On ratio and product methods with certain known population parameters of auxiliary variable in sample surveys

Size: px
Start display at page:

Download "On ratio and product methods with certain known population parameters of auxiliary variable in sample surveys"

Transcription

1 Statistics & Operatios Research Trasactios SORT 34 July-December 010, ISSN: Statistics & Operatios Research c Istitut d Estadística de Cataluya Trasactios sort@idescat.cat O ratio product methods with certai kow populatio parameters of auxiliary variable i sample surveys Housila. P. Sigh, Ritesh Tailor 1 Rajesh Tailor School of Studies i Statistics Vikram Uiversity, Ujjai , M.P., Idia Abstract This paper proposes two ratio product-type estimators usig trasformatio based o kow miimum maximum values of auxiliary variable. The biases mea squared errors of the suggested estimators are obtaied uder large sample approximatio. Coditios are obtaied uder which the suggested estimators are superior to the covetioal ubiased estimator, usual ratio product estimators of populatio mea. The superiority of the proposed estimators are also established through some atural populatio data sets. MSC: 94A0 Keywords: Study variate, auxiliary variate bias, mea squared error, simple rom samplig without replacemet 1. Itroductio The use of supplemetary iformatio o a auxiliary variable for estimatig the fiite populatio mea of the variable uder study has played a emiet role i samplig theory practices. Out of may ratio, product regressio methods of estimatio are good illustratios i this cotext. Whe the correlatio betwee the study variable y the auxiliary variable x is positive high, the ratio method of estimatio is employed. O the other h if this correlatio is egative high, the product method of estimatio ivestigated by Robso 1957 Murthy 1964, is quite effective. 1 Lokmaya Tilak Mahavidyalaya, Ujjai , M.P., Idia Received: August 009 Accepted: September 010

2 158 O ratio product methods with certai kow populatio... It is a well-established fact that the ratio estimator is most effective whe the relatio betwee y x is straight lie through the origi the variace of y about this lie is proportioal to x, for istace, see Cochra I may practical situatios, the regressio lie does ot pass through the origi. Also due to stroger ituitive appeal survey statisticias are more iclied towards the use of ratio product estimators. Keepig these facts i mid several authors icludig Srivastava 1967, 1983, Reddy 1973,74, Walsh 1970, Gupta 1978, Vos 1980, Naik Gupta 1991, Mohaty Sahoo 1995, Sahai Sahai 1985, Upadhyaya Sigh 1999, Srivekataramaa 1980, Byopadhyaya 1980, Mohaty Das 1971, Srivekataramaa 1978, Sisodia Dwivedi 1981 Sigh 003 have suggested various modificatios i ratio product estimators. Suppose we have populatio of N idetifiable uits o which the two variates y x are defied. For estimatig the populatio mea Y = N y i /N of the study variate y, a simple rom sample of size is draw without replacemet. It is assumed that the N populatio mea X = x i /N of the auxiliary variate x is kow. The the classical ratio product estimators of populatio mea Y are respectively defied by where y= y i / x= y R = yx/x 1.1 y p = yx/x 1. x i / are the sample meas of variates y x respectively. Let x m x M be the miimum maximum values of a kow positive variate x respectively. Usig these values i.e. x m x M, Mohaty Sahoo 1995 suggested to trasform auxiliary variable x to ew variables z u such that z i = x i+ x m x M + x m 1.3 u i = x i+ x M x M + x m,,,...,n. 1.4 Usig these trasformed variables z u, Mohaty Sahoo 1995 proposed the followig ratio estimators for populatio mea Y as t 1R = yz/z 1.5

3 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 159 t R = yu/u, 1.6 where z= 1 xi + x m x M + x m x+xm = x M + x m u= 1 xi + x M x M + x m x+xm = x M + x m are sample meas of z u respectively, Z = 1 N N xi + x m x M + x m X+ xm = x M + x m U = 1 N N xi + x M x M + x m X+ xm = x M + x m are the populatio meas of z u respectively. Whe the correlatio betwee y x is egative, the product estimator based o trasformed variables z u are defied by t 1p = yz/z 1.7 t p = yu/u 1.8 It is well kow uder simple rom samplig without replacemet SRSWOR that the mea squared error or variace of y is MSEy= Vary=θS y =θ Y C y 1.9 whereθ =N /N,C y = S y : the coefficiet of variatio of the study variate y. Y To the first degree of approximatio, the biases mea squared errors MSEs of the ratio-type estimators y R, t 1R, t R, product-type estimators y p, t 1p t p are respectively give by By R =θ Y C x1 K 1.10 Bt 1R =θ YC x/c 1 {1/C 1 K} 1.11 Bt R =θ YC x/c {1/C K} 1.1 By p =θ Y C x K 1.13

4 160 O ratio product methods with certai kow populatio... Bt 1p =θ YC x/c 1 K 1.14 Bt p =θy C x/c K 1.15 MSEy R =θ Y [C y +C x1 K] 1.16 MSEt 1R =θ Y [C y +C x/c 1 {1/C 1 K}] 1.17 MSEt R =θ Y [C y +C x/c {1/C K}] 1.18 MSEy p =θ Y [C y +C x1+k] 1.19 MSEt 1p =θ Y [C y +C x/c 1 {1/C 1 +K}] 1.0 MSEt p =θ Y [C y +C x/c {1/C +K}] 1.1 where K =ρc y /C x,ρ= S yx /S x S y is the correlatio coefficiet betwee y x, S x = N C 1 = x i X /N 1, S y =, C = N y i Y /N 1, S xy = N x i Xy i Y/N 1, 1+ x m 1+ x M C x = S x : the coefficiet of variatio of the auxiliary X X X variate x. It is to be oted that the trasformatios deped o both maximum x M miimumx m values but the estimators t 1R t 1P t R t P geerated through these trasformatios deped oly o maximum value x M miimum value x m respectively. For istace, t 1R = y Z z = y X+ x m/x M + x m x+x m /x M + x m = y X+ x m x+x m 1.

5 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 161 I similar fashio it ca be show that the estimators t 1P t R, t P, deped oly o x m x M respectively. Expressios motivated authors to ivestigate some trasformatios which make use of both maximum valuex M miimum valuex m hece usig such trasformatios the costructed estimators should also deped o x M x m. Some ratio- product-type estimators of populatio mea Y have bee suggested their properties are studied. Numerical illustratios are give i support of the preset study.. The suggested trasformatios estimators Let x m x M be the miimum maximum values of a kow positive variate x respectively. Usig x m x M, it is suggested to trasform the auxiliary variable x to ew variables a b such that a i = x M x i + x m.1 b i =x M x m x i + x m,,...,n.. Usig the trasformed variates at.1. we defie the followig ratio-type estimators for populatio mea Y as d 1R = y A a B d R = y b.3.4 the product-type estimators for Y as d 1p = y a A.5 where a= d p = y b B a i /=x M x+x m b= b i /=x M x m x+x m.6

6 16 O ratio product methods with certai kow populatio... are the sample meas of a b respectively A= N a i /N = x M X+ x m B= N are the populatio meas of a b respectively. b i /N =x M x m X+ x m.1. Biases variaces of ratio-type estimators d 1R d R To obtai the biases variaces of d 1R d R, we write y= Y1+e 0 such that x=x1+e 1 Ee 0 =Ee 1 =0 Ee 0 =θ C y Ee 1 =θ C x Ee 0 e 1 =θkc x.7 Expressig d 1R d R i terms of e s we have A d 1R = Y1+e 0 { xm X1+e 1 +xm} A = Y1+e 0 { } xm X+ xm+ x M X e 1 A = Y1+e 0 { } A+xM X e 1 = Y1+e λ 1 e 1.8 B d R = Y1+e 0 { xm x m X1+e 1 +xm} B = Y1+e 0 { } xm x m X+ xm+x M x m X e 1

7 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 163 B d R = Y1+e 0 { } B+xM x m X e 1 = Y1+e 0 1+λ e where λ 1 = x M X = x M X x M X+ xm A = C 1 C 1+C λ = x M x m X = x M x m X x M x m X+ xm B = C C 1 C C 1 +C We ow assume that λ1 e 1 < 1 λ e < 1 so that we may exp1+λ 1 e λ e 1 1 as a series i power ofλ 1 e 1 λ e 1. Expig right h sides of.8.9, multiplyig out retaiig terms of e s to the secod degree, we obtai t 1R = Y 1+e 0 λ 1 e 1 λ 1 e 1 e 0 +λ 1 e 1 or or t 1R Y= Y e 0 λ 1 e 1 λ 1 e 1 e 0 +λ 1 e 1 t R = Y 1+e 0 λ e 1 λ e 1 e 0 +λ e 1 t R Y= Y e 0 λ e 1 λ e 1 e 0 +λ e Takig expectatios of both sides of.1.13 usig the results i.7 we get the biases of d 1R d R to the first degree of approximatio respectively as Bd 1R =θ YC x λ 1λ 1 K.14 Bd R =θ YC x λ λ K.15 It follows from that the biases Bd 1R Bd R are egligible, if the sample size is large eough.

8 164 O ratio product methods with certai kow populatio... Squarig both sides of.1.13 retaiig terms of e s to the secod degree we have d 1R Y = Y e 0+λ 1 e 1 λ 1 e 0 e 1.16 d R Y = Y e 0+λ e 1 λ e 0 e 1.17 Takig expectatio of both sides of usig the results i.7, we get the MSEs of d 1R d R to the first degree of approximatio respectively as MSEd 1R =θy [ C y +λ 1 C xλ 1 K ].18 MSE=d R =θy [ C y +λ C xλ K ].19.. Biases variaces of product-type estimators To obtai the biases MSEs of d 1P d P, we express d 1P d P i terms of e s as { xm X1+e 1 +x m} d 1P = Y1+e 0 x M X+ xm { = Y1+e 0 1+ x } MX e 1 x M X+ xm = Y1+e 0 1+λ 1 e 1 = Y1+e 0 +λ 1 e 1 +λ 1 e 0 e 1 or d 1P Y= Ye 0 +λ 1 e 1 +λ 1 e 0 e 1.0 d P = Y1+e 0 = Y1+e 0 { xm x m X1+e 1 +x m} { { xm x m X+ x m} 1+ x M x m X e { 1 xm x m X+ xm} } = Y1+e 0 1+λ e 1 = Y1+e 0 +λ e 1 +λ e 0 e 1

9 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 165 or d P Y= Ye 0 +λ e 1 +λ e 0 e 1,.1 whereλ 1 λ are respectively give by Takig expectatio of both sides of.19.0 usig the results i.7, we get the exact biases of d 1P d P as Bd 1P =θ Yλ 1 KC x. Bd P =θ Yλ KC x.3 Squarig both sides of.0.1 retaiig terms of e s to the secod degree, the takig expectios, we get the MSEs of d 1P d P respectively as MSEd 1P =θy [ C y +λ 1 C xλ 1 + K ].4 MSEd P =θy [ C y +λ C xλ + K ].5 3. Compariso of biases The absolute relative bias ARB of a estimator t of the populatio mea Y is defied by ARBt = Bt Y 3.1 where Bt sts for bias of the estimator t. The compariso of absolute relative biases of ratio-type product-type estimators have bee made the coditios are displayed i Tables respectively.

10 166 O ratio product methods with certai kow populatio... Estimator Table 3.1: Compariso of absolute relative biases of ratio-type estimators. Absolute Relative Bias of d 1R is less tha d R is tha y R t 1R t R d R if either K > 1+λ 1 or K < 1+λ 1 1+λ 1 if 1+λ 1 C 1 C 1 1+λ 1 C 1 < K < 1+λ 1C 1 C 1 if 1+λ 1 C C 1+λ 1 C < K < 1+λ 1C 1 C if λ 1 +λ λ 1 +λ < K <λ 1+λ if either K > 1+λ or K < 1+λ 1+λ if either 1+λ C 1 C 1 1+λ C 1 < K < 1+λ C 1 C 1, C 1 < 1 1+C or K < 1+λ C 1 C 1 1+λ C 1, C 1 < 1 1+C or K > 1+λ C 1 C 1, C 1 > 1 1+C if either 1+λ C C 1+C λ < K < 1+λ C C, λ C > 1 or K < 1+λ C C 1+λ C, λ C > 1 or K > 1+λ C C, λ C < 1 It ca be easily proved that d 1P has smaller absolute relative bias ARB tha the covetioal product estimator y p but larger tha that of Mohaty Sahoo s 1995 estimators t 1p t p. Table 3. clearly idicates that the proposed estimator d P has smaller absolute relative bias tha the covetioal product estimator y P as the coditio λ < 1 always holds.

11 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 167 Table 3.: Compariso of absolute relative biases of product-type estimators. Estimator y P ifλ < 1 Absolute Relative Bias of d P is less tha t 1P ifλ < 1 C 1, C 1 > 1+C t P if C 1 +C 1 C 3 C C 1+1 > 0 d 1P ifλ <λ 1 4. Efficiecy compariso The efficiecy comparisos of ratio-type d 1R d R product-type d 1P d P estimators have bee made with y, y R, t 1R t R ; show i Tables respectively. Estimator Table 4.1: Compariso of mea squared errors of ratio-type estimators. Mea squared error of d 1R d R y if K > λ 1 1+λ 1 y R if K < 1+λ 1 C 1 t 1R if K > C 1 1+λ C if K > λ if K < 1+λ if either K < 1+C 1λ C 1, λ < 1 C 1 or K > 1+C 1λ C 1, λ > 1 C 1 t R if K > C if either K < 1+λ C C, λ < 1 C or K > 1+λ C C, λ > 1 C

12 168 O ratio product methods with certai kow populatio... Table 4.: Compariso of mea squared errors of product-type estimators. Mea squared error of Estimator d 1P is less tha d P is less tha y if K < λ 1 y P if K > 1+λ 1 t 1P if K < 1+λ 1C 1 C 1 t P if K < 1+λ 1C C if K < λ if K > 1+λ if either K < 1 1+λ C 1, λ C > 1 1 C 1 or K > 1 1+λ C 1, λ C < 1 1 C 1 if either K < 1 1+λ C, λ C > 1 C or K > 1 1+λ C, λ C < 1 C Table 4.1 exhibits that the ratio type estimator d 1R is better tha y, y R, t 1R t R if 1+λ1 C 1 < K < 1+λ 1 C 1 We also ote that the estimator d 1R is more efficiet tha d R if 4.1 K > λ 1+λ 4. It is observed from Table 4.1 that the product-type estimator d 1P is more efficiet tha y, y P, t 1P t P if 1+λ 1 < K < 1+λ 1C 1 C Further it ca be proved that the product-type estimator d 1P is better tha the producttype estimator d P if K < λ 1+λ 4.4

13 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor Ubiased versios of the suggested estimators I this sectio we will obtai the ubiased versios of the suggested estimators i Sectio, usig two well kow procedures: i Iterpeetratig subsamples desig ii Jack-kife techique Iterpeetratig sub-sample desig Let the sample i the form of idepedet iterpeetratig subsamples be draw. Let y i x i be ubiased estimates of the populatio totals Y= NY X= NX respectively based o the i th idepedet iterpeetratig subsample,,,...,. We ow cosider followig ratio product-type estimators of the populatio mea Y : d 1 = y A / a 5.1 d 1 = A / / yi ai 5. d = y B / b 5.3 d = B / / yi bi 5.4 d 3 = y a / A 5.5 d 3 = y i a i / A 5.6 d 4 = y b / B 5.7 d 4 = y i b i / B 5.8 where a, b, A, B, a i b i are same as defied i Sectio.

14 170 O ratio product methods with certai kow populatio... It is easy to verify that Bd 1 =Bd Bd =Bd 5.10 Bd 3 =Bd Bd 4 =Bd Thus we get the followig ratio product-type ubiased estimators of Y as d 1u = d 1 d d u = d d d 3u = d 3 d d 4u = d 4 d The properties of these ubiased estimators d ju, j = 1 to 4 ca be studied o the lies of Murthy Najamma Remark 5.1. I the case of simple rom samplig without replacemet SRSWOR, let y i x i deote respectively the y x values of the sample of uit,,,...,. We have d 1 = y A / a d 1 = A / yi / ai d = y B / b

15 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 171 d = B / / yi bi d 3 = y a / A d 3 = / y i a i A d 4 = y b / B d 4 = y i b i / B It ca be show uder SRSWOR scheme that the followig ratio-type estimators are ubiased for populatio mea Y as d 1u = N 1 N 1 y A a N A N 1 yi / ai 5.17 d u = N 1 N 1 y B b N B N 1 yi / bi 5.18 d 3u = d 4u = N 1 N 1 y N 1 N 1 y a A b B N N 1 N N 1 To the first degree of approximatio, it ca be show that 1 A 1 B y i a i 5.19 y i b i 5.0 Vard 1u= Vard 1R 5.1 Vard u= Vard R 5. Vard 3u= Vard 1p 5.3

16 17 O ratio product methods with certai kow populatio... Vard 4u= Vard p. 5.4 Thus the ubiased estimators d1u, d u, d 3u d 4u estimators d 1R, d R, d 1p d p respectively. are to be preferred over biased 5.. Jack-kife techique We may take =m split the sample at rom ito two subsamples of m uits each. Let y i, x i, be ubiased estimators of populatio mea Y X respectively based o the subsamples y, x the meas based o the etire sample. Thus a i, b i ;, are ubiased estimators based o the sub-samples a, b the meas based o the etire sample i.e., a i = x M x i + x m, b i = { x M x m x i + x m}, a = xx M + x m, b= { x M x m x+x m}, Thus motivated by Queoulle 1956 we defie the followig ratio product-type ubiased estimators of populatio mea Y as d u 1J N N { } = d 1 d d 1 N N 5.5 d u J N N { } = d d 1 + d N N 5.6 d u 3J d u 4J N N { } = d 3 d d 3 N N N N { } = d 4 d d 4 N N

17 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 173 where d 1, d, d 3 d 4 are same as defied i Sectio 5, d i 1 = y i A / ai, d i = y i B / bi, d i 3 = y i ai / A d i 4 = y i bi / B,,. Followig the procedure outlied i Sukhatme Sukhatme [1970, pp ], it ca be show to the first degree of approximatio that the variace expressios of d u l J, l = 1,,3,4 variace expressios of d 1R, d R, d 1p d p respectively are same. Thus we advocate that oe ca prefer the ubiased estimators d u l J, l = 1,,3,4 as compared to biased estimators d 1R, d R, d 1p d p. 6. Empirical study 6.1. Whe the variates y x are positively correlated To see the performaces of the suggested estimators d 1R d R over y, y R, t 1R t R, we have cosidered eight atural populatio data sets. Descriptios of the populatios are give below:

18 174 O ratio product methods with certai kow populatio... Pop. No. Table 6.1: Descriptio of populatios. Source N Y X ρ C x C y C 1 C K 1 Sahoo Swai Uit: 0.,0.6, 0.9,0.8 Uit: 0.1,0., 0.3, Murthy 1967, p Number of cattle Survey Number of cattle Cesus Murthy 1967, p Number of Absetees Number of Workers Pase Sukhatme 1967, p Paretal plot mea mm Paretal plat value mm Pase Sukhatme 1967, p Paretal plot mea mm Paretal plat value mm Pase Sukhatme 1967, p Progey mea mm Paretal plat value mm Sigh Chaudhary p No. of Cows i milk Survey No. of Cows i milk Cesus Sigh Chaudhary p No. of ihabitats 000 i No. of ihabitats 000 i Samford 196, p Acreage uder oats i 1957 Acreage of crops gross i To assess the biasedess of the ratio-type estimators y R, t 1R, t R, d 1R d R, we have computed the followig quatities for the populatio give i Table 6.1 usig the formulae: B 1 = By R θy Cx = 1 K 6.1

19 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 175 B = Bt 1R θ Y Cx = 1 1 K C 1 C 1 6. B 3 = Bt R θ Y Cx = 1 1 K C C 6.3 B 4 = Bd 1R θ Y Cx =λ 1 λ 1 K 6.4 B 5 = Bd R θ Y Cx =λ λ K 6.5 The fidigs are listed i Table 6.. Values of B i s i= 1 to 5 Table 6.: Values of B 1, B, B 3, B 4 B 5. Populatio B B B B B Table 6. exhibits that the proposed estimator d R has least bias for all data sets except i populatio III cosidered here. I populatio III, the proposed estimator d 1R has least bias. Usig the followig formulae: [ PREy R,y= MSEy MSEy R 100= 1+ Cx C y 1 K ] [ PREt 1R,y= MSEy MSEt 1R 100= 1+ 1 ] 1 Cx 1 K C 1 C y C 1 [ PREt R,y= MSEy MSEt R 100= 1+ 1 ] 1 Cx 1 K C C y C

20 176 O ratio product methods with certai kow populatio... [ PREd 1R,y= MSEy MSEd 1R 100= 1+ [ PREd R,y= MSEy MSEd R 100= 1+ Cx C y Cx C y λ 1λ 1 K λ 1λ K ] ] We have computed the percet relative efficiecies PREs of y R, t 1R, t R, d 1R d R with respect to usual ubiased estimator y compiled i Table 6.3. Table 6.3: Percet relative efficiecies of y R, t 1R, t R, d 1R d R with respect to y. Estimator PRE., y Populatio y y R t 1R t R d 1R d R Table 6.3 shows that the proposed estimator d R has largest gai i efficiecy for all populatio data sets except i populatio III, where the proposed estimator d 1R has maximum gai i efficiecy. We also ote that the proposed estimator d 1R domiates over the estimators y, y R, t 1R t R i populatio I, II, III, IV, VII VIII. Thus the proposed estimators d 1R d R are to be preferred over other estimators. Fially, from Tables we recommed the use of the proposed estimator d R i practice as it has largest gai i efficiecy also fewer bias i all populatio data sets except i populatio III, where the proposed estimator d 1R has largest gai i efficiecy as well as less bias hece d 1R is to be recommeded for this populatio data set. 6.. Whe the variates y x are egatively correlated To assess the biasdeess efficiecy of the product-type estimators y p, t 1p, t p, d 1p d p we have cosidered atural populatio data sets.

21 Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 177 Pop. No. Table 6.4: Descriptio of the populatios. Source N Y X ρ C x C y C 1 C K 1 Maddla, G.S. 1977, p Capita Cosumptio Deflated price Gupta, S.P. Gupta, A p Artificial Populatio To observe the biasedess of the estimators y p, t 1p, t p, d 1p d p, we use the followig formulae: B By 1 = p θ YCx = K 6.11 B By = 1p θ YCx = K C B 3 = Bt p θ YCx = K C 6.13 B 4 = Bd 1p θ YCx =λ 1 K 6.14 B 5 = Bd p θyc x =λ K 6.15 The quatities B s to 5 have bee computed fidigs are give i Table 6.5. Populatio Table 6.5: Values of B 1, B, B 3, B 4 B 5. Values of B i s, i = 1 to 5 B 1 B B 3 B 4 B

22 178 O ratio product methods with certai kow populatio... Usig the followig formulae: [ PREy p,y= MSEy MSEy P 100= 1+ Cx C y 1+K ] [ PREt 1p,y= MSEy MSEt 1P 100= 1+ Cx C y 1 C 1 ] K C 1 [ PREt p,y= MSEy MSEt P 100= 1+ Cx C y 1 C ] K C [ PREd 1p,y= MSEy MSEy P 100= 1+ [ PREd p,y= MSEy MSEy P 100= 1+ Cx C y Cx C y λ 1 λ1 + K ] λ λ + K ] We have computed the percet relative efficiecies PREs of y p, t 1p, t p, d 1p d p with respect to usual ubiased estimator y the results are show i Table 6.6. Table 6.6: Percet relative efficiecies of y p, t 1p, t p, d 1p d p with respect to y. Estimators y y p t 1p t p d 1p d p PRE.,y Populatio Populatio Tables show that the proposed estimators d 1p d p are more efficiet with substatial gai tha usual ubiased estimator y, product estimator y p the estimators t 1p t p reported by Sahoo Mohaty 1995, but these two estimators d 1p d p are more biased tha t 1p t p. Thus if the variace / MSE s criterio of judgig the performace of the estimators are adopted also the biasedess of the estimators are ot of primary cocer the the proposed estimators d 1p d p are recommeded for their use i practice.

23 Ackowledgemet Housila. P. Sigh, Ritesh Tailor Rajesh Tailor 179 Authors are thakful to the referee for his valuable suggests ios regardig improvemet of the earlier draft of the paper. Refereces Byopadhyaya, S Improved ratio product estimators. Sakhya, 4,C, Cochra, W. G Samplig Techiques. Joh Wiley Sos Ic., New York, II Editio. Gupta, P. C O some quadratic higher degree ratio product estimators. Joural of the Idia Society of Agricultural Statistics, 30, Gupta, S. P. Gupta, A Statistical Methods. Sulta Ch & Sos, 6.5, New Delhi. Maddala, G. S Ecoometrics. McGraw Hills pub.co., New York. Mohaty, S. Das, M. N Use of trasformatio i samplig. Joural of the Idia Society of Agricultural Statistics, 3,, Mohaty, S. Sahoo, J A ote o improvig the ratio method of estimatio through liear trasformatio usig certai kow populatio parameters. Sakhya, 57, B, Murthy, M. N Product method of estimatio. Sakhya, Murthy, M. N Samplig Theory Methods. Statistical Publishig Society, Calcutta. Murthy, M. N. Najamma, N. S Almost ubiased ratio estimators based o iterpeetratig sub samples. Sakhya, 1, Naik, V. D. Gupta, P. C A geeral class of estimators for estimatig populatio meas usig auxiliary iformatio. Metrika, 38, Pase, V. G. Sukhatme, P. V Statistical methods for agricultural workers. Idia Coucil of Agricultural Research, New Delhi. Reddy, V. N O ratio product methods of estimatio. Sakhya, B, 35, Reddy, V. N O a trasformed ratio method of estimatio. Sakhya, C, 36, Robso, D. S Applicatio of multivariate Polykays to the theory of ubiased ratio-type estimatio. Joural of the America Statistical Associatio, 59, Rueda, M. Gozález, S A ew ratio-type imputatio with rom disturbace. Applied Mathematics Letters, 1, 9, Sahai, A. Sahai, A O efficiet use of auxiliary iformatio. Joural of Statistical Plaig Iferece, 1, Sahoo, L. N. Swai, A. K. P. C Some modified ratio estimators. Metero, Sampford, M. R A Itroductio to Samplig Theory. Oliver Boyd. Samawi, H. M. Al-Saleh, M. F O bivariate raked set samplig for ratio regressio estimators. Iteratioal Joural of Modellig Simulatio, 7, 4, Sigh, D. Chaudhary, F. S Theory Aalysis of Sample Survey Desigs. Wilely easter Lt., New Delhi. Sigh, S Advaced Samplig Theory with Applicatios. How Michael selected Amy, 1-147, Kluwer Academic Publishers, The Netherls. Sisodia, B. V. Dwivedi, V. K A modified ratio estimator usig coefficiet of variatio of auxiliary variable. Joural of the Idia Society of Agricultural Statistics, 33, Srivastava, S. K A estimator usig auxiliary iformatio i sample surveys. Cal. Statist. Assoc. Bull., 6,

24 180 O ratio product methods with certai kow populatio... Srivastava, S. K Predictive estimatio of fiite populatio mea usig product estimator. Metrika, 30, Srivekataramaa, T Chage of origi scale i ratio differece method of estimatio i samplig. The Caadia Joural of Statistics, 6,1, Srivekataramaa, T A dual to ratio estimator i sample surveys. Biometrika, 67, Sukhatme, P. V. Sukhatme, B. V Samplig Theory of Surveys with Applicatios, d ed. Ames. Iowa State Uiversity Press. Upadhyaya, L. N. Sigh, H. P Use of trasformed auxiliary variable i estimatig the fiite populatio mea. Biometrical Joural 41, 5, Vos, J. W. E Mixig of direct ratio product method estimators. Statistica Neerlica, 34, Walsh, J. E Geeralizatio of ratio estimate for populatio total. Sakhya, A., 3,

Research Article An Alternative Estimator for Estimating the Finite Population Mean Using Auxiliary Information in Sample Surveys

Research Article An Alternative Estimator for Estimating the Finite Population Mean Using Auxiliary Information in Sample Surveys Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume 01, Article ID 65768, 1 pages doi:10.50/01/65768 Research Article A Alterative Estimator for Estimatig the Fiite Populatio Mea

More information

Improved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling

Improved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA

More information

Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable

Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya

More information

Modified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis

Modified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry

More information

Some Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation

Some Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation ; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 097-70)]; ISSN 0975-556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product

More information

Varanasi , India. Corresponding author

Varanasi , India. Corresponding author A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet

More information

Improved exponential estimator for population variance using two auxiliary variables

Improved exponential estimator for population variance using two auxiliary variables OCTOGON MATHEMATICAL MAGAZINE Vol. 7, No., October 009, pp 667-67 ISSN -5657, ISBN 97-973-55-5-0, www.hetfalu.ro/octogo 667 Improved expoetial estimator for populatio variace usig two auxiliar variables

More information

Abstract. Ranked set sampling, auxiliary variable, variance.

Abstract. Ranked set sampling, auxiliary variable, variance. Hacettepe Joural of Mathematics ad Statistics Volume (), 1 A class of Hartley-Ross type Ubiased estimators for Populatio Mea usig Raked Set Samplig Lakhkar Kha ad Javid Shabbir Abstract I this paper, we

More information

A Family of Unbiased Estimators of Population Mean Using an Auxiliary Variable

A Family of Unbiased Estimators of Population Mean Using an Auxiliary Variable Advaces i Computatioal Scieces ad Techolog ISSN 0973-6107 Volume 10, Number 1 (017 pp. 19-137 Research Idia Publicatios http://www.ripublicatio.com A Famil of Ubiased Estimators of Populatio Mea Usig a

More information

Estimation of the Population Mean in Presence of Non-Response

Estimation of the Population Mean in Presence of Non-Response Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of No-Respose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,

More information

AClassofRegressionEstimatorwithCumDualProductEstimatorAsIntercept

AClassofRegressionEstimatorwithCumDualProductEstimatorAsIntercept Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 15 Issue 3 Versio 1.0 Year 2015 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic.

More information

Jambulingam Subramani 1, Gnanasegaran Kumarapandiyan 2 and Saminathan Balamurali 3

Jambulingam Subramani 1, Gnanasegaran Kumarapandiyan 2 and Saminathan Balamurali 3 ISSN 1684-8403 Joural of Statistics Volume, 015. pp. 84-305 Abstract A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of

More information

Chain ratio-to-regression estimators in two-phase sampling in the presence of non-response

Chain ratio-to-regression estimators in two-phase sampling in the presence of non-response ProbStat Forum, Volume 08, July 015, Pages 95 10 ISS 0974-335 ProbStat Forum is a e-joural. For details please visit www.probstat.org.i Chai ratio-to-regressio estimators i two-phase samplig i the presece

More information

A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)

A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s) Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig

More information

Alternative Ratio Estimator of Population Mean in Simple Random Sampling

Alternative Ratio Estimator of Population Mean in Simple Random Sampling Joural of Mathematics Research; Vol. 6, No. 3; 014 ISSN 1916-9795 E-ISSN 1916-9809 Published by Caadia Ceter of Sciece ad Educatio Alterative Ratio Estimator of Populatio Mea i Simple Radom Samplig Ekaette

More information

Improved Ratio Estimators of Population Mean In Adaptive Cluster Sampling

Improved Ratio Estimators of Population Mean In Adaptive Cluster Sampling J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) 1 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.18576/jsapl/030101 Improved Ratio Estimators of Populatio

More information

A Generalized Class of Unbiased Estimators for Population Mean Using Auxiliary Information on an Attribute and an Auxiliary Variable

A Generalized Class of Unbiased Estimators for Population Mean Using Auxiliary Information on an Attribute and an Auxiliary Variable Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN 89-4966 Volume, Number 07, pp. -8 Research Idia ublicatios http://www.ripublicatio.com A Geeralized Class of Ubiased Estimators for opulatio

More information

REVISTA INVESTIGACION OPERACIONAL VOL. 35, NO. 1, 49-57, 2014

REVISTA INVESTIGACION OPERACIONAL VOL. 35, NO. 1, 49-57, 2014 EVISTA IVESTIGAIO OPEAIOAL VOL. 35, O., 9-57, 0 O A IMPOVED ATIO TYPE ESTIMATO OF FIITE POPULATIO MEA I SAMPLE SUVEYS A K P Swai Former Professor of Statistics, Utkal Uiversit, Bhubaeswar-7500, Idia ABSTAT

More information

Dual to Ratio Estimators for Mean Estimation in Successive Sampling using Auxiliary Information on Two Occasion

Dual to Ratio Estimators for Mean Estimation in Successive Sampling using Auxiliary Information on Two Occasion J. Stat. Appl. Pro. 7, o. 1, 49-58 (018) 49 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.18576/jsap/070105 Dual to Ratio Estimators for Mea Estimatio i Successive

More information

Enhancing ratio estimators for estimating population mean using maximum value of auxiliary variable

Enhancing ratio estimators for estimating population mean using maximum value of auxiliary variable J.Nat.Sci.Foudatio Sri Laka 08 46 (: 45-46 DOI: http://d.doi.org/0.408/jsfsr.v46i.8498 RESEARCH ARTICLE Ehacig ratio estimators for estimatig populatio mea usig maimum value of auiliar variable Nasir Abbas,

More information

Double Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution

Double Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.

More information

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

More information

Element sampling: Part 2

Element sampling: Part 2 Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

More information

SYSTEMATIC SAMPLING FOR NON-LINEAR TREND IN MILK YIELD DATA

SYSTEMATIC SAMPLING FOR NON-LINEAR TREND IN MILK YIELD DATA Joural of Reliability ad Statistical Studies; ISS (Prit): 0974-804, (Olie):9-5666 Vol. 7, Issue (04): 57-68 SYSTEMATIC SAMPLIG FOR O-LIEAR TRED I MILK YIELD DATA Tauj Kumar Padey ad Viod Kumar Departmet

More information

Use of Auxiliary Information for Estimating Population Mean in Systematic Sampling under Non- Response

Use of Auxiliary Information for Estimating Population Mean in Systematic Sampling under Non- Response Maoj K. haudhar, Sachi Malik, Rajesh Sigh Departmet of Statistics, Baaras Hidu Uiversit Varaasi-005, Idia Floreti Smaradache Uiversit of New Mexico, Gallup, USA Use of Auxiliar Iformatio for Estimatig

More information

Estimation of Population Ratio in Post-Stratified Sampling Using Variable Transformation

Estimation of Population Ratio in Post-Stratified Sampling Using Variable Transformation Ope Joural o Statistics, 05, 5, -9 Published Olie Februar 05 i SciRes. http://www.scirp.org/joural/ojs http://dx.doi.org/0.436/ojs.05.500 Estimatio o Populatio Ratio i Post-Stratiied Samplig Usig Variable

More information

Journal of Scientific Research Vol. 62, 2018 : Banaras Hindu University, Varanasi ISSN :

Journal of Scientific Research Vol. 62, 2018 : Banaras Hindu University, Varanasi ISSN : Joural of Scietific Research Vol. 6 8 : 3-34 Baaras Hidu Uiversity Varaasi ISS : 447-9483 Geeralized ad trasformed two phase samplig Ratio ad Product ype stimators for Populatio Mea Usig uiliary haracter

More information

New Ratio Estimators Using Correlation Coefficient

New Ratio Estimators Using Correlation Coefficient New atio Estimators Usig Correlatio Coefficiet Cem Kadilar ad Hula Cigi Hacettepe Uiversit, Departmet of tatistics, Betepe, 06800, Akara, Turke. e-mails : kadilar@hacettepe.edu.tr ; hcigi@hacettepe.edu.tr

More information

Research Article A Two-Parameter Ratio-Product-Ratio Estimator Using Auxiliary Information

Research Article A Two-Parameter Ratio-Product-Ratio Estimator Using Auxiliary Information Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume, Article ID 386, 5 pages doi:.54//386 Research Article A Two-Parameter Ratio-Product-Ratio Estimator Usig Auxiliary Iformatio

More information

Bayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function

Bayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume 12, Number 4 (217), pp. 791-796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E- Bayesia Method of Estimatio of Parameter

More information

A Generalized Class of Estimators for Finite Population Variance in Presence of Measurement Errors

A Generalized Class of Estimators for Finite Population Variance in Presence of Measurement Errors Joural of Moder Applied Statistical Methods Volume Issue Article 3 --03 A Geeralized Class of Estimators for Fiite Populatio Variace i Presece of Measuremet Errors Praas Sharma Baaras Hidu Uiversit, Varaasi,

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.

It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable. Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig

More information

ESTIMATION OF FINITE POPULATION MEAN WITH KNOWN COEFFICIENT OF VARIATION OF AN AUXILIARY CHARACTER

ESTIMATION OF FINITE POPULATION MEAN WITH KNOWN COEFFICIENT OF VARIATION OF AN AUXILIARY CHARACTER STATISTICA, ao LXV,. 3, 2005 ESTIMATION OF FINITE POPULATION MEAN WITH KNOWN COEFFICIENT OF VARIATION OF AN AUXILIAR CHARACTER H.P. Sigh, R. Tail 1. INTRODUCTION AND THE SUGGESTED ESTIMATOR It is well

More information

Investigating the Significance of a Correlation Coefficient using Jackknife Estimates

Investigating the Significance of a Correlation Coefficient using Jackknife Estimates Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN 2307-4531 (Prit & Olie) http://gssrr.org/idex.php?joural=jouralofbasicadapplied ---------------------------------------------------------------------------------------------------------------------------

More information

Estimation of Gumbel Parameters under Ranked Set Sampling

Estimation of Gumbel Parameters under Ranked Set Sampling Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com

More information

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

More information

Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation

Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece

More information

Method of Estimation in the Presence of Nonresponse and Measurement Errors Simultaneously

Method of Estimation in the Presence of Nonresponse and Measurement Errors Simultaneously Joural of Moder Applied Statistical Methods Volume 4 Issue Article 5--05 Method of Estimatio i the Presece of Norespose ad Measuremet Errors Simultaeousl Rajesh Sigh Sigh Baaras Hidu Uiversit, Varaasi,

More information

On stratified randomized response sampling

On stratified randomized response sampling Model Assisted Statistics ad Applicatios 1 (005,006) 31 36 31 IOS ress O stratified radomized respose samplig Jea-Bok Ryu a,, Jog-Mi Kim b, Tae-Youg Heo c ad Chu Gu ark d a Statistics, Divisio of Life

More information

Control Charts for Mean for Non-Normally Correlated Data

Control Charts for Mean for Non-Normally Correlated Data Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

Modeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy

Modeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral

More information

Maximum likelihood estimation from record-breaking data for the generalized Pareto distribution

Maximum likelihood estimation from record-breaking data for the generalized Pareto distribution METRON - Iteratioal Joural of Statistics 004, vol. LXII,. 3, pp. 377-389 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Maximum likelihood estimatio from record-breakig data for the geeralized Pareto distributio

More information

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity

More information

GUIDELINES ON REPRESENTATIVE SAMPLING

GUIDELINES ON REPRESENTATIVE SAMPLING DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue

More information

Generalized Exponential Type Estimator for Population Variance in Survey Sampling

Generalized Exponential Type Estimator for Population Variance in Survey Sampling Revista Colombiaa de Estadística Juio 2014, volume 37, o. 1, pp. 211 a 222 Geeralized Expoetial Type Estimator for Populatio Variace i Survey Samplig Estimadores tipo expoecial geeralizado para la variaza

More information

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram. Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios

More information

In this section we derive some finite-sample properties of the OLS estimator. b is an estimator of β. It is a function of the random sample data.

In this section we derive some finite-sample properties of the OLS estimator. b is an estimator of β. It is a function of the random sample data. 17 3. OLS Part III I this sectio we derive some fiite-sample properties of the OLS estimator. 3.1 The Samplig Distributio of the OLS Estimator y = Xβ + ε ; ε ~ N[0, σ 2 I ] b = (X X) 1 X y = f(y) ε is

More information

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the

More information

ON POINTWISE BINOMIAL APPROXIMATION

ON POINTWISE BINOMIAL APPROXIMATION Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece

More information

Study on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm

Study on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm Joural of ad Eergy Egieerig, 05, 3, 43-437 Published Olie April 05 i SciRes. http://www.scirp.org/joural/jpee http://dx.doi.org/0.436/jpee.05.34058 Study o Coal Cosumptio Curve Fittig of the Thermal Based

More information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

The Sample Variance Formula: A Detailed Study of an Old Controversy

The Sample Variance Formula: A Detailed Study of an Old Controversy The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace

More information

Enhancing the Mean Ratio Estimators for Estimating Population Mean Using Non-Conventional Location Parameters

Enhancing the Mean Ratio Estimators for Estimating Population Mean Using Non-Conventional Location Parameters evista Colombiaa de Estadística Jauary 016, Volume 39, Issue 1, pp. 63 to 79 DOI: http://dx.doi.org/10.15446/rce.v391.55139 Ehacig the Mea atio Estimators for Estimatig Populatio Mea Usig No-Covetioal

More information

A new distribution-free quantile estimator

A new distribution-free quantile estimator Biometrika (1982), 69, 3, pp. 635-40 Prited i Great Britai 635 A ew distributio-free quatile estimator BY FRANK E. HARRELL Cliical Biostatistics, Duke Uiversity Medical Ceter, Durham, North Carolia, U.S.A.

More information

Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution

Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical

More information

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n. Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

More information

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determine sample size to estimate characteristic value of soil parameters A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig

More information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

11 Correlation and Regression

11 Correlation and Regression 11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record

More information

Ismor Fischer, 1/11/

Ismor Fischer, 1/11/ Ismor Fischer, //04 7.4-7.4 Problems. I Problem 4.4/9, it was show that importat relatios exist betwee populatio meas, variaces, ad covariace. Specifically, we have the formulas that appear below left.

More information

ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY

ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,

More information

Estimation of Population Mean in Presence of Non-Response in Double Sampling

Estimation of Population Mean in Presence of Non-Response in Double Sampling J. Stat. Appl. Pro. 6, No. 2, 345-353 (2017) 345 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.18576/jsap/060209 Estimatio of Populatio Mea i Presece of No-Respose

More information

Confidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation

Confidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology

More information

Bivariate Sample Statistics Geog 210C Introduction to Spatial Data Analysis. Chris Funk. Lecture 7

Bivariate Sample Statistics Geog 210C Introduction to Spatial Data Analysis. Chris Funk. Lecture 7 Bivariate Sample Statistics Geog 210C Itroductio to Spatial Data Aalysis Chris Fuk Lecture 7 Overview Real statistical applicatio: Remote moitorig of east Africa log rais Lead up to Lab 5-6 Review of bivariate/multivariate

More information

A Relationship Between the One-Way MANOVA Test Statistic and the Hotelling Lawley Trace Test Statistic

A Relationship Between the One-Way MANOVA Test Statistic and the Hotelling Lawley Trace Test Statistic http://ijspccseetorg Iteratioal Joural of Statistics ad Probability Vol 7, No 6; 2018 A Relatioship Betwee the Oe-Way MANOVA Test Statistic ad the Hotellig Lawley Trace Test Statistic Hasthika S Rupasighe

More information

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

An Improved Warner s Randomized Response Model

An Improved Warner s Randomized Response Model Iteratioal Joural of Statistics ad Applicatios 05, 5(6: 63-67 DOI: 0.593/j.statistics.050506.0 A Improved Warer s Radomized Respose Model F. B. Adebola, O. O. Johso * Departmet of Statistics, Federal Uiversit

More information

1 Inferential Methods for Correlation and Regression Analysis

1 Inferential Methods for Correlation and Regression Analysis 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet

More information

o <Xln <X2n <... <X n < o (1.1)

o <Xln <X2n <... <X n < o (1.1) Metrika, Volume 28, 1981, page 257-262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow

More information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

A New Mixed Randomized Response Model

A New Mixed Randomized Response Model Iteratioal Joural of Busiess ad Social Sciece ol No ; October 00 A New Mixed adomized espose Model Aesha Nazuk NUST Busiess School Islamabad, Paksta E-mail: Aeshaazuk@bsedupk Phoe: 009-5-9085-367 Abstract

More information

EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR

EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Joural of Statistical Research 26, Vol. 37, No. 2, pp. 43-55 Bagladesh ISSN 256-422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics

More information

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated

More information

COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun

COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun Korea J. Math. 23 2015) No. 3 pp. 371 377 http://dx.doi.org/10.11568/kjm.2015.23.3.371 COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized

More information

A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca

A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity

More information

A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos

A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos .- A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES by Deis D. Boos Departmet of Statistics North Carolia State Uiversity Istitute of Statistics Mimeo Series #1198 September,

More information

Self-normalized deviation inequalities with application to t-statistic

Self-normalized deviation inequalities with application to t-statistic Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric

More information

4.5 Multiple Imputation

4.5 Multiple Imputation 45 ultiple Imputatio Itroductio Assume a parametric model: y fy x; θ We are iterested i makig iferece about θ I Bayesia approach, we wat to make iferece about θ from fθ x, y = πθfy x, θ πθfy x, θdθ where

More information

Stat 319 Theory of Statistics (2) Exercises

Stat 319 Theory of Statistics (2) Exercises Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.

More information

Journal of Multivariate Analysis. Superefficient estimation of the marginals by exploiting knowledge on the copula

Journal of Multivariate Analysis. Superefficient estimation of the marginals by exploiting knowledge on the copula Joural of Multivariate Aalysis 102 (2011) 1315 1319 Cotets lists available at ScieceDirect Joural of Multivariate Aalysis joural homepage: www.elsevier.com/locate/jmva Superefficiet estimatio of the margials

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

Discrete Orthogonal Moment Features Using Chebyshev Polynomials

Discrete Orthogonal Moment Features Using Chebyshev Polynomials Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical

More information

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

Estimation of Backward Perturbation Bounds For Linear Least Squares Problem

Estimation of Backward Perturbation Bounds For Linear Least Squares Problem dvaced Sciece ad Techology Letters Vol.53 (ITS 4), pp.47-476 http://dx.doi.org/.457/astl.4.53.96 Estimatio of Bacward Perturbatio Bouds For Liear Least Squares Problem Xixiu Li School of Natural Scieces,

More information

Generalization of Samuelson s inequality and location of eigenvalues

Generalization of Samuelson s inequality and location of eigenvalues Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,

More information

Correlation Regression

Correlation Regression Correlatio Regressio While correlatio methods measure the stregth of a liear relatioship betwee two variables, we might wish to go a little further: How much does oe variable chage for a give chage i aother

More information

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.

More information

Evapotranspiration Estimation Using Support Vector Machines and Hargreaves-Samani Equation for St. Johns, FL, USA

Evapotranspiration Estimation Using Support Vector Machines and Hargreaves-Samani Equation for St. Johns, FL, USA Evirometal Egieerig 0th Iteratioal Coferece eissn 2029-7092 / eisbn 978-609-476-044-0 Vilius Gedimias Techical Uiversity Lithuaia, 27 28 April 207 Article ID: eviro.207.094 http://eviro.vgtu.lt DOI: https://doi.org/0.3846/eviro.207.094

More information

Module 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation

Module 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio

More information

This is an introductory course in Analysis of Variance and Design of Experiments.

This is an introductory course in Analysis of Variance and Design of Experiments. 1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class

More information

Precise Rates in Complete Moment Convergence for Negatively Associated Sequences

Precise Rates in Complete Moment Convergence for Negatively Associated Sequences Commuicatios of the Korea Statistical Society 29, Vol. 16, No. 5, 841 849 Precise Rates i Complete Momet Covergece for Negatively Associated Sequeces Dae-Hee Ryu 1,a a Departmet of Computer Sciece, ChugWoo

More information

Orthogonal Gaussian Filters for Signal Processing

Orthogonal Gaussian Filters for Signal Processing Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios

More information

Lecture 12: September 27

Lecture 12: September 27 36-705: Itermediate Statistics Fall 207 Lecturer: Siva Balakrisha Lecture 2: September 27 Today we will discuss sufficiecy i more detail ad the begi to discuss some geeral strategies for costructig estimators.

More information

Control chart for number of customers in the system of M [X] / M / 1 Queueing system

Control chart for number of customers in the system of M [X] / M / 1 Queueing system Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.

More information

January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS

January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we

More information