Estimation of Population Variance Using the Coefficient of Kurtosis and Median of an Auxiliary Variable Under Simple Random Sampling

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1 Estimatio of Populatio Variace Usig the Coefficiet of Kurtosis ad Media of a Auxiliary Variable Uder Simple Radom Samplig Toui Kiplagat Milto MS /2016 A Thesis Submitted to Pa Africa Uiversity, Istitute of Basic Scieces, Techology ad Iovatio i Partial fulfillmet of the requiremet for the award of Master of Sciece i Mathematics(Statistics Optio) of the Pa Africa Uiversity 2018

2 DECLARATION This thesis is my origial work ad has ot bee submitted to ay other Uiversity for ay award. Toui Kiplagat Milto MS /2016 Sigature...Date... This thesis has bee submitted for examiatio with our approval as Uiversity supervisors Prof. Romaus Otieo Odhiambo Departmet of Statistics ad Actuarial Scieces Jomo Keyatta Uiversity Of Agriculture ad Techology P.O. Box , Nairobi, Keya. Sigature...Date... Prof. George Otieo Orwa Departmet of Statistics ad Actuarial Scieces Jomo Keyatta Uiversity Of Agriculture ad Techology P.O. Box , Nairobi, Keya. Sigature...Date... i

3 ACKNOWLEDGMENTS First, I would like to thak God, The Almighty, for the gift of life, grace ad stregth that has eabled me reach this far. Secodly, I would like to express my sicere gratitude to Prof. Romaus Odhiambo for the cotiuous uwaverig support ad ecouragemet i my MSc. study ad related research. His guidace, motivatio ad immese kowledge helped me i all the time of research ad writig of this thesis. I could ot have imagied havig a better guide ad metor for my MSc. study. Besides my guide, I would like to thak Prof. George Orwa for his isightful commets ad ecouragemet. My sicere thaks also goes to Africa Uio Commissio, who provided me a opportuity to udertake my graduate studies. Without their precious support it would ot be possible to coduct this research. Last but ot least, I would like to thak my family: my mother ad my sibligs for supportig me spiritually throughout writig this proposal ad my life i geeral. Special thaks to my late father who tirelessly supported me i my educatio. ii

4 Cotets DECLARATION i ACKNOWLEDGMENTS ii DEDICATION v List of Tables vi List of Appedices vii ABSTRACT viii CHAPTER ONE 1 1 INTRODUCTION Backgroud of the study Problem Statemet Justificatio of the study Objectives of the study Geeral Objective Specific Objectives Sigificace of the study The Scope of the study CHAPTER TWO 7 2 LITERATURE REVIEW Existig Populatio Variace Estimators Statistical Properties(Bias ad MSE) Empirical studies iii

5 2.4 Taylor s Liearizatio Method CHAPTER THREE 18 3 METHODOLOGY Liearity of Expectatio Expected value of the ratio of correlated radom variables Taylor s approximatio method Proposed Estimator Expressios for Bias ad Mea Squared Errors of the Proposed ad Existig Estimators CHAPTER FOUR 29 4 RESULTS AND DISCUSSION Theoretical Evaluatio Numerical Studies Bias ad Mea Squared Errors Efficiecy Compariso CHAPTER FIVE 34 5 CONCLUSION AND RECOMMENDATIONS Coclusio Recommedatios REFERENCES 35 APPENDICES 40 iv

6 DEDICATION I dedicate this work to my frieds ad sibligs who have bee affected i every way possible by this quest. Thak you. My love for you all ca ever be quatified. GOD BLESS YOU. v

7 List of Tables 1 Summary of Expressios of Statistical Properties (Bias ad Mea Squared Errors(MSE)) Bias ad Mea Squared Errors(MSE) of Existig ad Proposed Estimators for Populatio Variace Percet Relative Efficiecies(PRE) of Existig ad Proposed Estimators.. 33 vi

8 List of Appedices Appedix I, Murthy Populatio Data Set Appedix II, Daroga Populatio Data Set Appedix III, Cochra Populatio Data Set vii

9 ABSTRACT I this study we have proposed a modified ratio type estimator for populatio variace of the study variable y uder simple radom samplig without replacemet makig use of coefficiet of kurtosis ad media of a auxiliary variable x. The estimator s properties have bee derived up to first order of Taylor s series expasio. The efficiecy coditios are derived theoretically uder which the proposed estimator performs better tha existig estimators. Empirical studies have bee doe usig real populatios to demostrate the performace of the developed estimator i compariso with the existig estimators. The proposed estimator as illustrated by the empirical studies performs better tha the existig estimators uder some specified coditios i.e. it has the smallest Mea Squared Error ad the highest Percetage Relative Efficiecy. The proposed estimator is therefore suitable to be applied to situatios i which the variable of iterest has a positive correlatio with the auxiliary variable. viii

10 CHAPTER 0NE 1 INTRODUCTION 1.1 Backgroud of the study It is otable that the appropriate use of auxiliary iformatio i probability samplig desigs yields cosiderable reductio i the variace of the estimators of populatio parameters amely, populatio mea, media,variace,regressio coefficiet ad populatio correlatio coefficiet. Cochra (1940) was the first to show the cotributio of kow auxiliary iformatio i improvig the efficiecy of the estimator of the populatio mea Ȳ i survey samplig. I this study we are iterested i the estimatio of populatio variace usig kow auxiliary iformatio uder simple radom samplig without replacemet(srswor) samplig scheme. The precisio of estimators uder this situatio is always icreased, the ratio, product ad regressio estimators gives better outcome tha those of simple radom samplig. Variace estimatio has become a priority as may surveys require that the quality of the statistics be assessed. Samplig variace which is a estimate of the populatio variace is a key idicator of quality i sample surveys ad estimatio. Variace helps the user to draw more accurate coclusios about the statistics produced ad it is also importat for the desig ad estimatio phases of surveys. However due to the complexity of the methods used for the desig ad aalysis of the survey like the samplig desig, weightig, ad the type of estimators ivolved the calculatios are ot straightforward. Variace estimatio i sample survey is crucial for future surveys either for determiatio of sample size or stratificatio. Usually i the estimatio of the fiite populatio mea survey data is used, however i may situatios the mea may ot be a appropriate average sice 1

11 it fluctuates from large to small observatios or outliers i a set of data. Hece the eed for the populatio variace to overcome the difficulty. O regular istaces we ecouter surveys i which a auxiliary variable x is relatively cheap (with regard to time ad moey) to observe tha the study variable y. Use of auxiliary iformatio ca icrease the precisio of a estimator whe the study variable y is highly correlated with auxiliary variable x. I reality such situatios do occur whe iformatio is available i the form of auxiliary variable, which is highly correlated with study variable, for example: (a) Sex ad height of the persos, (b) Amout of milk produced ad a particular breed of the cow, (c) Amout of yield of wheat crop ad a particular variety of wheat etc. (d) Number of trees i a orchard ad the yield of fruits. May authors have come up with more precise estimators by employig prior kowledge of certai populatio parameter(s). For istace Searls (1964) used coefficiet of variatio of study variable at estimatio stage. I practice however, this coefficiet of variatio is seldom kow. Motivated by Searls (1964) work, Se (1978), Sisodia ad Dwivedi (1981) ad Upadhyaya ad Sigh (1984) used the kow coefficiet of variatio of the auxiliary variable for estimatig populatio mea of study variable i ratio method of estimatio. Reasoig alog the same path Hirao et al. (1973) used the prior value of coefficiet of kurtosis i estimatig the populatio variace of the study variable y. Kurtosis i most cases is ot reported or used i may research articles, i spite of the fact that virtually speakig every statistical package provides a measure of kurtosis. This maybe attributed to the likelihood that kurtosis is ot well uderstood or its importace i various aspects of statistical aalysis has ot bee explored fully. Kurtosis ca simply be expressed 2

12 as κ = E(x µ)4 (E(x µ) 2 ) 2 = µ4 σ 4 where E is the expectatio operator, µ is the mea, µ 4 is the fourth momet about the mea ad σ is the stadard deviatio. Media beig the middlemost value i a distributio (whe the values are arraged i ascedig or descedig order) has the advatage of beig less affected by the outliers ad skewed data, thus is preferred to the mea especially whe the distributio is ot symmetrical. We ca therefore utilize the media ad the coefficiet of kurtosis of the auxiliary variable to derive a more precise ratio type estimator for populatio variace. 1.2 Problem Statemet The theory ad applicatios of survey samplig have grow tremedously i the last 7 decades. May authors have cosidered the estimatio of populatio variace, from the iitial works of Evas (1951), Hase et al. (1953), Isaki (1983), Das ad Tripathi (1978), Srivastava ad Jhajj (1980), Upadhyaya ad Sigh (1983), Upadhyaya ad Sigh (1999), Sigh (2001), Sigh et al. (2003), Kadilar ad Cigi (2006), Gupta ad Shabbir (2008), Grover (2010), Sigh et al. (2011), Kha ad Shabbir (2013a), ad recetly Yadav et al. (2016). High umber of surveys are ow carried out every year i the various govermetal agecies, the private sector ad the academic commuity, both i Keya ad the etire world at large. For istace the atiowide surveys about health care, ecoomic activity, poverty(people s wellbeig), eergy usage ad uemploymet; market researches ad public opiio surveys; ad surveys associated with academic research studies. I the curret world, survey samplig touch almost every field of scietific study, icludig demography, educatio, eergy, trasportatio, health care, ecoomics, forestry, sociology, politics ad so o. I fact it is ot a exaggeratio to say that much of the data udergoig 3

13 ay form of statistical aalysis are collected i surveys. It is imperative to ote that as the umber ad uses of sample surveys icrease, so is the eed for methods of aalyzig ad iterpretig the resultig data. A cetral requiremet for early all forms of aalysis ad ideed the prime requiremet of good survey practice, is that measure of precisio be provided for each estimate derived from the survey data. The most commo ad widely used measure of precisio is the variace of the survey estimator. I reality however, populatio variaces are always ot kow but must be estimated from the survey data themselves. The problem of costructig such estimate of the populatio variace which is more efficiet usig both the coefficiet of kurtosis ad media has ot bee explored. As a result of the ecessity to offer solutios to fill the gap i methodological problems ecoutered i the estimatio of populatio variace of the study variable, this study is udertake utilizig the populatio coefficiet of kurtosis ad the media of the auxiliary variable. 1.3 Justificatio of the study The approach employed i the developmet of proposed estimator is umerical studies ad existig literature. We ot oly propose a theoretically more efficiet populatio variace estimator but also test its efficiecy usig real data from atural populatio existig i literature; as a cosequece of a umber of factors that a good estimator for the populatio variace estimator should possess; umerical studies stregthes, puts flesh o the boes of a survey estimator. 4

14 1.4 Objectives of the study Geeral Objective The mai objective of this study is to estimate the populatio variace usig the coefficiet of kurtosis ad media of a auxiliary Variable uder simple radom samplig Specific Objectives The above geeral objective is accomplished by fulfillig the followig research objectives: 1. To develop a modified ratio type populatio variace estimator usig the coefficiet of kurtosis ad media of the auxiliary variable. 2. To evaluate the bias ad Mea Squared Error (MSE) of the proposed modified ratio type populatio variace estimator. 3. To perform empirical study to assess the performace of the proposed estimator vis-a-vis the existig estimators usig Percetage Relative Efficiecies (PREs). 1.5 Sigificace of the study The mathematical results obtaied i this study adds value ad kowledge to the field of sample surveys, a ew more efficiet modified ratio type populatio variace estimator has bee developed makig useful use of coefficiet of kurtosis ad the media of the auxiliary variable. Further to the society cosiderig the fact that mathematics plays a importat role i our day to day activities that ivolve statistical aalyses. The greater demad for more precisio i the use of survey data justifies the eed to develop more efficiet estimators with high precisio. Thus, usig the approach of estimatio derived from this study achieves better results tha the existig estimators. 5

15 1.6 The Scope of the study This study focused o estimatio of populatio variace uder simple radom samplig utilizig the kowledge of kow coefficiet of kurtosis ad media of the auxiliary variable. Assumig simple radom samplig, Bias ad Mea squared error has bee obtaied up to first order of approximatios. Efficiecy compariso of existig ad proposed modified ratio type populatio variace estimators usig the MSEs has bee implemeted o the data from the atural populatios existig i the literature usig percet relative efficiecy (PRE). 6

16 CHAPTER TWO 2 LITERATURE REVIEW 2.1 Existig Populatio Variace Estimators I this sectio we have reviewed some of the existig estimators available i literature which will help i the costructio ad developmet of the proposed estimator. Whe there is o auxiliary iformatio the usual ubiased estimator to the populatio variace of the study variable is t 1 = s 2 y (1) Populatio variace, S 2 y estimatio usig auxiliary iformatio was cosidered by Isaki (1983), ad proposed ratio type populatio variace estimator, give by t 2 = s 2 Sx 2 y s 2 x (2) Usage of prior value of coefficiet of kurtosis i estimatig populatio variace of study variable y was first doe by Hirao et al. (1973). Later, the coefficiet of kurtosis was used by Se (1978), Upadhyaya ad Sigh (1984), Searls ad Itarapaich (1990) i the estimatio of populatio mea of study variable. Srivastava ad Jhajj (1980), proposed a geeral class of ratio type estimators for estimatig the fiite populatio variace Sy 2 as Ŝ 2 SJ = s 2 G(u, v) (3) where u = x X, v = s2 x S 2 x ad G(u, v) is a fuctio of u ad v such that (i) The poit (u, v) assumes a value i a closed covex subset R 2 of two dimesioal real space cotaiig the poit (1, 1). (ii) The fuctio G(u, v) is cotiuous ad bouded i R 2 7

17 (iii) G(1, 1) = 1 (iv) The first ad secod order partial derivatives of G(u, v) exist ad are cotiuous ad bouded i R 2. We ote that all ratio or product type estimators of populatio variace cosidered by Das ad Tripathi (1978) ad Kaur ad Sigh (1982) are special cases of class of estimators of Srivastava ad Jhajj (1980). The kowledge of coefficiet of kurtosis of a variable uder study is seldom available. However, the coefficiet of kurtosis of a auxiliary variable ca be obtaied easily. I order to have the survey estimate for populatio mea Ȳ of the study variable y for istace assumig the kowledge of populatio mea X of the auxiliary variable x we have the well kow ratio estimator. ˆȲ R = ȳ( X x ) (4) where ȳ ad x are the uweighted sample mea of the variable y ad x respectively. The Bias ad MSE of ˆȲ R to first order approximatio are give by B( ˆȲ R ) = θȳ C2 x(1 K) MSE( ˆȲ R ) = θȳ 2 [C 2 y + C 2 x(1 2K)] where θ = 1 Cy, K = ρ( N C x ), C y ad C x are coefficiets of variatio of y ad x respectively ad ρ is the correlatio coefficiet betwee y ad x. Prasad ad Sigh (1990) cosidered a ratio type for estimatig the fiite populatio variace by improvig o the Isaki s estimator(1983) i terms of bias ad precisio. Sigh (1991) cosidered a geeral class of estimators for estimatig the fiite populatio variace S 2 y ad defied his estimator, Ŝ 2 S91 as Ŝ 2 S91 = s 2 yg(u, v) (5) where u = x x, v = s2 x s 2 x coditios: (i) G(1, 1) = 1 ad G(u, v) is a parametric fuctio satisfyig the followig regularity 8

18 (ii) The first ad secod order partial derivatives of G with respect to u ad v exist ad are cotiuous ad kow costats. Upadhyaya ad Sigh (1999) usig the kow iformatio o both S 2 x ad κ x suggested modified ratio type populatio variace estimator for S 2 y as t 3 = s 2 y[ S2 x + κ x s 2 x + κ x ] (6) Upadhyaya ad Sigh (2001) utilized the mea of the auxiliary variable ad proposed the followig modified ratio estimator of populatio variace Ŝ 2 U01 = s 2 y[ X x ] (7) Sigh et al. (2004) assumig kow coefficiet of kurtosis κ x ad usig the trasformatio µ i = x i +κ x,(i=1,2,...,n) suggested the followig modified ratio estimator for the populatio mea Ȳ as ˆȲM = ȳ( X + κ x x + κ x ) (8) To first order approximatio the bias ad MSE of ˆȲ M was obtaied by lettig ȳ = Ȳ (1 + ξ 0), x = X(1+ξ 1 ) so that E(ξ 0 )=E(ξ 1 )=0 ad V (ξ 0 ) = C2 y, V (ξ 1 ) = C2 x ad Cov(ξ 0, ξ 1 ) = ρc yc x. Assumptio is made that the sample size is large eough to make ξ 0 ad ξ 1 < 1 so as to validate the first degree approximatio i.e. the terms ivolvig ξ 0 ad/or ξ 1 havig powers greater tha two will be egligible. The ˆȲ M = Ȳ (1 + ξ 0)(1 + λξ 1 ) 1 (9) where λ = X x+κ x. Suppose that λξ 1 < 1 so that (1 + λξ 1 ) 1 coverges. The the Bias ad MSE of ˆȲ M to first degree of approximatio, respectively are give by Bias( ˆȲ M ) = 1 f Ȳ λc2 x(λ K) (10) MSE( ˆȲ M ) = 1 f Ȳ 2 [C 2 y + λc 2 x(λ 2K)] (11) 9

19 Arcos et al. (2005) also came up with aother type of modified ratio estimator that improved o Isaki s estimator (1983) which is less biased ad more precise tha the previous existig estimators, give by Ŝ 2 Ar = s 2 y + c(s 2 x s 2 x) + d( X x) (12) Kadilar ad Cigi (2006) suggested four modified ratio type variace estimators usig kow values of coefficiet of variatio variatio C x ad coefficiet of kurtosis κ x of a auxiliary variable X as follows t 4 = s 2 y{ S2 x C x s 2 x C x } (13) t 5 = s 2 y{ S2 x κ x s 2 x κ x } (14) t 6 = s 2 y{ S2 xκ x C x s 2 xκ x C x } (15) t 7 = s 2 y{ S2 xc x κ x s 2 xc x κ x } (16) Sigh et al. (2011) improved Bahl ad Tuteja (1991) expoetial ratio type estimator for the populatio mea defied as, Ȳ = ȳ exp[ X x ] ad proposed the followig expoetial ratio X+ x type estimator for the populatio variace as: Ŝ 2 S11 = s 2 yexp S2 x s 2 x S 2 x + s 2 x (17) Usig the kow value of populatio media M x of the auxiliary variable x Subramai ad Kumarapadiya (2012a) have suggested the modified ratio type estimator of the populatio variace S 2 y of study variable as t 8 = s 2 y{ S2 x + M x s 2 x + M x } (18) Subramai ad Kumarapadiya (2012b) have proposed the modified ratio type estimators of populatio variace S 2 y usig the kow quartiles of the auxiliary variable x as t 9 = s 2 y{ S2 x + Q 1 s 2 x + Q 1 } (19) t 10 = s 2 y{ S2 x + Q 3 s 2 x + Q 3 } (20) 10

20 Motivated by Kadilar ad Cigi (2006) ad Subramai ad Kumarapadiya (2012a), Subramai ad Kumarapadiya (2013) cosidered the estimatio of fiite populatio variace usig kow coefficiet of variatio ad media of a auxiliary variable, proposed a estimator, give as: t 11 = s 2 y[ C xs 2 x + M x C x s 2 x + M x ] (21) Kha ad Shabbir (2013b) gave a ratio type estimator of populatio variace usig coefficiet of correlatio ad upper quartile of auxiliary variable x. The problem herei was built o Isaki s kow parameter variace estimator. The estimator postulated is give as: [ ] S t 12 = s 2 2 x ρ xy + Q 3 y s 2 xρ xy + Q 3 (22) Kha (2015) proposed a improved modified ratio type estimator for fiite populatio variace usig the trasformatio of variables. Ŝ 2 K15 = s 2 y[α{2 ( S2 x + κ x S 2 x + κ x )} + (1 α){2 ( S2 x + κ x S 2 x + κ x )}] (23) The mea squared error of his proposed estimator is less tha the mea squared errors of previously suggested existig estimators meaig that it got some good gai i efficiecy. Yadav et al. (2016) cosidered a efficiet dual to ratio ad product estimator of the populatio variace, makig use of the coefficiet of kurtosis ad mea of the auxiliary variable ad proposed the followig improved ratio type estimator of the populatio variace Ŝ 2 Y 16 = s 2 y[ x + α X X + α x ] (24) where α is a suitably chose characterizig costat ad is obtaied by miimizig the MSE of the proposed estimator t Y ad x = N X x N = (1 + g) X g x, g = N. Bhat et al. (2017) estimated variace usig Tri-mea(TM) ad semi-quartile rage of the auxiliary variable x, defied as T M = Q 1+2M x+q 3 4 ad Q a = Q 3+Q 1 2 respectively. The estimator is give by: t 13 = s 2 y [ ] S 2 x + (T M + Q a ) s 2 x + (T M + Q a ) 11 (25)

21 2.2 Statistical Properties(Bias ad MSE) First, we defie the otatios we are usig i this sectio: µ rs = 1 N 1 Σ i=1(y i ȳ) r (x i x) s, λ rs = µrs. Thus we ote the followig; µ r 2 20 µ 2 s 02 µ 20 = Sy, 2 µ 02 = Sx, 2 ad µ 11 = S xy ; λ 22 = µ 22 µ 20 µ 02, λ 21 = µ 21 such that; µ 20 µ C y = S2 y = µ 20 Ȳ 2 Ȳ 2 coefficiet of variatio for the study variable y, C x = S2 x X = µ 02 2 X 2 of variatio for the auxiliary variable x ad ρ xy = Sxy S xs y = µ 11 µ20 µ02 coefficiet coefficiet of correlatio betwee x ad y, κ (y) = λ 40 = µ 40 coeffciet of kurtosis for the study variable, κ µ 2 (x) = λ 04 = µ µ 2 02 coefficiet of kurtosis for the auxiliary variable ad M x populatio media of the auxiliary variable. The bias ad variace of t 1 to first order approximatio are give by: Bias(t 1 ) = 1 f S2 y{(κ x 1)Ψ 1 (Ψ 1 λ 22 1 )} = 0 (26) κ x 1 MSE(t 1 ) = V ar(t 1 ) = 1 f S4 y{(κ y 1) + (κ x 1)Ψ 1 (Ψ 1 2( λ 22 1 κ x 1 ))} (1 f) = S 4 y(κ y 1) where Ψ 1 = 0 (27) Prasad ad Sigh (1990) obtaied the bias ad Mea Squared Error of Isaki s estimator, to first order of approximatio as follows Bias(t 2 ) = 1 f S2 y{(κ x 1)Ψ 2 (Ψ 2 λ 22 1 (1 f) )} = S 2 κ x 1 y[(κ x 1) (λ 22 1)] (28) where Ψ 2 = 1 MSE(t 2 ) = 1 f S4 y{(κ y 1) + (κ x 1)Ψ 2 (Ψ 2 2( λ 22 1 κ x 1 ))} (1 f) = S 4 y[(κ y 1) + (κ x 1) 2(λ 22 1)] (29) Upadhyaya ad Sigh (1999) estimator usig the kow iformatio o both S 2 x ad κ x obtaied the bias ad MSE of their estimator t 3 to first order of approximatio Bias(t 3 ) = 1 f S2 y[({κ x 1})Ψ 3 (Ψ 3 λ 22 1 )] (30) κ x 1 12

22 where Ψ 3 = MSE(t 3 ) = 1 f S4 y[{κ y 1} + {κ x 1}Ψ 3 (Ψ 3 2( λ 22 1 ))] (31) κ x 1 S2 x S 2 x+κ x Upadhyaya ad Sigh (2001) obtaied the bias ad MSE of their modified ratio type populatio variace Ŝ2 U01 estimator up to first order approximatios Bias(Ŝ2 U01) = 1 f S2 y[c 2 x λ 21 C x ] (32) MSE(Ŝ2 U01) = 1 f S4 y[(λ 40 1) + C 2 x 2λ 21 C x ] (33) Kadilar ad Cigi (2006), derived the biases ad MSE of their four modified ratio type variace estimators to first order approximatios to get; where; Ψ 4 = S2 x ; Ψ Sx 2 Cx 5 = Bias(t 4 ) = 1 f S2 y(κ x 1){Ψ 4 (Ψ 4 λ 22 1 )} (34) κ x 1 MSE(t 4 ) = 1 f S4 y{(κ y 1) + Ψ 4 (κ x 1)(Ψ 4 2( λ 22 1 ))} κ x 1 (35) Bias(t 5 ) = 1 f S2 y(κ x 1){Ψ 5 (Ψ 5 ( λ 22 1 ))} κ x 1 (36) MSE(t 5 ) = 1 f S4 y{(κ y 1) + Ψ 5 (κ x 1)(Ψ 5 2( λ 22 1 ))} κ x 1 (37) Bias(t 6 ) = 1 f S2 y(κ x 1){Ψ 6 (Ψ 6 ( λ 22 1 ))} κ x 1 (38) MSE(t 6 ) = 1 f S4 y{(κ y 1) + Ψ 6 (κ x 1)(Ψ 6 2( λ 22 1 ))} κ x 1 (39) Bias(t 7 ) = 1 f S2 y(κ x 1){Ψ 7 (Ψ 7 ( λ 22 1 ))} κ x 1 (40) MSE(t 7 ) = 1 f S4 y{(κ y 1) + Ψ 7 (κ x 1)(Ψ 7 2( λ 22 1 ))} κ x 1 (41) S2 x ; Ψ Sx 2 κx 6 = S2 x κx ; Ψ Sx 2κx Cx 7 = S2 x Cx. Sx 2Cx κx Subramai ad Kumarapadiya (2013) obtaied the bias ad MSE of their estimator t 8 to first order approximatio as: Bias(t 8 ) = S2 y(κ x 1){Ψ 8 (Ψ 8 ( λ 22 1 ))} κ x 1 MSE(t 8 ) = S4 y{(κ y 1) + Ψ 8 (κ x 1)(Ψ 8 2( λ 22 1 ))} κ x 1 13

23 where, Ψ 8 = S2 x. Sx 2+Mx This estimator is more efficiet i terms of bias ad mea squared error tha the traditioal ratio type ad precedig modified ratio type populatio variace estimators uder specified coditios. Subramai ad Kumarapadiya (2012b) i their proposed modified ratio type populatio variace estimators usig the kow quartiles of the auxiliary variable x (upper ad lower quartile Q 3 ad Q 1 respectively) came up with the bias ad MSE of their estimators t 9 ad t 10 as follows Bias(t 9 ) = S2 y(κ x 1){Ψ 9 (Ψ 9 ( λ 22 1 ))} κ x 1 MSE(t 9 ) = S4 y{(κ y 1) + Ψ 9 (κ x 1)(Ψ 9 2( λ 22 1 ))} κ x 1 Bias(t 10 ) = S2 y(κ x 1){Ψ 10 (Ψ 10 ( λ 22 1 ))} κ x 1 MSE(t 10 ) = S4 y{(κ y 1) + Ψ 10 (κ x 1)(Ψ 10 2( λ 22 1 ))} κ x 1 where Ψ 9 = S2 x S 2 x+q 1 ad Ψ 10 = S2 x S 2 x+q 3. The modified ratio type estimator by Subramai ad Kumarapadiya (2013) takig motivatio from Kadilar ad Cigi (2006) ad Subramai ad Kumarapadiya (2012a) obtaied bias ad MSE of their proposed populatio variace estimator that utilizes the coefficiet of variatio ad media of auxiliary variable as follows: where Ψ 11 = Bias(t 11 ) = 1 f S2 y(κ x 1){Ψ 11 (Ψ 11 ( λ 22 1 ))} (42) κ x 1 MSE(t 11 ) = 1 f S4 y{(κ y 1) + Ψ 11 (κ x 1)(Ψ 11 2( λ 22 1 ))} (43) κ x 1 CxS2 x C xsx+m 2 x. The bias ad MSE of t 12 to first order of approximatios is give by: where Ψ 12 = Bias(t 12 ) = 1 f MSE(t 12 ) = 1 f S4 y S2 x ρxy. Sx 2ρxy+Q 3 S2 y [(κ x 1)Ψ 12 ( Ψ 12 [ (κ y 1) + Ψ 12 (κ x 1) ( ))] λ22 1 κ x 1 ( λ22 1 κ x 1 ( Ψ 12 2 ))] Yadav et al. (2016) derived the bias ad MSE of their efficiet dual to ratio ad product 14 (44) (45)

24 estimator of the populatio variace, Ŝ 2 Y 16 to first order approximatios as where F = 1 α 1+α which is miimum for F = λ 21 gc x value of F is, Bias(Ŝ2 Y 16) = 1 f S2 y[f gλ 21 C x + F α 1 + α g2 C 2 x] (46) MSE(Ŝ2 Y 16) = 1 f S4 y[(λ 40 1) + F 2 g 2 C 2 x 2F gλ 21 C x ] (47) ad the miimum mea squared error of t Y for this optimum MSE mi (Ŝ2 Y 16) = 1 f S4 y[(λ 40 1) λ 2 21] (48) The Bias ad MSE of t 13 to first order of approximatios is give by: where Ψ 13 = Bias(t 13 ) = 1 f MSE(t 13 ) = 1 f S4 y S 2 x S 2 x+t M+Q a S2 y [(κ x 1)Ψ 13 ( Ψ 13 [ (κ y 1) + Ψ 13 (κ x 1) ( ))] λ22 1 κ x 1 ( λ22 1 κ x 1 ( Ψ 13 2 ))] (49) (50) 2.3 Empirical studies The performace of proposed modified ratio type variace estimator is always assessed by may authors usig empirical studies comparig it with the traditioal ad existig modified ratio type variace estimators. Subramai ad Kumarapadiya (2013) used real data from the Italia Bureau for Eviromet Protectio (APAT) 2004 Report o Waste 2004 to assess the performace of their estimator. Their results showed that the bias ad mea squared error of their proposed estimator is less tha the biases ad mea squared errors of the traditioal ad existig estimators. Kha ad Shabbir (2013a) cosidered two atural populatios from the literature of survey to perform efficiecy compariso of their proposed estimator with the existig estimators. 15

25 Populatio 1 from Das (1988) ad populatio 2 from (Cochra, 1977, p.325). They coclude out of their empirical studies that their estimator uder optimizig coditios was more efficiet tha the existig estimators. 2.4 Taylor s Liearizatio Method Applyig the Taylor Liearizatio method, o-liear statistics are approximated by liear forms of the observatios (by takig the first-order terms i a appropriate Taylor-series expasio). Secod or eve higher-order approximatios could be developed by extedig the Taylor series expasio. However, i practice, the first-order approximatio usually yields satisfactory results, with the exceptio of highly skewed populatios Wolter (2007). After applyig Taylor s approximatio, stadard variace estimatio techiques ca the be applied to the liearized statistic. This implies that Taylor Liearizatio is ot i itself method for variace estimatio, it simply provides approximate liear forms of the statistics of iterest (e.g. a weighted total) ad the other methods should be deployed for the estimatio of variace itself (either aalytic or approximate oes). Taylor liearizatio method is a widely applied method because it is quite straightforward for ay case where a estimator already exists for totals. However, the Taylor liearizatio variace estimator is a biased estimator. Its bias stems from its tedecy to uderestimate the true value ad it depeds o the size of the sample ad the complexity of the estimated parameter. Though, if the statistic is fairly simple, like the weighted sample mea, the bias is egligible eve for small samples, while it becomes il for large samples Sardal et al. (1992). O the other had for a complex estimator for a parameter like the variace, large samples are eeded for the bias to be small. It is the most popular method of variace estimatio for complex statistics such as ratio ad regressio estimators ad logistic regressio coefficiet estimators. Geerally applicable to ay samplig desig that permits ubiased variace estimatio for liear estimators. Its 16

26 advatage is that it is computatioally simpler ad more compatible with may existig programs ad softwares tha the resamplig methods such as the jackkife. 17

27 CHAPTER THREE 3 METHODOLOGY Cosider a fiite populatio V = {V 1, V 2, V 3,..., V N } of N distict idetifiable uits. Let Y be our study variable ad X be its correspodig auxiliary variable. Suppose we take a radom sample of size from this bivariate populatio (Y, X) that is (y i, x i ), for i = 1, 2, 3,..., usig a Simple Radom Samplig Without Replacemet (SRSW OR) method. Let Ȳ ad X be the populatio meas of the study ad auxiliary variable respectively ad their correspodig sample meas be ȳ ad x. This study cosiders the problem of estimatig the populatio variace,defied as S 2 y = 1 N 1 ΣN i=1(y i Ȳ )2 ad uses auxiliary iformatio to improve the efficiecy of the populatio variace estimator. We defie the followig otatios that we will make use of throughout the thesis. For the populatio observatios we have; Ȳ = 1 N ΣN i=1y i, X = 1 N ΣN i=1x i, S 2 y = 1 N 1 ΣN i=1(y i Ȳ )2, S 2 x = 1 N 1 ΣN i=1(x i X) 2, S xy = 1 N 1 ΣN i=1(y i Ȳ )(X i X). Also we defie the followig from the sample observatios: ȳ = 1 Σ i=1y i, x = 1 Σ i=1x i, s 2 y = 1 1 Σ i=1(y i ȳ) 2, s 2 x = 1 1 Σ i=1(x i x) 2, s xy = 1 1 Σ i=1(y i ȳ)(x i x). I geeral, we recall the followig parameters we defied i sectio (2.2): µ rs = 1 N 1 Σ i=1(y i ȳ) r (x i x) s, λ rs = µrs. Thus we ote the followig; µ r 2 20 µ 2 s 02 µ 20 = Sy, 2 µ 02 = Sx, 2 ad µ 11 = S xy ; λ 22 = µ 22 µ 20 µ 02, λ 21 = µ 21 such that; µ 20 µ C y = S2 y = µ 20 Ȳ 2 Ȳ 2 coefficiet of variatio for the study variable y, C x = S2 x X = µ 02 2 X 2 of variatio for the auxiliary variable x ad ρ xy = Sxy S xs y = µ 11 µ20 µ02 coefficiet coefficiet of correlatio betwee x ad y, κ (y) = λ 40 = µ 40 coeffciet of kurtosis for the study variable, κ µ 2 (x) = λ 04 = µ µ 2 02 coefficiet of kurtosis for the auxiliary variable ad M x populatio media of the auxiliary 18

28 variable. 3.1 Liearity of Expectatio Followig the works of Karr (1993) we have Theorem 1 Let X ad Y be radom variables. We have that E(X + Y ) = E(X) + E(Y ) (51) ad ote that true for ay X ad Y eve whe they are depedet. Proof We first show that E(X + Y ) = (i + j)p {X = i, Y = j} (52) i= j= E(X + Y ) = K.P {X + Y = K} (53) K= = K.( P {X = i, Y = K i}) (54) K= i= = K.( P {X = i, Y = K i}) (55) i= k= settig K i = j K = j + i = (i + j)( P {X = i, Y = j}) (56) We ow have i= j= E(X + Y ) = (i + j)( P {X = i, Y = j}) (57) i= j= = i.p {X = i, Y = j} + j.p {X = i, Y = j} (58) i= j= i= j= 19

29 Cosiderig the first part of equatio (58) i.p {X = i, Y = j} = i. P {X = i, Y = j} i= j= i= j= }{{} (59) = i.p {X = i, Y = j} (60) i= = E(X) (61) ad the secod part of equatio (58) j.p {X = i, Y = j} = j.( P {X = i, Y = j} ) (62) i= j= i= j= }{{} = j.p {X = i, Y = j} (63) i= = E(Y ) (64) Therefore E(X + Y ) = (i + j)p {X = i, Y = j} (65) i= j= = i.p {X = i, Y = j} + j.p {X = i, Y = j} (66) i= j= i= j= = E(X) + E(Y ) (67) 3.2 Expected value of the ratio of correlated radom variables Cosider radom variables m ad which are correlated. Suppose we defied them as m = E(m) + m = E() + (68) by simply iterchagig variables m ad with ew variables m ad hece still measurig the same thigs; we have just shifted the axes so that 0 is the expected value (for istace if 20

30 the expected umber of descedats is 2, the we measure the actual umber by how much it differs from 2; if the idividual eds up just leavig just 1 descedat, the m =-1). We ote that E( m ) is udefied for ay ozero probability that = 0. Therefore we calculate E( m 0), the expected value of the ratio coditioal o ot equalig zero. This coditio makes complete sese i evolutioary theory; sice = 0 iff the populatio goes extict- hece the case where the result become udefied. Usig the defiitios i equatio (68) we ca write: ( m ) ( ) ( E(m) + m E 0 E(m) = E = E E() + E() [1 + m E(m) ] [1 + E() ] Notig that the expected values E(m) ad E() are ot radom variables we ca remove ) (69) outside the expectatio o the right had side of equatio (69) yieldig: ( ) [ ( m ) E 0 = E(m) 1 + m ( E() E E(m) 1 + = E(m) E() E 1 + m ) ( 1 + ) ] 1 E(m) E() E() Multiplyig out the terms i the square brackets yields: [ ( m ) ( E 0 = E(m) E() E 1 + ) ] [ E() E() E m ( 1 + ) ] 1 E() (70) (71) By the defiitio of harmoic mea E( 1 ) = 1, where H() is the harmoic mea of. H() We ca use equatio (70) to fid E( 1 ) by settig m = 1 (so that E(m) = 1 ad m = 0). Thus we will obtai: ( m ) E 0 [ ( 1 H() = 1 E() E 1 + ) ] 1 E() (72) We ca rewrite ow the right had side of equatio (71) by usig equatio (72) [ ( m ) E 0 = E(m) H() + 1 ( E() E m 1 + ) ] 1 E() [ ( ) ] 1 Now we have to deal with the term E m 1 +. Provided that < E() i.e. E(m) ( 1 < 2E(), we ca expad 1 + E()) as a Taylor series i. If we defie: f = (73) ( 1 + ) 1 (74) E() 21

31 Taylor s theorem will yield: f = 1 + ( 1) i i (75) E() i i=1 Importatly we ote that the use of Taylor s theorem is ot applicable i all cases. Precisely, equatio (74) does ot coverge to equatio (75) if E(), hece i such situatios we fall back ad use the calculus of fiite differeces. Whe we ca apply the Taylor expasio i equatio (75), we will have: [ ( E m 1 + ) ] [ ] 1 = E m + ( 1) i m b i E() E() i From the defiitios of m ad i equatio (68), we kow that E(m ) = 0, E(m ) = cov(m, ) ad i geeral, E(m i ) is the mixed cetral momet defied as E[m E(m)][ E()] i, which ca simply use the otatio, i m. Hece we ca ow write equatio (76) as [ ( E m 1 + ) ] 1 = ( 1) i,i m (77) E() E() i Substitutig equatio (77) ito equatio (73) gives the equatio for the expected value of the ratio: i=1 i=1 (76) ( m ) E 0 = E(m) H() + ( 1) i,i m (78) E() i+1 For other situatios, it is useful to have a result i which the first term does ot ivolve the harmoic mea. To do this we simply substitute the series i equatio (75) directly ito the far right had part of equatio (70). Deotig the i th cetral momet of by i, so that 1 = 0. Thus i=1 ( m ) E 0 = E(m) E() + ( 1) i E(m) i +, i m (79) E() i+1 i=1 3.3 Taylor s approximatio method Suppose we have a estimator Z = g(x, Y ) a fuctio of two variables. Suppose that we ca measure X ad determie its populatio parameters such as mea ad variace but really be iterested i Y which is related to X i some way. We might be iterested to kow V ar(y ) 22

32 at least approximately i order to assess the accuracy of idirect measuremet process. Sice we caot i geeral fid E(y) = µ y ad V ar(y ) = σ y from E(X) = µ x ad V ar(x) = σ x uless the fuctio g is liear. As i our case of estimatig the populatio variace we ivolve ratio which is o-liear so let us suppose g is o-liear thus we have to liearize. Usig Taylors series expasio of g about µ = (µ x, µ y ) i order to approximate the mea or variace of Z. To first order The otatio g(µ) y (µ x, µ y ). Z = g(x, Y ) g(µ) + (X µ x ) g(µ) x ad g(µ) x + (Y µ y) g(µ) y (80) meas that the partial derivative is evaluated at the poit Z havig bee expressed approximately equal to a liear fuctio of X ad Y. The mea ad variace of this liear fuctio are easily calculated to be E(Z) µ ad V ar(z) = σ 2 x( g(µ) x )2 + σ 2 y( g(µ) y ) + 2σ xy( g(µ) x g(µ) )( ) y Illustratio Usig Isaki (1983) ratio type populatio variace estimator for our illustratio Let us defie ξ 0 = s2 y S 2 y t R = s 2 Sx 2 y s 2 x 1, ξ 1 = s2 x 1, ξ Sx 2 2 = ȳȳ 1, ξ 3 = x X 1, ξ 4 = sxy S xy E(ξ 0 ) = E(ξ 1 ) = E(ξ 2 ) = E(ξ 3 ) = E(ξ 4 ) = 0 ad E(ξ 2 2) = ( )C2 y, E(ξ3) 2 = ( )C2 x, E(ξ 2 ξ 3 ) = ( )ρ xyc y C x. 1 such that Expressig the estimator t R i terms of ξ 0 ad ξ 1 ca easily be writte as t R = S 2 y(1 + ξ 0 )(1 + ξ 1 ) 1 = S 2 y(1 + ξ 0 )(1 ξ 1 + ξ ) = S 2 y[1 + ξ 0 ξ 1 + ξ 2 1 ξ 0 ξ ] (81) To the first order of Taylor s approximatios we have; E(ξ0) 2 = ( )(λ 40 1), E(ξ1) 2 = ( )(λ 04 1), 23

33 E(ξ 2 4) = ( )( λ 22 ρ 2 xy 1), E(ξ 2 ξ 0 ) = ( E(ξ 3 ξ 0 ) = ( )C xλ 21, E(ξ 3 ξ 1 ) = ( )C xλ 03, E(ξ 3 ξ 4 ) = ( )C x λ 12 ( )( λ 13 ρ xy ρ xy, E(ξ 0 ξ 1 ) = ( )C yλ 30, E(ξ 2 ξ 1 ) = ( )C yλ 12, E(ξ 2 ξ 4 ) = ( )C y λ 21 ρ xy, )(λ 22 1), E(ξ 0 ξ 4 ) = ( )( λ 31 ρ xy 1), ad E(ξ 1 ξ 4 ) = 1), where f is the fiite populatio correctio (f.p.c) factor. Thus we have the followig theorems as stated by Sigh (2003) Theorem 2 Bias upto order O( 1 ) i the estimator of t R is Proof Takig the expectatio o both sides of (81) we have Bias(t R ) = 1 f S2 y(λ 04 λ 22 ) (82) E(t R ) = S 2 ye[1 + ξ 0 ξ 1 + ξ 2 1 ξ 0 ξ 1 ] = S 2 y[1 + ( )(λ 04 1) (λ 22 1)] ad usig the result B(t R ) = E(t R ) S 2 y we have (82). Theorem 3 The MSE of the estimator t R up to first order of approximatios is Proof We have MSE(t R ) = E(t R S 2 y) 2 E[S 2 y(1 + ξ 0 ξ 1 + ξ 2 1 ξ 0 ξ ) S 2 y] 2 S 4 ye(ξ 0 ξ 1 ) 2 = S 4 ye[ξ ξ 2 1 2ξ 0 ξ 1 ] = S4 y[(λ 40 1) + (λ 04 1) 2(λ 22 1)]. MSE(t R ) = ( 1 f )S4 y[λ 40 + λ 04 2λ 22 ] (83) 24

34 3.4 Proposed Estimator Motivated by the works of Kha ad Shabbir (2013a), Upadhyaya ad Sigh (1999), Sigh et al. (2004), Subramai ad Kumarapadiya (2013), Kadilar ad Cigi (2006), ad Yadav et al. (2016) i the improvemet of the performace of the populatio variace estimator of the study variable usig kow populatio parameters of a auxiliary variable. We propose the followig modified ratio type estimator for the populatio variace S 2 y usig kow values of populatio coefficiet of kurtosis κ x ad media M x of a auxiliary variable. ŜP 2 M = s 2 y{ S2 xκ x + Mx 2 } (84) s 2 xκ x + Mx 2 To obtai the bias ad the MSE of our proposed estimator Ŝ2 P M, We defie s 2 y = S 2 y(1 + ξ 0 ) ad s 2 x = S 2 x(1 + ξ 1 ) or ξ 0 = s2 y S 2 y 1 ad ξ 1 = s2 x S 2 x 1 such that E(ξ 0 ) = E( s2 y ) E(1) = 0 ad E(ξ Sy 2 1 ) = E( s2 x ) E(1) = 0 ad to the first degreee of Sx 2 approximatios we have E(ξ 2 0) = (λ 40 1), E(ξ1) 2 = (λ 04 1), E(ξ 0 ξ 1 ) = (λ 22 1). The above expectatios are obtaied followig the works of Sukhatme (1944), Sukhatme ad Sukhatme (1970), Srivastava ad Jhajj (1981), Tracy (1984) ad Withers ad Nadarajah (2014). Now expressig Ŝ2 P M i terms of ξ s we have Ŝ 2 P M = S2 y(1 + ξ 0 ){ κ xsx+m 2 x 2 κ xs } x 2(1+ξ 1)+Mx 2 = S 2 y(1 + ξ 0 )(1 + ϱ ξ 1 ) 1 (85) where ϱ = κ x S 2 x(κ x S 2 x +M 2 x) 1, we assume that ϱ ξ 1 < 1 so that (1+ϱ ξ 1 ) 1 is expadable. Expadig the right had side of (85) ad multiplyig out we have Ŝ 2 P M = S2 y(1 + ξ 0 )(1 ϱ ξ 1 + ϱ 2 ξ ) = S 2 y(1 + ξ 0 ϱ ξ 1 ϱ ξ 0 ξ 1 + ϱ 2 ξ ϱ 2 ξ 0 ξ ) Neglectig terms of ξ s havig power greater tha two we have 25

35 Ŝ 2 P M = S 2 y(1 + ξ 0 ϱ ξ 1 ϱ ξ 0 ξ 1 + ϱ 2 ξ 2 1) or Takig the expectatio o both sides of (86) Ŝ 2 P M S 2 y = S 2 y(ξ 0 ϱ ξ 1 ϱ ξ 0 ξ 1 + ϱ 2 ξ 2 1) (86) E(Ŝ2 P M S2 y) = E(S 2 y(ξ 0 ϱ ξ 1 ϱ ξ 0 ξ 1 + ϱ 2 ξ 2 1)) We get the bias of the estimator Ŝ2 P M to the first degree of approximatio as Bias(Ŝ2 P M) = 1 f S2 y(κ x 1)ϱ {ϱ (λ 22 1) (κ x 1) } (87) Squarig both sides of (86) ad eglectig terms of ξ s havig power greater tha two we have Takig the expectatio o both sides of (88) (Ŝ2 P M S 2 y) 2 = S 4 y (ξ ϱ 2 ξ 2 1 2ϱ ξ 0 ξ 1 ) (88) E((Ŝ2 P M S2 y) 2 ) = E(Sy(ξ ϱ 2 ξ1 2 2ϱ ξ 0 ξ 1 )) We get the ( S 2ˆ P M ) estimator s Mea Squared Error to first degree of approximatio as MSE(Ŝ2 P M) = 1 f S4 y{(κ y 1) + ϱ (κ x 1)(ϱ 2 (λ 22 1) )} (89) (κ x 1) Theoretical Coditios for our Proposed Estimator Cosider our proposed estimator Suppose we rewrite it as ŜP 2 M = s 2 y{ S2 xκ x + Mx 2 } (90) s 2 xκ x + Mx 2 Ŝ 2 P M = m (91) i.e. we let m = s 2 y(s 2 xκ x + M 2 x) ad = s 2 xκ x + M 2 x ad ivoke the coditio i fidig the expectatio of a ratio of correlated radom variables i equatio (69). We ote that E(Ŝ2 P M ) = E( m ) is udefied if there is ay ozero probability that = 0. 26

36 Thus we will calculate E( m 0) the expected value of the ratio, coditioal ot equalig zero. This meas that i order for our proposed estimator to be applicable. κ x s 2 x + M 2 x 0 (92) 27

37 3.5 Expressios for Bias ad Mea Squared Errors of the Proposed ad Existig Estimators Table 1: Summary of Expressios of Statistical Properties (Bias ad Mea Squared Errors(MSE)) ESTIMATOR BIAS(.) MEAN SQUARED ERROR(MSE) s 2 y s 2 y( S2 x ) s 2 x s 2 y( S2 x+κ x s 2 x+κ x ) s 2 y( S2 x Cx s 2 x Cx ) s 2 y( S2 x κ x s 2 x κx ) s 2 y( S2 x κx Cx s 2 xκ x C x ) s 2 y( S2 xc x κ x s 2 x Cx κx ) s 2 y( S2 x+m x s 2 x+m x ) s 2 y( S2 x +Q1 s 2 x +Q1 ) s 2 y( S2 x+q 3 s 2 x+q 3 ) s 2 y( S2 x Cx+Mx s 2 xc x+m x ) [ s 2 S 2 x ρ xy+q 3 y [ ] s 2 S 2 x +(T M+Q a) y s 2 x +(T M+Qa) s 2 x ρxy+q3 ] s 2 y{ S2 x κx+m 2 x s 2 x κx+m 2 x } S2 y{(κ x 1)Ψ 1 (Ψ 1 λ22 1 )} κ x 1 S2 y{(κ x 1)Ψ 2 (Ψ 2 λ22 1 )} κ x 1 S2 y[({κ x 1})Ψ 3 (Ψ 3 λ22 1 )] κ x 1 S2 y(κ x 1){Ψ 4 (Ψ 4 λ22 1 )} κ x 1 S2 y(κ x 1){Ψ 5 (Ψ 5 ( λ22 1 ))} κ x 1 S2 y(κ x 1){Ψ 6 (Ψ 6 ( λ22 1 ))} κ x 1 S2 y(κ x 1){Ψ 7 (Ψ 7 ( λ22 1 ))} κ x 1 S2 y(κ x 1){Ψ 8 (Ψ 8 ( λ22 1 ))} κ x 1 S2 y(κ x 1){Ψ 9 (Ψ 9 ( λ22 1 ))} κ x 1 S2 y(κ x 1){Ψ 10 (Ψ 10 ( λ22 1 ))} κ x 1 S2 y(κ x 1){Ψ 11 (Ψ 11 ( λ22 1 ( ( [(κ x 1)Ψ 12 S2 y S2 y Ψ 12 ( ( [(κ x 1)Ψ 13 Ψ 13 S2 y(κ x 1)ϱ {ϱ (λ22 1) κ x 1 ))} λ 22 1 κ x 1 λ 22 1 κ x 1 ))] ))] S4 y{(κ y 1) + (κ x 1)Ψ 1 (Ψ 1 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + (κ x 1)Ψ 2 (Ψ 2 2( λ22 1 ))} κ x 1 S4 y[{κ y 1} + {κ x 1}Ψ 3 (Ψ 3 2( λ22 1 ))] κ x 1 S4 y{(κ y 1) + Ψ 4 (κ x 1)(Ψ 4 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + Ψ 5 (κ x 1)(Ψ 5 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + Ψ 6 (κ x 1)(Ψ 6 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + Ψ 7 (κ x 1)(Ψ 7 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + Ψ 8 (κ x 1)(Ψ 8 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + Ψ 9 (κ x 1)(Ψ 9 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + Ψ 10 (κ x 1)(Ψ 10 2( λ22 1 ))} κ x 1 S4 y{(κ y 1) + Ψ 11 (κ x 1)(Ψ 11 2( λ22 1 [ ( ( S4 y S4 y } (κ x 1) (κ y 1) + Ψ 12 (κ x 1) [ (κ y 1) + Ψ 13 (κ x 1) Ψ 12 2 ( ( Ψ 13 2 κ x 1 ))} λ 22 1 κ x 1 λ 22 1 κ x 1 S4 y{(κ y 1) + ϱ (κ x 1)(ϱ 2 (λ22 1) (κ x 1) )} ))] ))] I geeral the Bias ad MSE of existig modified ratio estimators t j, j = 1, 2,..., 13 is Bias(t j ) = S2 y(κ x 1){Ψ j (Ψ j ( λ 22 1 ))} κ x 1 MSE(t j ) = S4 y[(κ y 1) + Ψ j (κ x 1)(Ψ j 2( λ 22 1 ))] κ x 1 where Ψ 1 = 0; Ψ 2 = 1; Ψ 3 = Ψ 6 = Ψ 10 = S2 xκ x Sxκ 2 x C x ; Ψ 7 = S2 x ; Ψ Sx 2+Q 11 = 3 S2 x Sx+κ 2 x ; Ψ 4 = S2 xc x SxC 2 x κ x ; Ψ 8 = CxS2 x ; Ψ C xsx 2+Mx 12 = S2 x Sx C 2 x ; Ψ 5 = S2 x Sx+M 2 x ; Ψ 9 = S2 x Sx κ 2 x ; S2 x Sx+Q 2 1 ; S2 xρ xy ; Ψ Sx 2ρxy+Q 13 = 3 Sx 2. Sx 2 +T M+Qa 28

38 CHAPTER FOUR 4 RESULTS AND DISCUSSION 4.1 Theoretical Evaluatio The theoretical coditios uder which the proposed modified ratio type estimators Ŝ2 P M is more efficiet tha the other existig estimators t j, j = 1, 2,..., 13, from MSE of t j, j = 1, 2,..., 13 give to first degree of approximatio i geeral as MSE(t j ) = 1 f S4 y[(κ y 1) + Ψ j (κ x 1)(Ψ j 2( λ 22 1 ))] (93) κ x 1 Usig equatio (89) ad (93) we have that MSE(Ŝ2 P M ) < MSE(t j), if ϱ (ϱ 2( λ 22 1 κ x 1 )) < Ψ j(ψ j 2( λ 22 1 κ x 1 )) 4.2 Numerical Studies Usig the data from Populatio I (Source:(Murthy, 1967, p.228)), Populatio II (source:(daroga ad Chaudhary, 1986, p.177)) ad Populatio III (source:(cochra, 1977, p.152)). We assess the performace of the proposed estimator whe simple radom samplig without replacemet (SRSWOR) scheme is used with that of sample variace ad existig estimators.we apply the proposed ad existig estimators to this data set ad the data summaries are give below: Populatio I(Dataset i Appedix I) X= Fixed capital Y = output of 80 factories N = 80, = 20 X = , Ȳ = , Sx 2 = , Sy 2 = , S xy = , λ 04 = κ x = 2.866, λ 40 = κ y = 2.267, λ 22 = 2.221, 29

39 ρ xy = 0.941, C y = 0.354, C x = M x = Q 1 = Q 3 = T M = , Q a = Populatio II X = acreage uder wheat crop i 1973 Y = acreage uder wheat crop i 1974, N = 70, = 25 X = , Ȳ = , Sx 2 = , Sy 2 = , λ 04 = κ x = , λ 40 = κ y = , λ 22 = , ρ xy = , C y = , C x = M x = Q 1 = Q 3 = T M = , Q a = Populatio III(Data Set i Appedix II) X = Total umber of ihabitats i the 196 cities i 1920 Y = Total umber of ihabitats i the 196 cities i 1930, N = 49, = 20 X = , Ȳ = , Sx 2 = , Sy 2 = , λ 04 = κ x = , λ 40 = κ y = , λ 22 = , ρ xy = , C y = , C x = M x = Q 1 = Q 3 = T M = 72.75, Q a =

40 Usig these summary values to obtai the Bias ad MSE of the existig estimators ad our proposed estimator we have Bias ad Mea Squared Errors Table 2: Bias ad Mea Squared Errors(MSE) of Existig ad Proposed Estimators for Populatio Variace Populatio I Populatio II Populatio III ESTIMATOR BIAS(.) MSE BIAS(.) MSE BIAS(.) MSE t 1 = s 2 y t 2 = s 2 y( S2 x ) s 2 x t 3 = s 2 y( S2 x+κ x ) s 2 x +κx t 4 = s 2 y( S2 x Cx s 2 x C x ) t 5 = s 2 y( S2 x κ x ) s 2 x κx t 6 = s 2 y( S2 xκ x C x s 2 xκ x C x ) t 7 = s 2 y( S2 x Cx κx ) s 2 xcx κx t 8 = s 2 y( S2 x+m x s 2 x+m x ) t 9 = s 2 y( S2 x +Q1 s 2 x+q 1 ) t 10 = s 2 y( S2 x+q 3 ) s 2 x +Q3 t 11 = s 2 y( S2 x Cx+Mx s 2 xc x+m x ) [ t 12 = s 2 S 2 x ρ xy+q 3 y [ ] t 13 = s 2 S 2 x +(T M+Q a) y s 2 x +(T M+Qa) s 2 x ρxy+q3 ] Ŝ 2 P M = s2 y{ S2 x κx+m 2 x s 2 xκ x+m 2 x } From the above table Mea Squared Errors it is clear that our proposed modified ratio type populatio variace estimator Ŝ2 P M has the least Mea Squared Error(MSE). 31

41 4.2.2 Efficiecy Compariso The performace of the proposed modified ratio type variace estimator evaluated agaist the usual ubiased estimator s 2 y ad the existig estimators t j, j = 1, 2,..., 13 usig real populatio from (Murthy, 1967, p.228), (source:daroga ad Chaudhary (1986)) ad (source:(cochra, 1977, p.152)). We have computed the Percet Relative Efficiecies (PREs) of the estimators t j, j = 1, 2,..., 13 usig the formulae P RE(t j, s 2 y) = MSE(s2 y) MSE(t j ) = { = 100 (94) () S4 y(κ y 1) S4 y[{κ y 1} + {κ x 1}Ψ j (Ψ j 2( λ 22 1 κ x 1 (κ y 1) The PRE for our proposed estimator is subsequetly, ))]} 100 (95) 100 (96) [{κ y 1} + {κ x 1}Ψ j (Ψ j 2( λ 22 1 κ x 1 ))] P RE(Ŝ2 P M, s 2 y) = MSE(s2 y) 100 MSE(Ŝ2 P M ) (97) = () S4 y(κ y 1) 100 S4 y{(κ y 1) + ϱ (κ x 1)(ϱ 2 (λ 22 1) (κ x 1) )} (98) = (κ y 1) 100 {(κ y 1) + ϱ (κ x 1)(ϱ 2 (λ 22 1) (κ x 1) )} (99) 32

42 Usig formula (96) ad (99) we computed the Percet Relative Efficiecies ad preseted i table 3 below Percet Relative Efficiecies Table 3: Percet Relative Efficiecies(PRE) of Existig ad Proposed Estimators ESTIMATOR POPULATION I POPULATION II POPULATION III t t t t t t t t t t t t t ŜP 2 M From the fidigs summarized i the table above it is clear that our proposed estimator Ŝ2 P M performed best, that is it has the highest PRE amog all the other estimators. 33

43 CHAPTER FIVE 5 CONCLUSION AND RECOMMENDATIONS 5.1 Coclusio I this study we have suggested a modified ratio type estimator of populatio variace Sy 2 of the study variable y usig kow populatio parameters of the auxiliary variable x, the coefficiet of kurtosis ad the media. The bias ad mea squared error of the proposed estimator has bee obtaied to first order degree of approximatio ad cosequetly compared with that of the usual ubiased estimator ad the estimators due to Isaki (1983), Kadilar ad Cigi (2006),Subramai ad Kumarapadiya (2013), Subramai ad Kumarapadiya (2012a), Subramai ad Kumarapadiya (2012b), Upadhyaya ad Sigh (1999) Kha ad Shabbir (2013b) ad Bhat et al. (2017). We have also assessed the performace of our proposed estimator usig kow atural populatio data sets ad foud out that the performace of our proposed estimator is better tha the other existig estimators for the data sets by comparig their Percet Relative Efficiecies. Based o the results of our studies, it is evidet that our proposed estimator has the highest Percet Relative Efficiecy. 5.2 Recommedatios We recommed that our proposed estimator ca be applied to practical applicatios, where kowledge of populatio parameters of auxiliary variable positively correlated with study variable is available. We further recommed that our proposed estimator ca be improved by extedig the umber of Taylor s series terms to be more tha order oe or be protracted to Stratified Samplig Scheme. 34