An Improved Estimator of Population Variance using known Coefficient of Variation

Transcription

1 J. Stat. Appl. Pro. Lett. 4, No. 1, (017) 11 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural A Improved Estimator of Populatio Variace usig kow Coefficiet of Variatio Sheela Misra, Dipika Kumari ad Dharmedra Kumar Yadav Departmet of Statistics, Uiversity of Luckow, Luckow, Idia Received: 7 ju. 016, Revised: 15 Nov. 016, Accepted: Nov. 016 Published olie: 1 Ja. 017 Abstract: I the preset article, a improved estimator (s y k ) over usual ubiased estimator of populatio variace (s y ) is proposed by usig kow coefficiet of variatio (C y ) of the study variable y. Asymptotic expressio for its bias ad mea square error (MSE) have bee obtaied. For more practical utility the study of proposed estimator uder estimated optimum value of k has also bee carried out. A comparative study has bee made betwee the proposed estimator ad the covetioal estimator. Numerical illustratio is also give i support of the preset study. Keywords: Bias, Coefficiet of Variatio, Efficiecy, Mea Square Error, Simple Radom Samplig. 1 Itroductio The problem of estimatig populatio variace arises i may practical situatios like agricultural, biological ad medical studies [Blad ad Altma (1986)] [3]. The problem has bee well dealt i literature i simple radom samplig. This problem is cosidered by Wakimoto (1971) [0] i stratified samplig. Variace estimatio i PPS ad geeral samplig desig was also cosidered by Das ad Tripathi(1977) [6], Liu (1974) [13], Chaudhury (1978) [5], Mukhopadhyay (1978) [14], Swai ad Mishra(1994) [18]. Estimatio of populatio variace uder super-populatio models has bee carried out by Mukhopadhyay (198) [15], Padmawar ad Mukhopadyay (1981) [16]. Takig advatage of high correlatio betwee study ad auxiliary variables, Isaki (1983) [10] proposed ratio ad regressio type estimators of populatio variace. Biradar ad Sigh (1998) [], Agrawal ad Pada (1999) [1] explored their discussio uder predictio approach. The assumptio of a kow coefficiet of variatio is actually commo i may agricultural, biological ad idustrial applicatios. If the situatio arises that the populatio mea is proportioal to the populatio stadard deviatio, the kowig the proportioality costat is equivalet to kowig the populatio coefficiet of variatio. For a more thorough discussio of this cocept we suggest Gleser ad Healy(1976) [8]. For estimatio of fiite populatio variace we assume that the fiite populatio cosists of N idetifiable uits(u 1,U,U 3,...,U N ) takig the values(y 1,Y,Y 3,...,Y N ) o study variable y. Let Ȳ = 1 N N i=1 Y i, σ y = 1 N N i=1 (Y i Ȳ) ad C y = σ ȳ Y be the populatio mea, variace ad coefficiet of variatio of y respectively. Similarly µ r = 1 N N i=1 (Y i Ȳ) r, β 1 = µ 3 µ 3, γ 1 = µ 3 3 µ, β = µ 4 µ, γ = µ 4 3 µ Correspodig author dipikascholar@gmail.com c 017 NSP

2 1 S. Misra et al.: A improved estimator of... Let, ȳ= 1 i=1 y i, s y = 1 1 i=1 (y i ȳ) be the sample mea ad variace of y based o a sample s =(1,,..,) take from U by simple radom samplig. The proposed estimator, usig kow coefficiet of variatio of study variable y for the estimatio of populatio variace σ y is s y k = s y where k is the characterizig scalar to be chose suitably. ( ȳ C y s y ) k (1) Bias ad mea square error of Proposed Estimator For the sake of simplicity we are assumig that the populatio size N is large as compared to sample size so that fiite populatio correctio (fpc) is igored. Let, ȳ= Ȳ(1+e 0 ), s y = σy(1+e 1 ), E(e 0 )=E(e 1 )=0, E(e 0 )= C y, E(e 1 )= γ y+ ( ) s y k = s y ȳ C y s y Now expressig proposed estimator i terms of ei s E(e 0 e 1 )= γ 1yC y From (1) we have s y k = σy (1+e 1) ( ) k Ȳ (1+e 0 ) )( Cy σy (1+e 1) = σ y[1+ke 0 (k 1)e 1 + k(k 1)e 0 k(k 1)e 0e 1 + k(k 1) e ] (s y k σy )=σ y [ke 0 (k 1)e 1 + k(k 1)e 0 k(k 1)e 0e 1 + k(k 1) e ] () O takig expectatio o both the sides of () ad usig first order of approximatio, we get the bias of proposed estimator s y as Bias(s y k )=(s y k σ y)= σ y [k(k 1)C y 4k(k 1)γ 1 C y + k(k 1)(γ y+)] (3) Agai, squarig () both sides ad takig expectatio, we have the mea square error of s y k up to first order of approximatio to be MSE(s y k )=E(s y k σy ) = σy 4[4kE(e 0 )+(k 1) E(e 1 ) 4k(k 1)E(e 0e 1 )] = σ 4 y (γ y+)+ σ 4 y [k (4C y 4γ 1 yc y + γ y+)+k(γ 1 yc y γ y )] (4) The optimum value of k which miimizes the mea square error of s y k i(4) is give by k 0 = (γ 1yC y γ y ) (4C y 4γ 1yC y + γ y + ) The miimum value of mea square error of proposed estimator s y k for k 0 is give by MSE(s y k ) mi = σ y 4 (γ y+ ) σ ( ) y 4 (γ 1y C y γ y ) 4Cy 4γ 1y C y + γ y + (5) (6) c 017 NSP

3 J. Stat. Appl. Pro. Lett. 4, No. 1, (017) / Estimator with Estimated optimum Value of k A alterative procedure for calculatig mea square error whe values of γ 1 y ad γ y or their good guessed values are ot available is to replace these values ivolved i the optimum k by their estimates ˆγ 1y ad ˆγ y based o sample values ad get the estimated optimum value of k deoted by ˆk as Where, ˆγ 1y = ˆµ 3 3 ˆµ ˆk= ( ˆγ 1yC y ˆγ y ) (4C y 4 ˆγ 1yC y + ˆγ y + ) ad ˆγ y = ˆµ 4 3 with ˆµ ˆµ 3 = 1 i=1 (y i ȳ), ˆµ = s y = 1 i=1 (y i ȳ), ˆµ 4 = 1 i=1 (y i ȳ) 4 Thus, replacig k by estimated optimum value of k i the estimator s y k i (1),we get for wider practical utility of the estimator based o the estimated optimum value ˆk is give as (7) To fid the bias ad mea square error of s yˆk, let s yˆk = s y (ȳc y s y )ˆk (8) ˆµ 3 = µ 3 (1+e ), ˆµ 4 = µ 4 (1+e 3 ) Alog with ȳ= Ȳ(1+e 0 ),s y = σ y (1+e 1) ˆk= µ 3 (1+e ) σy(1+e 3 1 ) 3 C y µ 4 (1+e 3 ) σ y 4(1+e 1 )+1 4C y 4 µ 3 (1+e ) σ 3 y (1+e 1 ) 3 C y + µ 4 (1+e 3 ) σ 4 y (1+e 1 ) 1 = γ 1yC y (1+e 3 e 1 3 e 1e + 15 e 1...) (γ y+3)(1+e 3 e 1 e 1 e 3 +3e 1...)+1 4C y 4γ 1 yc y (1+e 3 e 1 3 e 1e + 15 e 1...)+(γ y+3)(1+e 3 e 1 e 1 e 3 +3e 1 ) 1 ( )( γ 1 yc y γ y = 4Cy 1+ γ 1yC y (e 3 e 1 3 e 1e + 15 e 1...) (γ y+3)(e 3 e 1 e 1 e 3 + 3e 1...) ) 4γ 1 yc y + γ y+ γ 1 yc y γ y ( 1+ (γ y+3)(e 3 e 1 e 1 e 3 + 3e 1...) 4γ 1yC y (e 3 e 1 3 e 1e + 15 e 1...) ) 1 4Cy 4γ 1 yc y + γ y+ ( ){ γ 1 yc y γ y = 4Cy 1+ γ 1yC y (e 3 e 1 3 e 1e + 15 e 1...) (γ y+3)(e 3 e 1 e 1 e 3 + 3e 1...) } 4γ 1 yc y + γ y+ γ 1 yc y γ y { γ 1 yc y γ y (γ y+3)(e 3 e 1 e 1 e 3 + 3e ) 4γ 1yC y (e 3 e 1 3/e 1 e + 15 e 1...) } 4Cy 4γ 1yC y + γ y+ 4Cy 4γ (9) 1yC y + γ y+ Substitutig ȳ= Ȳ(1+e 0 ), s y = σy(1+e 1 ) ad ˆk from (9) i (), we have [ ( ) { } ] s σ yˆk y = σy γ 1y C y γ y e 1 4Cy e 0 e 1 e 4γ 1 yc y + γ y + 0+ e 0 e 1 e (10) Takig expectatio of (10) ad igorig terms of e i s greater tha power two, we ca easily check that the bias of s y k is of O( 1 ), hece the bias of s y k egligible for large value of, that is the estimator s y k is approximately ubiased estimator of populatio variace. Now squarig ad takig expectatio of(10), we have MSE(s )=σ 4 yˆk y[e 1 γ 1yC y γ y (4C y 4γ 1 yc y +γ y +) e 0 e 1 ] { } = σ y 4 (γ y+ ) σ y 4 (σ 1y C y γ y ) (4C y 4γ 1yC y + γ y + ) (11) c 017 NSP

4 14 S. Misra et al.: A improved estimator of... Which is same as mea square error for the optimum value of k that is estimator s y k based o estimated value of optimum k also has same mea square error as that of the estimator s y k based o optimum k. 4 Theoretical Efficiecy Compariso We compare the proposed estimator s y k with respect to usual ubiased estimator of populatio variace s y ad the coditio for which the proposed estimator will efficiet is give by MSE(s y k ) MSE(s y)<0 γ 1 yc y < γ y + (1) 5 Numerical Illustratio For umerical illustratio, we cosider the two data as (1) Data give i Cochra(1977, pg34) dealig with the weekly expediture of family o food(y) of 33 low-icome families, the required values are calculated from data are =33,ȳ=7.49,σ y = ,C y= ,γ 1y = ,γ y =.7146 Usig above values, we have MSE(s y )= (13) MSE ( s y k = (14) )mi The percet relative efficiecy (PRE) of the proposed estimator over the usual ubiased estimator for populatio variace is 133%. () Geerated populatio from ormal distributio by usig simulatio techique through R software. The Descriptio of this data is as follows Y = N(5,10),=5000,ȳ=4.95,σy = 99.38,C y=.014,γ 1 y=0.039,γ y= Usig above values, we have MSE(s y )=3.87 (15) MSE(s y k ) mi = 3.5 (16) The percet relative efficiecy (PRE) of the proposed estimator over the usual ubiased estimator for populatio variace is 109%. Hece from both the data set we ca coclude that proposed estimator is better the usual ubiased estimator for populatio variace. 6 Cocludig Remarks (a) From (6) it is observed that the proposed estimator will perform better tha usual ubiased estimator of populatio variace. (b) The estimator s y k with optimum value k 0 ad the estimator based o estimated optimum ˆk have same mea square error give by (γ 1 yc y γ y ) (4C y 4γ 1yC y +γ y+) MSE(s )=MSE(s y ) mi = σ y 4 yˆk (γ y+) σ y 4 (c) For ormal populatio (,i.e.for γ 1y = 0 ad β y=3),the optimum value of k from(5),reduces to k= 1 1+C y For which mea square error of proposed estimator becomes c 017 NSP

5 J. Stat. Appl. Pro. Lett. 4, No. 1, (017) / 15 MSE(s y k )= σ 4 y (C y) (1+C y) showig that the proposed estimator is more efficiet tha usual ubiased estimator i ormal paret populatio also. (d) If for ay dataset (1) holds the proposed estimator will be better tha the usual ubiased estimator of populatio variace. (e) From umerical illustratio (1) it is observed that proposed estimator is 133% more efficiet tha the usual ubiased estimator for populatio variace. (f) From simulatio data aalysis it is observed that proposed estimator is 109% more efficiet tha the usual ubiased estimator for populatio variace. Ackowledgemet The authors are grateful to the aoymous referee for the valuable suggestio ad helpful commets that improved this paper. Authors are also thakful to editor i chief for his/her support ad suggestios. Refereces [1] Agrawal, M. C. ad Pada, K. B., A predictive justificatio for variace estimatio usig auxiliary iformatio, Jour. Id. Soc. Ag. Stat., 1999,, 5, [] Biradar, R. S. ad Sigh, H. P., Predictive estimatio of fiite populatio variace, Cal. Statist. Assoc. Bull., 1998, 48, [3] Blad, J. M. ad Altma, D. G., Statistical method for assessig agreemet betwee two methods of cliical measuremet, Lace, 1986, 1, 8476, [4] Cochra, W.G., Samplig Techiques, Wiley Easter Private Limited, New Delhi., 1963, Secod Editio, [5] Chaudhury, A., O estimatig the variace of a fiite populatio., Metrika, 1978, 5, [6] Das, A. K. ad Tripathi, T. P., Admissible estimators for quadratic forms i fiite populatios., Bull. Iter. Stat. Ist., 1977, Secod Editio, 47,4, [7] Das, A. K. ad Tripathi, T. P., Use of auxiliary iformatio i estimatig the fiite populatio variace., Sakhya, 1978, 4, c, [8] Gleser, L. J., ad Healy, J. D., Estimatig the Mea of a Normal Distributio with Kow Coefficiet of Variatio, Joural of the America Statistical Associatio, 1976, 71, [9] Gupta, S. ad Shabbir, J., Variace estimatio i simple radom samplig usig auxiliary iformatio, Hacettepe Joural of Mathematics ad Statistics, 008, 37, [10] Isaki, C.T., Variace estimatio usig Auxiliary Iformatio, Jour. Amer. Statist. Asssoct, 1983, 78, [11] Kadilar, C. ad Cigi, H., Improvemet i variace estimatio usig auxiliary iformatio, Hacettepe Joural of Mathematics ad Statistics, 006a, 35, 1, [1] Kadilar, C. ad Cigi, H., Ratio estimators for populatio variace i simple ad stratified samplig, Applied Mathematics ad Computatio, 006b, [13] Liu, T. P., A geeralized ubiased estimator for the variace of a fiite populatio, Sakhya, 1974, c, [14] Mukhopadhyay, P., Estimatig a fiite populatio variace uder a super populatio model, Metrika, 1978, [15] Mukhopadhyay, P., Optimum Strategies for estimatig the variace of a fiite populatio uder a super populatio model, Metrika, 198, [16] Padmwar, V. R. ad Mukhopadhya, Y. P., Estimatio of symmetric fuctios of a fiite populatio, Metrika, 1981, [17] Sukhatme P.V., Sukhatme B.V., Sukhatme, S. ad Ashok, C., Samplig Theory of Surveys with Applicatios, Iowa State Uiversity Press, Ams, [18] Swai, A. K. P. C. ad Mishra, G., Estimatio of populatio variace uder uequal probability samplig, Sakhya, 1994, B, [19] Tripathi, T. P., Sigh, H. P. ad Upadhyaya, L. N., A geeral method of estimatio ad its applicatio to the estimatio of coefficiet of variatio, Statistics i Trasitio, 00, 5, [0] Wakimoto, K., Stratified radom samplig (I): Estimatio of Populatio variace, A. Ist. Stat. Math., 1971, 3, [1] Wolter, K. M., Itroductio to variace estimatio, New York, NY: Spriger- Verlag. c 017 NSP

6 16 S. Misra et al.: A improved estimator of... Sheela Misra is Professor ad Head of the Departmet of Statistics, Uiversity of Luckow, Luckow, Idia. She is Ph.D. Supervisor of Dipika ad Dharmedra Kumar Yadav. She has a lot of cotributio i the field of Samplig Theory, Geder Statistics ad Biostatistics. She has successfully orgaized may Natioal ad Iteratioal Cofereces ad may Traiig Programmes i the Departmet. Dipika Kumari is workig as Research Scholar at the Departmet of Statistics, Uiversity of Luckow. She has preseted may research papers i Natioal ad Iteratioal Cofereces. Her area of research is Samplig theory Dharmedra Kumar Yadav is Research Scholar at Departmet of Statistics, Uiversity of Luckow. He has preseted may research papers i Natioal ad Iteratioal Cofereces. His area of research is No-Samplig Errors. c 017 NSP