Applied Mathematics and Computation
|
|
- Hugo McDowell
- 6 years ago
- Views:
Transcription
1 Appled Mathematcs ad Computato 215 (2010) Cotets lsts avalable at SceceDrect Appled Mathematcs ad Computato joural homepage: Improvemet estmatg the populato mea smple radom samplg usg formato o auxlar attrbute A.M. Abd-Elfattah *, E.A. El-Sherpe, S.M. Mohamed, O.F. Abdou Isttute of Statstcal Studes ad Research, Caro Uverst, Dokk, Gza 12613, Egpt artcle fo abstract Kewords: Rato-tpe estmator Smple radom samplg Auxlar attrbute Effcec Ths paper proposes some estmators for the populato mea b adaptg the estmator Sgh et al. (2008) [5] to the rato estmators preseted Kadlar ad Cg 2006 [2]. We obta mea square error (MSE) equato for all proposed estmators, ad show that all proposed estmators are alwas more effcet tha rato estmator Nak ad Gupta (1996) [3], ad Sgh et al. (2008) [5]. The results have bee llustrated umercall b takg some emprcal populato cosdered the lterature. Ó 2009 Elsever Ic. All rghts reserved. 1. Itroducto Cosder a sample of sze draw b smple radom sample wthout replacemet from a populato of sze N. Let ad u deoted the observato o varable ad u, respectvel, for th ut ð ¼ 1; 2; 3;...; NÞ. Suppose there s a complete dchotom the populato wth respect to the presece or absece of a attrbute, sa u, ad t s assumed that attrbute u takes ol the two values 0 ad 1 accordg as u ¼ 1; f th ut of the populato possesses attrbute u ¼ 0; f otherwse: Let A ¼ P N ¼1u ad a ¼ P ¼1u deoted the total umber of uts the populato ad sample possessg attrbute u, respectvel. Let P ¼ A ad P b ¼ a deoted the proporto of uts the populato ad sample, respectvel, possessg attrbute u. Takg to cosderato the pot bseral correlato coeffcet betwee auxlar attrbute ad stud varable, N Nak ad Gupta [3] defed rato estmator of populato mea whe the pror formato of populato proporto of uts, possessg the same attrbute s avalable, as follows: t NG ¼ P b P ; ð1:1þ where s the sample mea of stud varable. The MSE of t NG up to the frst order of approxmato s MSEðt NG Þ¼ 1 f h S 2 þ R2 1 S2 u 2R 1S u ; ð1:2þ where f ¼ ; s the sample sze; N s the umber of uts the populato; R N 1 ¼ Y ; P S2 u s the populato varace of auxlar attrbute u, ad S u s the populato covarace betwee varable of terest ad auxlar attrbute u. * Correspodg author. Address: Departmet of Statstcs, Facult of Scece, Kg Abdul Azz Uverst Box 80203, Jedda 21589, Sauda Araba. E-mal address: a_afattah@hotmal.com (A.M. Abd-Elfattah) /$ - see frot matter Ó 2009 Elsever Ic. All rghts reserved. do: /j.amc
2 A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) Sgh et al. [5] suggested the followg rato estmators for estmatg the populato mea Y of the stud varable smple radom samplg usg kow parameters of auxlar attrbute u, such as, coeffcet of varato C P, coeffcet of kurtoss B 2 ðuþ, ad pot bseral correlato coeffcet q Pb as: t 1 ¼ þ b uðp PÞ b P; ð1:3þ t 2 ¼ þ b uðp PÞ b ½P þ B 2 ðuþš; þ B 2 ðuþ ð1:4þ t 3 ¼ þ b uðp PÞ b ½P þ C P Š; þ C P ð1:5þ t 4 ¼ þ b uðp b PÞ B 2 ðuþþc P ½PB 2 ðuþþc P Š; ð1:6þ t 5 ¼ þ b uðp b PÞ C P þ B 2 ðuþ ½PC P þ B 2 ðuþš; where C P ad B 2 ðuþ are the populato coeffcet of varato ad the populato coeffcet of kurtoss of auxlar attrbute, respectvel, ad b u ¼ su s the regresso coeffcet. Here s 2 s 2 u s the sample varace of auxlar attrbute ad s u s the sample covarace betwee the auxlar attrbute ad the stud varable. u I Sgh et al. [5], mea square error MSE equato of these rato estmators were gve b MSEðt Þ¼ 1 f h R 2 S2 u þ S2 ð1 q2 Pb Þ ; ð ¼ 1; 2; 3;...; 5Þ; ð1:8þ where R 1 ¼ Y ; R P 2 ¼ Y ; R PþB 2 ðuþ 3 ¼ Y PþC P ; R 4 ¼ YB 2ðuÞ PB 2 ðuþþc P ad R 5 ¼ YC P. PC P þb 2 ðuþ Sgh et al. [5] cocluded that the rato estmators t ð ¼ 1; 2;...; 5Þ whch uses some kow value of populato proporto were more effcet tha the sample mea ad rato estmator Nak ad Gupta [3]. I the ext secto, we develop ew estmators combg rato estmators Sgh et al. [5] ad obta the MSE equatos of these ew estmators. I Secto 3, we compare the effceces, theoretcall, based o MSE equatos, betwee the proposed estmators ad the rato estmators preseted Sgh et al. [5]. I Secto 4, we also dscuss the comparso amog all the suggested estmators umercall. I Secto 5, we gve a ht to obta dfferet estmators b a smlar method preseted ths stud. 2. Suggested estmators We propose the estmator usg the procedure preseted Kadlar ad Cg 2006 [2] combg rato estmators (1.3) ad (1.4) as follows: ð1:7þ t pro1 ¼ m 1 P þ m b 2 ðp þ B 2 ðuþþ; þ B 2 ðuþ ð2:1þ where m 1 ad m 2 are weghts that satsf the codto m 1 þ m 2 ¼ 1. The MSE of ths estmator ca be foud usg the frst degree approxmato the Talor seres method defed b MSEðt pro1 Þffd X d 0 ; ð2:2þ where d s a vector defed as ; P s the varace covarace matrx as P " # ¼ 1 f S 2 S u Y;P Y;P S u S 2 (see Wolter [7]). Here hða; bþ ¼hð; PÞ¼t b pro1. Accordg to ths defto, we obta d for the proposed estmator as u follows: d ¼ 1 m 1 ðr 1 þ B u Þ m 2 ðr 2 þ B u Þ ; where B u ¼ Su ¼ q Pb S. Note that we omt the dfferece : b B (Cochra [1]). S 2 u Su We obta the MSE of the proposed estmator usg (2.2) as MSEðt pro1 Þ¼ 1 f S 2 2gS u þ g 2 S 2 u ; ð2:3þ where g ¼ m 1 ðr 1 þ B u Þþm 2 ðr 2 þ B u Þ: ð2:4þ We also propose the estmator combg rato estmators (1.3) ad (1.5) as t pro2 ¼ m b 1 P þ m b 2 ðp þ C P Þ: ð2:5þ þ C P The MSE of ths estmator s the same as (2.3) but R 2 (2.4) s replaced wth R 3.
3 4200 A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) I addto, we propose the followg estmator combg rato estmators (1.3) ad (1.6) as t pro3 ¼ m 1 P þ m 2 B 2 ðuþþc P ½PB 2 ðuþþc P Š: ð2:6þ The mea square error of ths estmator s aga the same as (2.3) but R 2 (2.4) s replaced wth R 4. Lastl, we propose the followg estmator combg rato estmators (1.3) ad (1.7) as t pro4 ¼ m b 1 P þ m b 2 C P þ B 2 ðuþ ½PC P þ B 2 ðuþš: ð2:7þ The mea square error of ths estmator s aga the same as (2.3) but R 2 (2.4) s replaced wth R 5. The optmal values of m 1 ad m 2 to mmze (2.3) ca easl be foud as follows: m 1 ¼ R 2 R 2 R 1 ad m 2 ¼ R 1 R 1 R 2 ; ð2:8þ whe we use m 1 ad m 2 stead of m 1 ad m 2 (2.4), we get g ¼ B u.asgs depedet of R 2, all proposed estmators have the same mmum MSE as follows: MSE m ðt pro Þ¼ 1 f S 2 2B us u þ B 2 u S2 u ; ¼ 1; 2; 3; 4: We ca also wrte ths expresso as MSE m ðt pro Þ¼ 1 f S2 ð1 q2 PbÞ: ð2:9þ 2.1. New rato estmators We suggest followg estmator: t pro ¼ ðm m 1P b 1 P þ m 2 Þ; þ m2 where m 1 ad m 2 are ether real umber or the fucto of the kow parameter of auxlar attrbute such as C P ; B 2 ðuþ ad q Pb, ote that the sum of m 1 ad m 2 ot ecessarl equal to oe. The followg scheme presets some of the mportat estmators of the populato mea, whch ca be obtaed b sutable choce of costats m 1 ad m 2 : ð2:10þ Estmator Values of m 1 m 2 t pro1sd ¼ b PþCP PþCP 1 C P t pro2sk ¼ PþB 2ðuÞ b PþB2ðuÞ 1 B 2 ðuþ t pro3us1 ¼ PB 2ðuÞþC P b PB2ðuÞþC P B 2 ðuþ C P t pro3us2 ¼ PCPþB 2ðuÞ b PCPþB 2ðuÞ C P B 2 ðuþ t pro4st ¼ Pþq Pb b PþqPb 1 q Pb We obta the MSE equato for these proposed estmators as MSEðt pro Þ¼ 1 f h Y2 C 2 þ C2 P w ðw 2K Pb Þ ¼ 1; 2;...; 5; where w 1SD ¼ P PþC P ; w 2SK ¼ P ; w PþB 2 ðuþ 3US1 ¼ PB 2ðuÞ PB 2 ðuþþc P ; w 3US2 ¼ PC P ad w PC P þb 2 ðuþ 4ST ¼ Pþq P. Pb ð2:11þ 3. Effcec comparso I ths secto, frstl, we compare MSE of proposed estmators, gve (2.9), wth the MSE of rato estmator preseted Sgh et al. [5], gve (1.8). As we obta the followg codto b these comparso: R 2 S2 u > zero: ð3:1þ
4 A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) Table 1 Percet relatve effceces of ; t NG; t ð ¼ 1; 2;...; 5Þ ad t pro wth respect to. Estmator PREs (.,Þ Populato I II t NG t t t t t t pro We ca fer that all proposed estmators are more effcet tha all rato estmators preseted Sgh et al. [5] all codtos, because the codto gve (3.1) s alwas satsfed. Secodl, we compare the MSE of the ew estmators gve (2.11) wth the varace of sample mea, so we have the followg codto: MSEðt pro Þ < VðÞ; ¼ 1; 2;...; 5; f; w 2q Pb C C P < zero; C 2q Pb > w C ; P ) q Pb > 1 C P w 2 C ; ¼ 1; 2;...; 5: ð3:2þ Whe ths codto s satsfed, proposed estmators are more effcet tha the sample mea. 4. Emprcal stud We ow compare the performace of varous estmators cosdered here usg the two data sets as prevousl used b Shabbr ad Gupta [4]. Populato I (Source: Sukhatme ad Sukhatme [6], p. 256). = Number of vllages the crcles. u = A crcle cosstg more tha fve vllages. N ¼ 89; Y ¼ 3:36; P ¼ 0:124; q Pb ¼ 0:766; C ¼ 0:601; C P ¼ 2:678; ¼ 23; B 2 ðuþ ¼ 6:162; R 1 ¼ 27:18; R 2 ¼ 0:534; R 3 ¼ 1:199; R 4 ¼ 6:019; R 5 ¼ 1:386. Populato II (Source: Sukhatme ad Sukhatme [6], p. 256). = Area ( acres) uder wheat crop the crcles. u = A crcle cosstg more tha fve vllages. N ¼ 89; Y ¼ 1102; P ¼ 0:124; q Pb ¼ 0:624; C ¼ 0:65; C P ¼ 2:678; ¼ 23; B 2 ðuþ ¼6:162; R 1 ¼ 8915; R 2 ¼ 175:31; R 3 ¼ 393:31; R 4 ¼ 6:019; R 5 ¼ 454:468. We have computed the percet relatve effceces (PREs) of ; t NG ; t ð ¼ 1; 2;...; 5Þ ad t pro wth respect to usual ubased estmator ad dsplaed Table 1. From Table 1 t ca be cocluded that all proposed estmators t pro ð ¼ 1; 2; 3; 4Þ are more effcet tha the usual ubased estmator, rato estmators of Nak ad Gupta [3], ad the rato estmators preseted Sgh et al. [5]. 5. Cocluso We have developed ew estmators combg rato estmators cosdered Sgh et al. [5] ad obtaed the mmum MSE equato for the proposed estmators. Theoretcall, we have demostrated that all proposed estmators are alwas more effcet tha rato estmators. I addto, we support ths theoretcal result umercall usg the data used b Shabbr ad Gupta [4].
5 4202 A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) Some other estmators ca also be derved combg rato estmators gve (1.4) (1.7) the form (2.1), but all these estmators have aga the same mmum MSE equato gve (2.9). We would lke to recall that R 1 ad R 2 (2.4) ad (2.8) should be chaged accordg to rato estmators that are combed. Ackowledgemets The authors are deepl grateful to the referee ad the edtor of the joural for ther extremel helpful commets ad valued suggestos that led to ths mproved verso of the paper. Refereces [1] W.G. Cochra, Samplg Techques, Joh Wle ad Sos, New York, [2] C. Kadlar, H. Cg, Improvemet estmatg the populato mea smple radom samplg, Appled Mathematcs Letters 19 (2006) [3] V.D. Nak, P.C. Gupta, A ote o estmato of mea wth kow populato of a auxlar character, Joural of Ida Socet Agrcultural Statstcs 48 (2) (1996) [4] J. Shabbr, S. Gupta, O estmatg the fte populato mea wth kow populato proporto of a auxlar varable, Paksta Joural of Statstcs 23 (1) (2007) 1 9. [5] R. Sgh, P. Chauha, N. Sawa, F. Smaradache, Rato estmators smple radom samplg usg formato o auxlar attrbute, Paksta Joural of Statstcs ad Operato Research IV (1) (2008) [6] P.V. Sukhatme, B.V. Sukhatme, Samplg Theor of Surves wth applcatos, Iowa State Uverst Press, Ames, USA, [7] K.M. Wolter, Itroducto to Varace Estmato, secod ed., Sprger-Verlag, 1985.
Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean
Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationA Generalized Class of Dual to Product-Cum-Dual to Ratio Type Estimators of Finite Population Mean In Sample Surveys
Appl Math If Sc Lett 4 o 5-33 (6) 5 Appled Mathematcs & Ifmato Sceces Letters A Iteratoal Joural http://ddog/8576/amsl/45 A Geeralzed lass of Dual to Product-um-Dual to Rato Tpe stmats of Fte Populato
More informationDepartment of Statistics, Banaras Hindu University Varanasi , India 2 Chair of Department of Mathematics, University of New Mexico, Gallup, USA
A Famly of eda Based Estmators Smple Radom Samplg Hemat K.Verma, Rajesh Sgh ad Floret Smaradache Departmet of Statstcs, Baaras Hdu Uversty Varaas-5, Ida har of Departmet of athematcs, Uversty of e exco,
More informationA Generalized Class of Ratio-Cum-Dual to Ratio Estimators of Finite Population Mean Using Auxiliary Information in Sample Surveys
Math Sc Lett 5 o 3- (6) 3 Mathematcal Sceces Letters A Iteratoal Joural http://ddog/8576/msl/55 A Geeralzed lass of ato-um-dual to ato Estmats of Fte Populato Mea Usg Aular Ifmato Sample Surves Housla
More informationSampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION
Samplg Theor MODULE V LECTUE - 4 ATIO AND PODUCT METHODS OF ESTIMATION D. SHALABH DEPATMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOG KANPU A mportat objectve a statstcal estmato procedure
More informationApplication of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design
Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud Applcato of Calbrato Approach for Regresso Coeffcet Estmato uder Two-stage Samplg Desg Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud
More informationStudy of Correlation using Bayes Approach under bivariate Distributions
Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 Stud of Correlato usg Baes Approach uder bvarate Dstrbutos N.S.Padharkar* ad. M.N.Deshpade** *Govt.Vdarbha Isttute of
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationA Note on Ratio Estimators in two Stage Sampling
Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue, December 0 ISS 0- A ote o Rato Estmators two Stage Samplg Stashu Shekhar Mshra Lecturer Statstcs, Trdet Academy of Creatve Techology (TACT),
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationADAPTIVE CLUSTER SAMPLING USING AUXILIARY VARIABLE
Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03 I: 549-3644 03 cece Publcatos do:0.3844/jmssp.03.49.55 Publshed Ole 9 (3) 03 (http://www.thescpub.com/jmss.toc) ADAPTIVE CLUTER AMPLIG UIG AUXILIARY VARIABLE
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationLECTURE - 4 SIMPLE RANDOM SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR
amplg Theory MODULE II LECTURE - 4 IMPLE RADOM AMPLIG DR. HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR Estmato of populato mea ad populato varace Oe of the ma objectves after
More informationON ESTIMATION OF POPULATION MEAN IN THE PRESENCE OF MEASUREMENT ERROR AND NON-RESPONSE
Pak. J. Statst. 015 ol. 31(5), 657-670 ON ESTIMATION OF POPLATION MEAN IN THE PRESENCE OF MEASREMENT ERROR AND NON-RESPONSE Muhammad Azeem 1 ad Muhammad Haf Natoal College of Busess Admstrato & Ecoomcs,
More informationChapter 3 Sampling For Proportions and Percentages
Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationPart 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
More informationSTRATIFIED SAMPLING IN AGRICULTURAL SURVEYS
3 STRATIFIED SAMPLIG I AGRICULTURAL SURVEYS austav Adtya Ida Agrcultural Statstcs Research Isttute, ew Delh-00 3. ITRODUCTIO The prme objectve of a sample survey s to obta fereces about the characterstc
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationChapter -2 Simple Random Sampling
Chapter - Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More informationMidterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uverst Mchael Bar Sprg 5 Mdterm xam, secto Soluto Thursda, Februar 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes exam.. No calculators of a kd are allowed..
More informationEstimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
More informationChapter 11 Systematic Sampling
Chapter stematc amplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of
More informationBAYESIAN ESTIMATOR OF A CHANGE POINT IN THE HAZARD FUNCTION
Mathematcal ad Computatoal Applcatos, Vol. 7, No., pp. 29-38, 202 BAYESIAN ESTIMATOR OF A CHANGE POINT IN THE HAZARD FUNCTION Durdu Karasoy Departmet of Statstcs, Hacettepe Uversty, 06800 Beytepe, Akara,
More informationChapter -2 Simple Random Sampling
Chapter - Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More informationECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity
ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationBias Correction in Estimation of the Population Correlation Coefficient
Kasetsart J. (Nat. Sc.) 47 : 453-459 (3) Bas Correcto Estmato of the opulato Correlato Coeffcet Juthaphor Ssomboothog ABSTRACT A estmator of the populato correlato coeffcet of two varables for a bvarate
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More information( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model
Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationBayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information
Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst
More informationFourth Order Four-Stage Diagonally Implicit Runge-Kutta Method for Linear Ordinary Differential Equations ABSTRACT INTRODUCTION
Malasa Joural of Mathematcal Sceces (): 95-05 (00) Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method for Lear Ordar Dfferetal Equatos Nur Izzat Che Jawas, Fudzah Ismal, Mohamed Sulema, 3 Azm Jaafar
More informationA Combination of Adaptive and Line Intercept Sampling Applicable in Agricultural and Environmental Studies
ISSN 1684-8403 Joural of Statstcs Volume 15, 008, pp. 44-53 Abstract A Combato of Adaptve ad Le Itercept Samplg Applcable Agrcultural ad Evrometal Studes Azmer Kha 1 A adaptve procedure s descrbed for
More informationLecture 2: The Simple Regression Model
Lectre Notes o Advaced coometrcs Lectre : The Smple Regresso Model Takash Yamao Fall Semester 5 I ths lectre we revew the smple bvarate lear regresso model. We focs o statstcal assmptos to obta based estmators.
More informationMidterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationBootstrap Method for Testing of Equality of Several Coefficients of Variation
Cloud Publcatos Iteratoal Joural of Advaced Mathematcs ad Statstcs Volume, pp. -6, Artcle ID Sc- Research Artcle Ope Access Bootstrap Method for Testg of Equalty of Several Coeffcets of Varato Dr. Navee
More informationChapter 10 Two Stage Sampling (Subsampling)
Chapter 0 To tage amplg (usamplg) I cluster samplg, all the elemets the selected clusters are surveyed oreover, the effcecy cluster samplg depeds o sze of the cluster As the sze creases, the effcecy decreases
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationMultivariate Transformation of Variables and Maximum Likelihood Estimation
Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty
More informationSampling Theory MODULE X LECTURE - 35 TWO STAGE SAMPLING (SUB SAMPLING)
Samplg Theory ODULE X LECTURE - 35 TWO STAGE SAPLIG (SUB SAPLIG) DR SHALABH DEPARTET OF ATHEATICS AD STATISTICS IDIA ISTITUTE OF TECHOLOG KAPUR Two stage samplg wth uequal frst stage uts: Cosder two stage
More informationChapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance
Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss
More informationBayes Interval Estimation for binomial proportion and difference of two binomial proportions with Simulation Study
IJIEST Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue 5, July 04. Bayes Iterval Estmato for bomal proporto ad dfferece of two bomal proportos wth Smulato Study Masoud Gaj, Solmaz hlmad
More informationExponential Type Product Estimator for Finite Population Mean with Information on Auxiliary Attribute
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 10, Issue 1 (June 015), pp. 106-113 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Exponental Tpe Product Estmator
More informationMaximum Likelihood Estimation
Marquette Uverst Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Coprght 08 b Marquette Uverst Maxmum Lkelhood Estmato We have bee sag that ~
More informationClass 13,14 June 17, 19, 2015
Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationSome Notes on the Probability Space of Statistical Surveys
Metodološk zvezk, Vol. 7, No., 200, 7-2 ome Notes o the Probablty pace of tatstcal urveys George Petrakos Abstract Ths paper troduces a formal presetato of samplg process usg prcples ad cocepts from Probablty
More informationInvestigation of Partially Conditional RP Model with Response Error. Ed Stanek
Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More informationLecture 8: Linear Regression
Lecture 8: Lear egresso May 4, GENOME 56, Sprg Goals Develop basc cocepts of lear regresso from a probablstc framework Estmatg parameters ad hypothess testg wth lear models Lear regresso Su I Lee, CSE
More information2SLS Estimates ECON In this case, begin with the assumption that E[ i
SLS Estmates ECON 3033 Bll Evas Fall 05 Two-Stage Least Squares (SLS Cosder a stadard lear bvarate regresso model y 0 x. I ths case, beg wth the assumto that E[ x] 0 whch meas that OLS estmates of wll
More informationDerivation of 3-Point Block Method Formula for Solving First Order Stiff Ordinary Differential Equations
Dervato of -Pot Block Method Formula for Solvg Frst Order Stff Ordary Dfferetal Equatos Kharul Hamd Kharul Auar, Kharl Iskadar Othma, Zara Bb Ibrahm Abstract Dervato of pot block method formula wth costat
More informationUnimodality Tests for Global Optimization of Single Variable Functions Using Statistical Methods
Malaysa Umodalty Joural Tests of Mathematcal for Global Optmzato Sceces (): of 05 Sgle - 5 Varable (007) Fuctos Usg Statstcal Methods Umodalty Tests for Global Optmzato of Sgle Varable Fuctos Usg Statstcal
More informationJournal of Mathematical Analysis and Applications
J. Math. Aal. Appl. 365 200) 358 362 Cotets lsts avalable at SceceDrect Joural of Mathematcal Aalyss ad Applcatos www.elsever.com/locate/maa Asymptotc behavor of termedate pots the dfferetal mea value
More informationComparison of Parameters of Lognormal Distribution Based On the Classical and Posterior Estimates
Joural of Moder Appled Statstcal Methods Volume Issue Artcle 8 --03 Comparso of Parameters of Logormal Dstrbuto Based O the Classcal ad Posteror Estmates Raja Sulta Uversty of Kashmr, Sragar, Ida, hamzasulta8@yahoo.com
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationEvaluating Polynomials
Uverst of Nebraska - Lcol DgtalCommos@Uverst of Nebraska - Lcol MAT Exam Expostor Papers Math the Mddle Isttute Partershp 7-7 Evaluatg Polomals Thomas J. Harrgto Uverst of Nebraska-Lcol Follow ths ad addtoal
More informationCan we take the Mysticism Out of the Pearson Coefficient of Linear Correlation?
Ca we tae the Mstcsm Out of the Pearso Coeffcet of Lear Correlato? Itroducto As the ttle of ths tutoral dcates, our purpose s to egeder a clear uderstadg of the Pearso coeffcet of lear correlato studets
More informationMultiple Choice Test. Chapter Adequacy of Models for Regression
Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to
More information12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model
1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed
More informationChapter 8: Statistical Analysis of Simulated Data
Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample
More informationChapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:
Chapter 3 3- Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic. Chap 2- Learg Objectves I ths chapter, you lear:
More informationSome Statistical Inferences on the Records Weibull Distribution Using Shannon Entropy and Renyi Entropy
OPEN ACCESS Coferece Proceedgs Paper Etropy www.scforum.et/coferece/ecea- Some Statstcal Ifereces o the Records Webull Dstrbuto Usg Shao Etropy ad Rey Etropy Gholamhosse Yar, Rezva Rezae * School of Mathematcs,
More information9.1 Introduction to the probit and logit models
EC3000 Ecoometrcs Lecture 9 Probt & Logt Aalss 9. Itroducto to the probt ad logt models 9. The logt model 9.3 The probt model Appedx 9. Itroducto to the probt ad logt models These models are used regressos
More informationVOL. 3, NO. 11, November 2013 ISSN ARPN Journal of Science and Technology All rights reserved.
VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. http://www.ejouralofscece.org Usg Square-Root Iverted Gamma Dstrbuto as Pror to Draw Iferece o the Raylegh Dstrbuto
More informationGeneral Families of Estimators for Estimating Population Mean in Stratified Random Sampling under Non-Response
J. tat. Appl. Pro. Lett. 3, No., 7-7 06 7 Joural of tatstcs Applcatos & Probablty Letters A Iteratoal Joural http://dx.do.org/0.8576/jsapl/0300 Geeral Famles of Estmators for Estmatg Populato Mea tratfed
More informationEFFICIENT ESTIMATOR IN SUCCESSIVE SAMPLING USING POST-STRATIFICATION
EFFICIET ETIMATOR I UCCEIVE AMPLIG UIG POT-TRATIFICATIO M. Trved* ad D. hula ** ABTRACT It s ofte see that a populato havg large umber of elemets remas uchaged several occasos but the value of uts chages.
More informationSome Applications of the Resampling Methods in Computational Physics
Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 Some Applcatos of the Resampl Methods Computatoal Physcs Sotraq Marko #, Lorec Ekoom * # Physcs Departmet, Uversty of Korca, Albaa,
More informationMAX-MIN AND MIN-MAX VALUES OF VARIOUS MEASURES OF FUZZY DIVERGENCE
merca Jr of Mathematcs ad Sceces Vol, No,(Jauary 0) Copyrght Md Reader Publcatos wwwjouralshubcom MX-MIN ND MIN-MX VLUES OF VRIOUS MESURES OF FUZZY DIVERGENCE RKTul Departmet of Mathematcs SSM College
More informationChapter Two. An Introduction to Regression ( )
ubject: A Itroducto to Regresso Frst tage Chapter Two A Itroducto to Regresso (018-019) 1 pg. ubject: A Itroducto to Regresso Frst tage A Itroducto to Regresso Regresso aalss s a statstcal tool for the
More informationModule 7. Lecture 7: Statistical parameter estimation
Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato
More informationMEASURES OF DISPERSION
MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda
More informationESS Line Fitting
ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here
More informationDISTURBANCE TERMS. is a scalar and x i
DISTURBANCE TERMS I a feld of research desg, we ofte have the qesto abot whether there s a relatoshp betwee a observed varable (sa, ) ad the other observed varables (sa, x ). To aswer the qesto, we ma
More informationECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model
ECON 48 / WH Hog The Smple Regresso Model. Defto of the Smple Regresso Model Smple Regresso Model Expla varable y terms of varable x y = β + β x+ u y : depedet varable, explaed varable, respose varable,
More informationCubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem
Joural of Amerca Scece ;6( Cubc Nopolyomal Sple Approach to the Soluto of a Secod Order Two-Pot Boudary Value Problem W.K. Zahra, F.A. Abd El-Salam, A.A. El-Sabbagh ad Z.A. ZAk * Departmet of Egeerg athematcs
More informationEstimation of Population Total using Local Polynomial Regression with Two Auxiliary Variables
J. Stat. Appl. Pro. 3, o., 9-36 04) 9 Joural of Statstcs Applcatos & Probablty A Iteratoal Joural http://dx.do.org/0.785/jsap/03003 Estmato of Populato Total usg Local Polyomal Regresso wth Two Auxlary
More informationMultiple Linear Regression Analysis
LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple
More informationSimple Linear Regression - Scalar Form
Smple Lear Regresso - Scalar Form Q.. Model Y X,..., p..a. Derve the ormal equatos that mmze Q. p..b. Solve for the ordary least squares estmators, p..c. Derve E, V, E, V, COV, p..d. Derve the mea ad varace
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections
ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty
More informationBeam Warming Second-Order Upwind Method
Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet
More informationEstimation and Testing in Type-II Generalized Half Logistic Distribution
Joural of Moder Appled Statstcal Methods Volume 13 Issue 1 Artcle 17 5-1-014 Estmato ad Testg Type-II Geeralzed Half Logstc Dstrbuto R R. L. Katam Acharya Nagarjua Uversty, Ida, katam.rrl@gmal.com V Ramakrsha
More informationLecture Notes Forecasting the process of estimating or predicting unknown situations
Lecture Notes. Ecoomc Forecastg. Forecastg the process of estmatg or predctg ukow stuatos Eample usuall ecoomsts predct future ecoomc varables Forecastg apples to a varet of data () tme seres data predctg
More informationSPECIAL CONSIDERATIONS FOR VOLUMETRIC Z-TEST FOR PROPORTIONS
SPECIAL CONSIDERAIONS FOR VOLUMERIC Z-ES FOR PROPORIONS Oe s stctve reacto to the questo of whether two percetages are sgfcatly dfferet from each other s to treat them as f they were proportos whch the
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationSTA302/1001-Fall 2008 Midterm Test October 21, 2008
STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from
More information4. Standard Regression Model and Spatial Dependence Tests
4. Stadard Regresso Model ad Spatal Depedece Tests Stadard regresso aalss fals the presece of spatal effects. I case of spatal depedeces ad/or spatal heterogeet a stadard regresso model wll be msspecfed.
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More informationWu-Hausman Test: But if X and ε are independent, βˆ. ECON 324 Page 1
Wu-Hausma Test: Detectg Falure of E( ε X ) Caot drectly test ths assumpto because lack ubased estmator of ε ad the OLS resduals wll be orthogoal to X, by costructo as ca be see from the momet codto X'
More informationComplete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables
Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationObjectives of Multiple Regression
Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of
More information