Dual to Ratio Estimators for Mean Estimation in Successive Sampling using Auxiliary Information on Two Occasion
|
|
- Loren Newton
- 5 years ago
- Views:
Transcription
1 J. Stat. Appl. Pro. 7, o. 1, (018) 49 Joural of Statistics Applicatios & Probability A Iteratioal Joural Dual to Ratio Estimators for Mea Estimatio i Successive Samplig usig Auxiliary Iformatio o Two Occasio azeema Beevi Departmet of Statistics, Uiversity of Calicut, Kerala , Idia. Received: 14 Ja. 016, Revised: 30 Dec. 017, Accepted: Ja. 018 Published olie: 1 Mar. 018 Abstract: I this paper, cosider dual to ratio estimator for estimatig mea usig auxiliary iformatio o both occasios i successive samplig scheme. Dual to ratio estimators have bee developed by Srivekataramaa (1980) uder simple radom samplig. Usig this estimator uder successive samplig scheme, the bias ad mea squared error are obtaied upto the first order of approximatio ad show theoretically that the proposed estimator is more efficiet tha the Cochra s estimator usig o auxiliary variable ad simple mea per uit estimator. Optimum replacemet strategy is also discussed. Results have bee justified through empirical iterpretatio. Keywords: Auxiliary Iformatio, Dual to Ratio Estimator, Optimum Replacemet, Successive Samplig. 1 Itroductio ow a days, it is ofte see that sample surveys are ot limited to oe time iquiries. A survey carried out o a fiite populatio is subject to chage overtime if the value of study character of a fiite populatio is subject to chage (dyamic) overtime. A survey carried out o a sigle occasio will provide iformatio about the characteristics of the surveyed populatio for the give occasio oly ad ca ot give ay iformatio o the ature or the rate of chage of the characteristics over differet occasios ad the average value of the characteristics over all occasios or most recet occasio. A part of the sample is retaied beig replaced for the ext occasio ( or samplig o successive occasios, which is also called successive samplig or rotatio samplig). The successive method of samplig cosists of selectig sample uits o differet occasios such that some uits are commo with samples selected o previous occasios. If samplig o successive occasios is doe accordig to a specific rule, with replacemet of samplig uits, it is kow as successive samplig. Replacemet policy was examied by Jesse (194) who examied the problem of samplig o two occasios, without or with replacemet of part of the sample i which what fractio of the sample o the first occasio should be replaced i order that the estimator of Ȳ may have maximum precisio. Yates (1949) exteded Jesse s scheme to the situatio where the populatio mea of a character is estimated o each of (h > ) occasios from a rotatio sample desig. These results were geeralized by Patterso (1950) ad arai (1953), amog others. Rao ad Mudhdkar (1983) ad Das (198), used the iformatio collected o the previous occasios for improvig the curret estimate. Data regardig chagig properties of the populatio of cities or couties ad uemploymet statistics are collected regularly o a sample basis to estimate the chages from oe occasio to the ext or to estimate the average over a certai period. A importat aspect of cotiuous surveys is the structure of the sample o each occasio. To meet these requiremets, successive samplig provide a strog tool for geeratig the reliable estimates at differet occasios. Se (1971) developed estimators for the populatio mea o the curret occasio usig iformatio o two auxiliary variables available o previous occasio. Se (197, 1973) exteded his work for several auxiliary variables. Sigh et.al (1991) ad Sigh ad Sigh (001) used the auxiliary iformatio o curret occasio for estimatig the curret populatio mea i two occasios successive samplig ad Sigh (003) exteded his work for h occasios successive samplig. Feg ad Zou (1997) ad Biradar ad Sigh (001) used the auxiliary iformatio o both the occasios for Correspodig author azeemathaj@gmail.com
2 50. Beevi : Dual to ratio estimators for mea estimatio i... estimatig the curret mea i successive samplig. I may situatios, iformatio o a auxiliary variate may be readily available o the first as well as the secod occasios; for example, toage (or seat capacity) of each vehicle or ship is kow i survey samplig of trasportatio ad umber of beds i hospital surveys. Most of the studies related to dual to ratio estimators have bee developed by Srivekataramaa (1980). He cosidered, the relatioship betwee the respose y ad the subsidiary variate x, is liear through the origi ad variace of y is proportioal to x. It is assumed that X is kow. Motivated with the above argumet ad utilizig the iformatio o a additio auxiliary variable is available o the both occasios, the dual to ratio estimator for estimatig the populatio mea o curret occasio i successive samplig has bee proposed. It has bee assumed that the additioal auxiliary variable over two occasios. The paper is spread over te sectios. Sample structure ad otatios have bee discussed i sectio ad sectio 3 respectively. I sectio 4, the proposed estimators have bee formulated. Properties of proposed icludig mea square error are derived uder sectio 5. I sectio 6, optimum replacemet policy is discussed. Sectio 7 cotais compariso of the proposed estimator with Cochra (1977) ad simple mea per uit whe there is o matchig from the previous occasio ad the estimator whe o additioal auxiliary iformatio has bee used. I Sectio 8 ad 9, the theoretical results are supported by a umerical iterpretatio ad give coclusio i Sectio 10. Selectio of the sample Cosider a fiite populatio U = (U 1,U...U ) which has bee sampled over two occasios. Let x ad y be the study variables o the first ad secod occasios respectively, further assumed that the iformatio o the auxiliary variable z, whose populatio mea is kow which is closely related (positively related) to x ad y o the first ad secod occasios respectively available o the first as well as o the secod occasio. For coveiece, it is assumed that the populatio uder cosideratio is large eough. Allowig SRSWOR (Simple Radom Samplig without Replacemet) desig i each occasios, the successive samplig scheme as follows is carried out: uits which costitutes the sample o the first occasio. A radom sub sample of m = λ (0 < λ < 1) uits is retaied (matched) for use o the secod occasio. I the secod occasio u = µ (= m ) (0 < µ < 1) uits are draw from the remaiig ( ) uits of the populatio. Where µ is the fractio of fresh sample o the curret occasio. So that the sample size o the secod occasio is also (= λ + µ). 3 Descriptio of otatios The followig otatios i this paper. X: The populatio mea of the study variable o the first occasio. Ȳ : The populatio mea of the study variable o the secod occasio. Z: The populatio mea of the auxiliary variable o both occasios. S y: Populatio mea square of y. z : The sample mea based o uits draw o the first occasio. z u : The sample mea based o u uits draw o the secod occasio. x : The sample mea based o uits draw o the first occasio. x m : The sample mea based o m uits observed o the secod occasio ad commo with the first occasio. ȳ u : The sample mea based o u uits draw afresh o the secod occasio. ȳ m : The sample mea based o m uits commo to both occasios ad observed o the first occasio. ρ yx : The correlatio coefficiet betwee the variables y o x. ρ xz : The correlatio coefficiet betwee the variables x o z. ρ yz : The correlatio coefficiet betwee the variables y o z. m : The sample uits observed o the secod occasio ad commo with the first occasio. u : The sample size of the sample draw afresh o the secod occasio. : Total sample size.
3 J. Stat. Appl. Pro. 7, o. 1, (018) / Proposed Product Ratio Estimators i Successive samplig I this sectio some dual to ratio estimators usig oe auxiliary variable have bee proposed. To estimate the populatio mea Ȳ o the secod occasio, two differet estimators are suggested. The first estimator is dual to ratio estimator based o sample of size u (= µ) draw afresh o the secod occasio ad is give by: t u = ȳ u z u Z, where z u =(1+g) Z g z ad g=. The secod estimator is a chai dual to ratio estimator based o the sample of size m (= λ) commo with both the occasios ad is defied as, t m = ȳ m x m x z Z, where x m =(1+g) X g x, x =(1+g) X g x ad g=. Combiig the estimators t u ad t m, the fial estimator t dr as follows t dr = ψt u +(1 ψ)t m, where ψ is a ukow costat to be determied such that MSE(t dr ) is miimum ad prove theoretically that the estimator is more efficiet tha the proposed estimator by (i) Cochra (1977) whe o auxiliary variables are used at ay occasio.this classical differece estimator is a widely used estimator to estimate the populatio mea Ȳ, i successive samplig. It is give by y = φ ȳ u +(1 φ )ȳ m, where φ is a ukow costat to be determied such that V( ˆȲ) opt is miimum ad y u = y u is the sample mea of the umatched portio o the secod occasio ad ȳ m = ȳ m+ b( y 1 ȳ 1m ) is based o matched portio. The variace of this estimator is V( ˆȲ) opt =[1+ (1 ρyx )] S y. Similarly, the variace of the mea per uit estimator is give by V(ȳ)= S y. (4.1) (4.) (4.3) 4.1 Properties of t dr Sice t u ad t m both are biased estimators of t dr, therefore, resultig estimator t dr is also a biased estimator. The bias ad MSE up to the first order of approximatio are derived as usig large sample approximatio give below: y u = Ȳ(1+eȳu ), ȳ m = Ȳ(1+eȳm ), x m = X(1+e xm ), x = X(1+e x ), z = Z(1+e z ), z u = Z(1+e zu ) where eȳu,eȳm,e xm,e x,e z, z u are samplig errors ad are of very small quatities. We assume that E(eȳu )=E(eȳm ) = E(e xm )=E(e x )=E(e z )=E(e zu )=0. The for simple radom samplig without replacemet for both first ad secod ) occasios, we write) by usig occasio wise operatio of expectatio as: E(eȳ u )=( 1u 1 Sy,E(e ȳ m )=( 1 m 1 Sy ), E(e x m )=( 1 m 1 Sx,E(e x )= ) 1 S x, E(e z )= ) 1 S z, E(e ) yu e zu )=( 1u 1 S yz, ) S yx, E(eȳm e xm )=( 1 m 1 ( ) E(eȳm e x )= 1 m 1 S yx, E(eȳm e x )= ) 1 Syx, E(eȳm e z )= ) 1 Syz, E(e xm e x )= ) 1 S x, E(e xm e z )= ) 1 Sxz, E(e x e z )= ) 1 Sxz. Derive the bias of t u i lemma 4.1. Lemma 4.1The bias of ) t u deoted by B(t u ) is give by B(t u )= Ȳ( 1u 1 gρ yz S y S z
4 5. Beevi : Dual to ratio estimators for mea estimatio i... Proof.Expressig (4.1) i terms of e s ad get t u = Ȳ(1+eȳu )[(1+g) g(1+e zu )] = (1+eȳu )(1 ge zu ). Takig expectatio o both sides ad igorig higher orders, E(t u Ȳ) = Ȳg 1 ) E(eȳu e zu ) u B(t u ) = Ȳ u 1 ) gρ yz S y S z. (4.4) The bias of t m is derived i lemma 4.. Lemma 4.The bias) of t m is deoted by B(t m )give by B(t m )= Ȳ( 1 m 1 (Sx ρ yx S y S x )+ ) 1 [ gρyz S y S z ] Proof.Expressig (4.) i terms of e s, get t m = Ȳ(1+eȳm )(1+e x ) 1 (1 ge xm )(1 ge z ). Expadig the right had side ad eglectig the terms with power two or greater ad get t m = Ȳ[(1+eȳm e x + e x eȳm e x ) (1 geȳm ge z + g eȳm e z )]. Takig expectatio (4.5) o both sides, B(t m )= Ȳ 1 ) (Sx m ρ yxs y S x )+ 1 ) [ gρ yz S y S z ]. (4.5) Usig 4.1 ad 4., derive the bias of t dr. Theorem 4.1Bias of the estimator t dr to the first order approximatio is, B(t dr ) = ψb(t u )+(1 ψ)b(t m ), (4.6) where B(t u ) = Ȳ ad B(t m )= Ȳ u 1 ) gρ yz S y S z. 1 ) (S m x ρ yx S y S x )+ 1 ) ( gρ yz S y S z ). Proof.The bias of the estimator t dr is give by B(t dr ) = E(t dr Ȳ) B(t dr ) = ψe(t u Ȳ)+(1 ψ)e(t m Ȳ), (4.7) Usig lemma (4.1) ad (4.) ito equatio (4.7), the expressio for the bias of the estimator t dr as show i (4.6) We derive the MSE of t u i lemma 4.3. Lemma 4.3The ( mea ) square error of t u deoted by M(t u ) is give by M(t u )= Ȳ 1u [S 1 y + g Sz ] gρ yz S y S z
5 J. Stat. Appl. Pro. 7, o. 1, (018) / 53 Proof.Expressig t u i terms of e s, get t u = Ȳ(1+eȳu )(1 ge zu ). Expadig ad squarig(4.8), the right had side ad eglectig the terms with power two or greater, t u = Ȳ(1+eȳu ge zu ) (4.8) t u = Ȳ(1+e ȳ u + g e z u geȳu e zu ). (4.9) Takig expectatio (4.9) o both sides ad get M(t u ) M(t u )= Ȳ 1 ) [S u y+ g Sz ] gρ yz S y S z. (4.10) The MSE of t m is derived i lemma 4.4 Lemma 4.4The mea square error of t m deoted by M(t m ) is give by M(t m )= Ȳ [ m 1 1 ) (g S ) z gρ yz S y S z ]. ) Sy + 1 ) (g S ) x gρ yxs y S x + m Proof.Expressig t m i terms of e s, {[ (1+g) X g(1+eȳm ) X ] t m = Ȳ(1+eȳm ) (1+e x ) X } [(1+g) Z g(1+e z ) Z], Z = Ȳ(1+eȳm )(1 ge xm )(1 e x )(1 e z ). (4.11) Expadig (4.10), the right had side ad eglectig the terms with power two or greater, get t m = Ȳ(1+eȳm ge x ge xm ge z ). (4.1) Squarig o both sides (4.11) ad takig expectatio, MSE of the estimator t m upto first order of approximatio as, M(t m )= Ȳ [ 1 ) Sy m + 1 ) (g S ) x gρ yxs y S x + m 1 ) (g Sz gρ ) yzs y S z ]. (4.13) Usig lemma 4.3 ad 4.4, we derive the MSE of t dr Theorem 4.The mea square error of the estimator t dr to the first order approximatio is, M(t dr )=ψ M(t u )+(1 ψ) M(t m )+ψ(1 ψ)cov(t u,t m ) (4.14) where M(t u ) = Ȳ u 1 M(t m )= Ȳ [ m 1 1 ) (g Sz gρ ) yzs y S z ]. ad Cov(t u,t m )=0. ) [S y+ g S z + gρ yz S y S z ] ) Sy + 1 ) (g S ) x gρ yxs y S x + m
6 54. Beevi : Dual to ratio estimators for mea estimatio i... Proof.The mea square error of the estimator t dr is give by M(t dr )=E(t dr Ȳ) M(t dr )=E[ψ(t u Ȳ)+(1 ψ)(t m Ȳ)], (4.15) M(t dr )=ψ M(t u )+(1 ψ) M(t m )+ψ(1 ψ)cov(t u,t m ), usig lemma (4.3) ad (4.4) ito the equatio (4.15), the expressio for the MSE of the estimator t dr as show i (4.14) Remark 4.3The estimators, t u ad t m are based o two idepedet samples of sizes u ad m respectively, therefore the covariace term has bee vaished. 5 Miimum Mea Square Error of t dr To obtai the optimum value of ψ, partially differetiate the expressio (4.14) with respect to ψ, ad put it equal to zero, we get ψ opt = M(t m ) M(t u )+M(t m ) substitutig the values of M(t u ) ad M(t m ) from (4.10) ad (4.13) i (5.1), get ψ opt = (k 1+µk ) (k 1 + µ k ) = µ[(k 1+µk )] (k 1 + µ k ). Substitutio of ψ opt from (5.1) ito (4.14) gives optimum value of MSE of t dr as: (5.1) M(t dr ) opt = M(t m )M(t u ) M(t u )+M(t m ). Substitutig the values of M(t m ) ad M(t u ) from (4.9) ad (4.10) i (5.), get M(t dr ) opt = 1 [ k 1 + µk 1 k k 1 + µ k ],, where k 1 = 1+g gρ yz, k = g(ρ yx ρ yz ), here µ(= u ) is the fractio of fresh sample draw o the secod occasio. Agai M(t dr ) opt derived i equatio (5.3) is the fuctio of µ. To estimate the populatio mea o each occasio the better choice of µ is 1 (o matchig). However, to estimate the chage i mea from oe occasio to the other, µ should be 0 (complete matchig). (5.) (5.3) 6 Replacemet Policy I order to estimate t dr with maximum precisio a optimum value of µ should be determied so as to kow what fractio of the sample o the first occasio should be replaced ad miimize, M(t dr ) opt i (5.3) with respect to µ, the optimum value of µ is obtaied as, ˆµ = k 1± k1 + k 1k, (6.1) k where k 1 = 1+g gρ yz, k = g(ρ yx ρ yz ). From (6.1) it is obvious that for ρ yz ρ yx two values of ˆµ are possible, therefore to choose a value of ˆµ, it should be remembered that 0 ˆµ 1. All other values of ˆµ are iadmissible. If both the real values of ˆµ are admissible, the lowest oe will be the best choice as it reduces the total cost of the survey. Substitutig the value of ˆµ from (6.1) i (5.3), M(t dr ) opt = 1 [ k 1 + ˆµk 1k k 1 + ˆ µ k ]. (6.)
7 J. Stat. Appl. Pro. 7, o. 1, (018) / Efficiecy Comparisos I this sectio, to compare t dr with respect to ȳ, (i) sample mea of y, whe a sample uits are selected at secod occasio without ay matched portio. (ii) differece estimator (Cochra 1977) whe o auxiliary iformatio is used at ay occasio, have bee obtaied for kow values of ρ yx ad ρ yz. Sice ȳ ad ˆȲ are ubiased estimators of Ȳ, their variaces for large are respectively give by V(ȳ)= S y, (7.1) V ˆ (Y)opt =[1+ (1 ρ yx )] S Y. (7.) For differet values ρ yx ad ρ yz, the below shows the optimum value of µ. That is ˆµ. The percet relative efficiecies, R 1 ad R of t opt with respect to ȳ ad ˆȲ respectively, where ad R 1 = V (y) M(t dr ) opt 100 R = V (Y) ˆ opt 100. M(t dr ) opt The estimator t pr (at optimal coditios) is also compared with respect to the estimators V(ȳ) ad V( ˆȲ), respectively. Where [ M(t pr ) opt = 1 k1 + ˆµk ] 1k k 1 + µ ˆ (7.3) k ad ˆµ = k 1± k 1 + k 1k k, where k 1 = 1+g gρ yz, k = g(ρ yx ρ yz ). (7.4) 7.1 Empirical Study The expressios of the optimum value of µ (i.e. ˆµ) ad the percet relative efficiecies R 1 ad R are i terms of populatio correlatio coefficiets ρ yx ad ρ yz. The Table 7.1. shows that the values of ˆµ, R 1 ad R for differet choices of ρ yx, ρ yz ad µ. Table 1: Table 7.1. optimum values of µ ad percet relative efficiecies of t dr with respect to ȳ ad ˆȲ ρ yz ρ yx ˆµ R R ˆµ R R ˆµ R R ˆµ R R
8 56. Beevi : Dual to ratio estimators for mea estimatio i... 8 umerical Illustratio The results obtaied i previous selectios are ow examied with the help of oe atural populatio set of data. Populatio Source: [Free access to the data by the Statistical Abstracts of the Uited States.] Let Y (study variable) be the level of cor productio (i per acre) ad Z (auxiliary variate) be the dosage of fertilizer usig i cor filed i 50 couties i the Uited states i 007 ad X be the cor productio i the year 006 i the States of Uited states. Based o the above descriptio, the values of the differet required parameters for populatio is, = 50, X = 139, Ȳ = 118, S Y = 3314., ρ yx = 0.987, ρ yz = 0.141, ˆµ = , ψ opt = Table : Table 8.1. Percet Relative Efficiecies of t dr with Respect to ȳ ad ˆȲ f g Relative Efficiecies R 1 = R = R 1 = R = R 1 = R = where R 1 = V(ȳ) M(t dr ) opt ad R = V ˆ (Y)opt M(t dr ) opt. ρ yz ρ yx ˆµ R R ˆµ R R ˆµ R R ˆµ R R
9 J. Stat. Appl. Pro. 7, o. 1, (018) / 57 Table 3: Table 8.. MSE ad Bias of Differet Estimators. Bias MSE Bias Ȳ M(t dr ) opt = V(ȳ) = V ˆ (Y)opt = M(t dr ) opt = V(ȳ) = V ˆ (Y)opt = M(t dr ) opt = V(ȳ) = V ˆ (Y)opt = Iterpretatios of Empirical Results of t dr From Table 7.1., the relative efficiecy is observed that the suggested estimator is compared with mea per uit estimator ad Cochra (1977) estimator. So, the use of auxiliary iformatio at both occasios is justified. 1.For fixed values of ρ yx, the value of R 1 ad R are icreasig with icreasig values of µ ad the icreasig ρ yz..the values of R 1, R ad µ are icreasig with icreasig values of ρ yz. This is a agreemet with the results Sukhatme et.al (1984), which justifies that higher the value of ρ yx, higher the fractio of fresh sample required at the secod (curret) occasio. 3.For fixed values of ρ yx ad ρ yz, there is appreciable gai i the performace of the proposed estimator t dr over ȳ ad ˆȲ with the icreasig value of µ. 10 Coclusio From Table 7.1. clearly see that the value of ˆµ (at optimum coditio) also exist for both the cosidered populatios. Hece, it justifies that the suggested family of estimators t dr is feasible uder optimal coditios. Tables 8.1. ad 8.. idicates that the suggested estimators t dr at optimum coditios is preferable over sample mea per uit estimator ad also performs better tha the Cochra s estimator. Hece, it may be cocluded that the estimatio of mea at curret usig auxiliary iformatio o both occasios i successive samplig is highly i terms of precisio ad reducig the cost of survey.clearly idicates that the proposed estimators is more efficiet tha simple arithmetic mea estimator ad Cochra (1977) estimator. The followig coclusio ca be formed from Tables 7.1. For fixed ρ yx, ρ yxz ad µ, the values of R 1 ad R are icreasig. This pheomeo idicates that smaller fresh sample at curret occasio is required, if a highly positively correlated auxiliary characters is available. That is the performace of precisio of the estimates also reduces the cost of the survey. 11 Perspective Table 7.1. clearly idicates that the suggested estimators is more efficiet tha simple arithmetic mea ad Cochra (1977) estimators. The followig coclusio ca be made from Table 7.1. Fixed ρ yx, the values of R 1 ad R are icreasig while ˆµ is decreasig with the icreasig values of ρ yz. This pheomeo idicates that smaller fresh sample at curret occasio is required, if a highly positively correlated auxiliary characters is available. For Fixed ρ yz, the values of R 1 ad ˆµ are icreasig while R is decreasig for iitial values of the icreasig values of ρ yx. Thus behavior is i agreemet with Cochra (1977) results, which explais that more the value of ρ yx, more fractio of fresh sample is required at curret occasio. That is the performace of precisio of the estimates as well as reduces the cost of the survey. Uder the give framework (Tables 8.1.ad 8..) tables it is possible to reduce the bias ad mea square error of the estimator, the aalytical ad empirical results support the theoretical justificatio of the work. The estimatio of populatio mea o successive occasios should be ecouraged as there are umerous practical situatios that require the estimate of mea at differet poits of time as the characters are time depedet. Hece, the proposed estimators should be recommeded for their use i practice.
10 58. Beevi : Dual to ratio estimators for mea estimatio i... Refereces [1] Biradar, R.S. ad Sigh, H.P., Successive samplig usig auxiliary iformatio o both occasios. Cal. Stat. Assoc. Bull., 51, 43-51, (001). [] Cochra, W.G.: Samplig Techiques. 3 rd editio, (1977). [3] Chaturvedi, D.K. ad Tripathi, T.P. (1983). Estimatio of Populatio ratio o two occasios usig multivariate auxiliary iformatio. Jour. Id. Stat. Asso., 1, [4] Das, K.: Estimatio of populatio ratio o two occasios. Jour. Id. Soc. Agri. Stat. 34(), 1-9, (198). [5] Feg, G.S, ad Zou, G.: Sample rotatio method with auxiliary variable. Commuicatio i Statistics-Theory ad Methods, 6, 6, , (1997). [6] Jesse, R.J : Statistical Ivestigatio of a sample survey for obtaiig farm facts. Iowa Agricultural Experimet Statistical Research Bulleti, 304, (194). [7] arai, R.D.: O the recurrece formula i samplig o successive occasios. Jour. Id. Soc. Agri. Stat., 5, 96-99, (1953). [8] Okafor, F.C. (199) The theory ad applicatio of samplig over two occasios for the estimatio of curret populatio ratio, Statistica1, [9] Patterso, H.D.: Samplig o successive occasios with partial replacemet of uits. Jour. Roy. Stat. Soc., B 1, 41-55, (1950). [10] Rao, J..K. ad Mudholkar, G.S.: Geeralized multivariate estimators for the mea of fiite populatio parameters. Jour. Amer. Stat. Asso., 6, , (1967). [11] Se, A.R.: Successive samplig with two auxiliary variables. Sakhya, Series B, 33, , (1971). [1] Se, A.R.: Theory ad applicatio of samplig o repeated occasios. Jour. Amer. Stat. Asso., 59, , (1973). [13] Se, A.R., Sellers, S. ad Smith, G.E.J.: The use of a ratio estimate i successive samplig.biometrics, 31, , (1975). [14] Srivekataramaa, T.: A dual to ratio estimator i sample surveys, Biometrika, 67(1), 199?04, (1980). [15] Sigh, G..: Estimatio of populatio mea usig auxiliary iformatio o recet occasio i h occasios successive samplig. Statistics i Trasitio, 6(4), 53-53, (003). [16] Sigh, G.. ad Sigh, V.K.: O the use of auxiliary iformatio i successive samplig. Jour. Id. Soc. Agri. Stat. 54 (1), 1-1, (001). [17] Yates, F.: Samplig Methods for Cesuses ad Surveys. Charles Griffl ad Co.,Lodo, (1949). azeema Beevi is a statistical ivestigator at Directorate of Ecoomics ad Statistics ad doe the Ph.D. i the Departmet of Statistics, Uiversity of Calicut. My research iterests are Samplig Theory (Two-Phase Samplig, Super Populatio Model, Predictive Model), Statistical Iferece ad Data Aalysis i particular, Small Area Estimatio ad Item Respose Theory. I am a lifelog member of Kerala Statistical Associatio (KSA) ad also Reviewer of Joural of Applied Probability ad Statistics, UK (016).
Improved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA
More informationChain ratio-to-regression estimators in two-phase sampling in the presence of non-response
ProbStat Forum, Volume 08, July 015, Pages 95 10 ISS 0974-335 ProbStat Forum is a e-joural. For details please visit www.probstat.org.i Chai ratio-to-regressio estimators i two-phase samplig i the presece
More informationEstimation of the Population Mean in Presence of Non-Response
Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of No-Respose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More informationEstimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya
More informationSome Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation
; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 097-70)]; ISSN 0975-556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product
More informationVaranasi , India. Corresponding author
A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet
More informationModified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis
America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry
More informationAClassofRegressionEstimatorwithCumDualProductEstimatorAsIntercept
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 15 Issue 3 Versio 1.0 Year 2015 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic.
More informationImproved exponential estimator for population variance using two auxiliary variables
OCTOGON MATHEMATICAL MAGAZINE Vol. 7, No., October 009, pp 667-67 ISSN -5657, ISBN 97-973-55-5-0, www.hetfalu.ro/octogo 667 Improved expoetial estimator for populatio variace usig two auxiliar variables
More informationAlternative Ratio Estimator of Population Mean in Simple Random Sampling
Joural of Mathematics Research; Vol. 6, No. 3; 014 ISSN 1916-9795 E-ISSN 1916-9809 Published by Caadia Ceter of Sciece ad Educatio Alterative Ratio Estimator of Populatio Mea i Simple Radom Samplig Ekaette
More informationImproved Ratio Estimators of Population Mean In Adaptive Cluster Sampling
J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) 1 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.18576/jsapl/030101 Improved Ratio Estimators of Populatio
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationAbstract. Ranked set sampling, auxiliary variable, variance.
Hacettepe Joural of Mathematics ad Statistics Volume (), 1 A class of Hartley-Ross type Ubiased estimators for Populatio Mea usig Raked Set Samplig Lakhkar Kha ad Javid Shabbir Abstract I this paper, we
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationResearch Article An Alternative Estimator for Estimating the Finite Population Mean Using Auxiliary Information in Sample Surveys
Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume 01, Article ID 65768, 1 pages doi:10.50/01/65768 Research Article A Alterative Estimator for Estimatig the Fiite Populatio Mea
More informationJambulingam Subramani 1, Gnanasegaran Kumarapandiyan 2 and Saminathan Balamurali 3
ISSN 1684-8403 Joural of Statistics Volume, 015. pp. 84-305 Abstract A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationSYSTEMATIC SAMPLING FOR NON-LINEAR TREND IN MILK YIELD DATA
Joural of Reliability ad Statistical Studies; ISS (Prit): 0974-804, (Olie):9-5666 Vol. 7, Issue (04): 57-68 SYSTEMATIC SAMPLIG FOR O-LIEAR TRED I MILK YIELD DATA Tauj Kumar Padey ad Viod Kumar Departmet
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationA Family of Unbiased Estimators of Population Mean Using an Auxiliary Variable
Advaces i Computatioal Scieces ad Techolog ISSN 0973-6107 Volume 10, Number 1 (017 pp. 19-137 Research Idia Publicatios http://www.ripublicatio.com A Famil of Ubiased Estimators of Populatio Mea Usig a
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationUse of Auxiliary Information for Estimating Population Mean in Systematic Sampling under Non- Response
Maoj K. haudhar, Sachi Malik, Rajesh Sigh Departmet of Statistics, Baaras Hidu Uiversit Varaasi-005, Idia Floreti Smaradache Uiversit of New Mexico, Gallup, USA Use of Auxiliar Iformatio for Estimatig
More informationMathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution
America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationA Generalized Class of Estimators for Finite Population Variance in Presence of Measurement Errors
Joural of Moder Applied Statistical Methods Volume Issue Article 3 --03 A Geeralized Class of Estimators for Fiite Populatio Variace i Presece of Measuremet Errors Praas Sharma Baaras Hidu Uiversit, Varaasi,
More informationEstimation of Population Ratio in Post-Stratified Sampling Using Variable Transformation
Ope Joural o Statistics, 05, 5, -9 Published Olie Februar 05 i SciRes. http://www.scirp.org/joural/ojs http://dx.doi.org/0.436/ojs.05.500 Estimatio o Populatio Ratio i Post-Stratiied Samplig Usig Variable
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More informationEstimation of Gumbel Parameters under Ranked Set Sampling
Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com
More informationModeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy
Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should be doe
More informationOn stratified randomized response sampling
Model Assisted Statistics ad Applicatios 1 (005,006) 31 36 31 IOS ress O stratified radomized respose samplig Jea-Bok Ryu a,, Jog-Mi Kim b, Tae-Youg Heo c ad Chu Gu ark d a Statistics, Divisio of Life
More informationResponse Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable
Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated
More informationLecture 3. Properties of Summary Statistics: Sampling Distribution
Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary
More informationOn ratio and product methods with certain known population parameters of auxiliary variable in sample surveys
Statistics & Operatios Research Trasactios SORT 34 July-December 010, 157-180 ISSN: 1696-81 www.idescat.cat/sort/ Statistics & Operatios Research c Istitut d Estadística de Cataluya Trasactios sort@idescat.cat
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationA Generalized Class of Unbiased Estimators for Population Mean Using Auxiliary Information on an Attribute and an Auxiliary Variable
Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN 89-4966 Volume, Number 07, pp. -8 Research Idia ublicatios http://www.ripublicatio.com A Geeralized Class of Ubiased Estimators for opulatio
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
More informationGUIDELINES ON REPRESENTATIVE SAMPLING
DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue
More informationLecture 33: Bootstrap
Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More information[412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION
[412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION BY ALAN STUART Divisio of Research Techiques, Lodo School of Ecoomics 1. INTRODUCTION There are several circumstaces
More informationNew Ratio Estimators Using Correlation Coefficient
New atio Estimators Usig Correlatio Coefficiet Cem Kadilar ad Hula Cigi Hacettepe Uiversit, Departmet of tatistics, Betepe, 06800, Akara, Turke. e-mails : kadilar@hacettepe.edu.tr ; hcigi@hacettepe.edu.tr
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationA LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!
A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited
More informationResearch Article A Two-Parameter Ratio-Product-Ratio Estimator Using Auxiliary Information
Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume, Article ID 386, 5 pages doi:.54//386 Research Article A Two-Parameter Ratio-Product-Ratio Estimator Usig Auxiliary Iformatio
More informationInvestigating the Significance of a Correlation Coefficient using Jackknife Estimates
Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN 2307-4531 (Prit & Olie) http://gssrr.org/idex.php?joural=jouralofbasicadapplied ---------------------------------------------------------------------------------------------------------------------------
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More informationBayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function
Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume 12, Number 4 (217), pp. 791-796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E- Bayesia Method of Estimatio of Parameter
More informationEconomics Spring 2015
1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationIf, for instance, we were required to test whether the population mean μ could be equal to a certain value μ
STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially
More information5. Fractional Hot deck Imputation
5. Fractioal Hot deck Imputatio Itroductio Suppose that we are iterested i estimatig θ EY or eve θ 2 P ry < c where y fy x where x is always observed ad y is subject to missigess. Assume MAR i the sese
More information10. Comparative Tests among Spatial Regression Models. Here we revisit the example in Section 8.1 of estimating the mean of a normal random
Part III. Areal Data Aalysis 0. Comparative Tests amog Spatial Regressio Models While the otio of relative likelihood values for differet models is somewhat difficult to iterpret directly (as metioed above),
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationCorrelation Regression
Correlatio Regressio While correlatio methods measure the stregth of a liear relatioship betwee two variables, we might wish to go a little further: How much does oe variable chage for a give chage i aother
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More informationJournal of Scientific Research Vol. 62, 2018 : Banaras Hindu University, Varanasi ISSN :
Joural of Scietific Research Vol. 6 8 : 3-34 Baaras Hidu Uiversity Varaasi ISS : 447-9483 Geeralized ad trasformed two phase samplig Ratio ad Product ype stimators for Populatio Mea Usig uiliary haracter
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationStudy on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm
Joural of ad Eergy Egieerig, 05, 3, 43-437 Published Olie April 05 i SciRes. http://www.scirp.org/joural/jpee http://dx.doi.org/0.436/jpee.05.34058 Study o Coal Cosumptio Curve Fittig of the Thermal Based
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More informationMOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationCHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION
CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION 4. Itroductio Numerous bivariate discrete distributios have bee defied ad studied (see Mardia, 97 ad Kocherlakota ad Kocherlakota, 99) based o various methods
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationPreponderantly increasing/decreasing data in regression analysis
Croatia Operatioal Research Review 269 CRORR 7(2016), 269 276 Prepoderatly icreasig/decreasig data i regressio aalysis Darija Marković 1, 1 Departmet of Mathematics, J. J. Strossmayer Uiversity of Osijek,
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationControl Charts for Mean for Non-Normally Correlated Data
Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies
More informationStatistical Properties of OLS estimators
1 Statistical Properties of OLS estimators Liear Model: Y i = β 0 + β 1 X i + u i OLS estimators: β 0 = Y β 1X β 1 = Best Liear Ubiased Estimator (BLUE) Liear Estimator: β 0 ad β 1 are liear fuctio of
More informationJ. Stat. Appl. Pro. Lett. 2, No. 1, (2015) 15
J. Stat. Appl. Pro. Lett. 2, No. 1, 15-22 2015 15 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.12785/jsapl/020102 Martigale Method for Rui Probabilityi
More informationBasics of Probability Theory (for Theory of Computation courses)
Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.
More informationII. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation
II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationComputing Confidence Intervals for Sample Data
Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios
More informationAn Improved Warner s Randomized Response Model
Iteratioal Joural of Statistics ad Applicatios 05, 5(6: 63-67 DOI: 0.593/j.statistics.050506.0 A Improved Warer s Radomized Respose Model F. B. Adebola, O. O. Johso * Departmet of Statistics, Federal Uiversit
More informationEECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationCSE 527, Additional notes on MLE & EM
CSE 57 Lecture Notes: MLE & EM CSE 57, Additioal otes o MLE & EM Based o earlier otes by C. Grat & M. Narasimha Itroductio Last lecture we bega a examiatio of model based clusterig. This lecture will be
More informationo <Xln <X2n <... <X n < o (1.1)
Metrika, Volume 28, 1981, page 257-262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow
More informationAnalytic Theory of Probabilities
Aalytic Theory of Probabilities PS Laplace Book II Chapter II, 4 pp 94 03 4 A lottery beig composed of umbered tickets of which r exit at each drawig, oe requires the probability that after i drawigs all
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationRATIO-TO-REGRESSION ESTIMATOR IN SUCCESSIVE SAMPLING USING ONE AUXILIARY VARIABLE
SAISICS IN RANSIION ew series, Suer 5 83 SAISICS IN RANSIION ew series, Suer 5 Vol. 6, No., pp. 83 RAIO-O-REGRESSION ESIMAOR IN SUCCESSIVE SAMPLING USING ONE AUXILIARY VARIABLE Zorathaga Ralte, Gitasree
More informationADVANCED SOFTWARE ENGINEERING
ADVANCED SOFTWARE ENGINEERING COMP 3705 Exercise Usage-based Testig ad Reliability Versio 1.0-040406 Departmet of Computer Ssciece Sada Narayaappa, Aeliese Adrews Versio 1.1-050405 Departmet of Commuicatio
More information