estmatos fo patal nfomaton case. Wold Appled Scence Jounal, 0(), Ahmad, Z., Maqsood, I. and Hanf, M. (0). Genealzed estmato of populaton mean fo two phase samplng usng mult-auxlay vaables n the pesence of nonesponse at fst phase fo no nfomaton case. J. Math. Sc. and App. E-Notes., (), Ahmad, Z., Afzal, M. and Hanf, M. (0). A geneal class of mean estmatos usng mxtue of auxlay vaables fo two-phase samplng n the pesence of nonesponse at second phase fo no nfomaton case. Pak. J. Statst., 0(), Bahl, S. and Tuteja, R.K. (99). Rato and poduct type exponental estmatos. Infomaton and Optmzaton Scences, (), Butt, N.S., Kamal, S. and Shahbaz, M.Q. (0). Multvaate estmatos fo two phase samplng. Wold Appled Scences Jounal, (0), 6-.

11 Ayesha Ayaz et al Chand, L. (97). Some ato type estmatos based on two o moe auxlay vaables. Unpublshed Ph.D. Thess, Iowa State Unvesty, Ames, Iowa (USA). 0. Cochan, W.G. (90). The estmaton of the yelds of ceeal expements by samplng fo the ato of gan to total poduce. J. Agcultual Sc., 0, Cochan, W.G. (977). Samplng Technques. John Wley & Sons, Inc., New Yok. p.. Hanf, M., Ahmad, Z. and Ahmad, M. (009). Genealzed Multvaate Rato Estmato usng Mult-Auxlay Vaables fo Mult-Phase Samplng. Pak. J. Statst., (), John, S. (969). On multvaate ato and poduct estmatos. Bometka, 6, -6.. Ngesa, O., Owa, G.O., Oteno, R.O. and Muay, H.M. (0). Multvaate ato estmato of the populaton total unde statfed andom samplng. Open Jounal of Statstcs,, Olkn, I. (98). Multvaate ato estmaton fo fnte populatons. Bometka,, Sanaullah, A., Khan. H., Al, H.A. and Sngh, R. (0). Impoved ato type estmatos n suvey samplng. Jounal of Relablty and Statstcal Studes, (), Sanaullah, A. (0). Genealzed exponental estmatos fo populaton mean unde statfed samplng n the pesence of non-esponse. Unpublshed Ph.D. thess, Natonal College of Busness Admnstaton and Economcs, Lahoe. 8. Samuddn, M. and Hanf, M. (007). Estmaton of populaton mean n sngle and two-phase samplng wth o wthout addtonal nfomaton. Pak. J. Statst., (), Sngh, H.P. and Solank, S.R. (0). An Effcent Class of estmatos fo the populaton mean usng auxlay nfomaton. Commun. n Statst., Theoy and Methods, (), Sngh, H.P. and Vshwakama, G.K. (007). Modfed exponental ato and poduct estmatos fo fnte populaton mean n two-phase samplng. Austan Jounal of Statstcs, 6, 7-.. Sngh, G.N. and Majh, D. (0). Some Chan-Type Exponental Estmatos of Populaton Mean n Two-Phase Samplng. Statstcs n Tanston New Sees, (), -0.. Svastava, S.K., Khae, B.B. and Svastava, S.R. (990). A genealzed chan ato estmato fo mean of fnte populaton. J. Ind. Soc. Ag. Statst.,, Tpath, T.P. (989). Optmum estmaton of mean vecto fo dynamc populaton. Invted pape n the Poceedng of Intenatonal Symposum on Optmzaton and Statstcs, held at AMU. Algah.. Tpath, T.P. and Khattee, R. (989). Smultaneous estmaton of seveal means usng multvaate auxlay nfomaton, Tech. Repot No. /89, Stat/Math, ISI, Calcutta.

12 0 Genealzed Multvaate Exponental Type (GMET) Estmato Table A Detal of Populatons S# Souce of Populatons Populaton census epot of Jhang dstct (998), Pakstan Populaton census epot of Fasalabad dstct (998), Pakstan. Populaton census epot of Gujat dstct (998), Pakstan. Populaton census epot of Kasu (998) Pakstan Populaton census epot of Salkot dstct (998), Pakstan. APPENDIX-A Table A Descpton of Vaables (Each vaable s taken fom Rual Localty) Y Lteacy ato X Populaton of pmay but below matc Y Populaton of cuently maed X Populaton of matc and above Y Total household X Populaton of 8 yeas old and above X Populaton of both sexes X Populaton of women -9 yeas old Table A Vaance Covaance Matces fo Dffeent Real Populaton Populaton-: Jhang Y Y Y X X X X X Y Y Y X X X X X Populaton-: Fasalabad Y Y Y X X X X X

13 Ayesha Ayaz et al. Table A Vaance Covaance Matces fo Dffeent Real Populaton (Cont ) Populaton-: Gujat Y Y Y X X X X X Y Y Y X X X X X Populaton-: Kasu Y Y Y X X X X X Populaton-: Salkot Y Y Y X X X X X

14 Salkot Kasu Gujat Fasalabad Jhang Genealzed Multvaate Exponental Type (GMET) Estmato Table A Vaance Covaance Matces and the Detemnants fo Poposed Estmato t p Populatons and Modfed Veson of Sngh and Majh (0) t g Σ p Σ g Σ p Σ g PRE t Σ p Σ g N n t PRE % % n N n % % n N n % % n N n % % n N n % % n

15 Ayesha Ayaz et al. Table B Vaance Covaance Matces fo Populaton Symmetc APPENDIX-B Y Y Y X X X X X Y Y Y X X X X X Table B Vaance Covaance Matces fo Populaton Skewed Y Y Y X X X X X Y Y Y X X X X X Table B Vaance Covaance Matces fo Symmetc of Fst Phase Sample of Sngle Replcaton Y Y Y X X X X X Y Y Y X X X X X

16 Genealzed Multvaate Exponental Type (GMET) Estmato Table B Vaance Covaance Matces fo Skewed of Fst Phase Sample of Sngle Replcaton Y Y Y X X X X X Y Y Y X X X X X Table B Vecto of Paametes and Bas (Optmum Values fom Populaton ) Bas( tg ) Paametes Bas( t p ) Bas( t g ) 00 / Bas( t ) Sym Shew Sym Shew Sym Shew Sym Shew Y % 6% Y % 8% Y % 7% Table B6 Vecto of Paametes and Bas (Optmum Values fom Fst Phase Sample ) Bas( tg ) Paametes Bas( t p ) Bas( t g ) 00 / Bas( t ) Sym Shew Sym Shew Sym Shew Sym Shew Y % 8% Y % % Y % % p p

17 Ayesha Ayaz et al. Table B7 Matces of Optmum Values of Unknown Constants A, B & A and A & C A B A A C Populaton Symmetc data Fst Phase Sample Symmetc data Populaton Skewed data Fst Phase Sample Skewed data

18 6 Genealzed Multvaate Exponental Type (GMET) Estmato Table B8 Vaance Covaance Matces and the Detemnants fo Poposed Estmato t p and Modfed Veson of Sngh and Majh (0) t g Populatons Σ p Σ g Σ p Σ g PRE t Σ p Σ g Symmetc Skewed Symmetc Skewed When Optmum Values ae Computed fom Populaton Paametes When Optmum Values ae Computed fom Fst Phase Sample t PRE % % % % % % e+ 6% %