Appraisal job has ever been critical in all the spheres of life. Effective planning depends upon clearcutness of the estimations that ‘s why research workers are ever in procedure of developing methods that can bring forth more precise estimations. Several methods are available in literature that can be used for efficient appraisal of the characteristic under survey, these methods are jointly called sampling methods. The scientific development in the field of study sampling has long history but the groundbreaking work in this field was done by Neyman ( 1934 ) . The work of Neyman ( 1934 ) guided the figure of statisticians for important development in assorted countries of study sampling. The historical work done by Hansen & A ; Hurwitz ( 1943 ) and by Horvitz & A ; Thompson ( 1952 ) in the development of unequal chance sampling is besides based upon the thoughts given by Neyman ( 1934 ) .

History of sampling has been discussed by many study statisticians, some noteworthy mentions are Chang ( 1976 ) , Dalenius ( 1962 ) , Duncan & A ; Shelton ( 1978 ) , Hansen ( 1987 ) , Kruskal & A ; Mosteller ( 1980 ) , Seng ( 1951 ) and Stephan ( 1948 ) . { Kiaer, 1895 # 164 } in a meeting of International Statistical Institute ( ISI ) put frontward the thought that a partial probe could supply utile information. Detailed treatments of { Kiaer, 1895 # 164 } work and its impact on trying methodological analysis may be found in Seng ( 1951 ) and Kruskal & A ; Mosteller ( 1980 ) . The initial reaction to { Kiaer, 1895 # 164 } work was negative and by and large non receptive ; nevertheless in 1901 and 1903 Kiaer was supported by C. D. Wright and subsequently by A. L. Bowley. Kiaer ( 1897 ) mentions the possibility of randomisation, in his words a sample ‘selected through the drawing of tonss ‘ , but does non develop the thought farther in his Hagiographas.

Like Kiaer, Bowley actively promoted his thoughts on sampling and randomisation specially. Bowley ( 1906 ) paper incorporating an empirical confirmation to simple random sampling, at this point Bowley has likely equated random trying to any trying strategy in which the inclusion chances are the same for every trying units. Bowley ( 1913 ) used a systematic sample of edifices of “ Reading ” from street listing in the local directory of residential edifices. Bowley ( 1913 ) besides checked the representativeness of his samples by comparing his sample consequences to cognize population counts. For two instances in which Bowley ( 1913 ) found a disagreement between his sample and official statistics, on farther look intoing it was discovered that the official statistics contained mistake. This work is discussed in inside informations by Seng ( 1951 ) and Kruskal & A ; Mosteller ( 1980 ) . Besides Bowley ( 1926 ) provided a theoretical monograph sum uping the known consequences in random and purposive choice.

The work of Neyman ( 1934 ) paper has been recognized as an of import part to the field of study trying. Kruskal & A ; Mosteller ( 1980 ) have discussed the work as “ the Neyman watershed ” and Hansen, Dalenius, & A ; Tepping ( 1985 ) have commented that the “ paper played a overriding function in advancing theoretical research, developments, and application of what is now known as chance sampling ” . The work of Neyman ( 1934 ) is considered as a authoritative work on two evidences ; Firstly Neyman was able to supply valid grounds, both theoretically and with practical illustrations that why randomisation gave a much more sensible solution than purposive choice. Second, the paper provides a paradigm in the history of sampling is that the theory of point and interval appraisal is provided under randomisation. Neyman ( 1938 ) introduced the usage of cost map into study trying in connexion with two-phase sampling.

In early 1940 ‘s Hansen & A ; Hurwitz ( 1943 ) made some cardinal part to theory of trying, they took an of import measure frontward by widening the thought of trying with unequal inclusion chances for units in different strata. This allowed the development of really complex multi-stage designs that are the anchor of big scale societal and economic study research.

## Introduction to Multiphase Sampling

Information related to the variable of involvement is termed as subsidiary information, which can be utilized to better the efficiency of the calculators. In Multiphase trying certain points of information are drawn from the whole trying units and certain other points of information are taken from the subsample. Multiphase sampling is used when it is expensive to roll up informations on the variable of involvement but it is comparatively cheap to roll up informations on variables that are correlated with the variables of involvement. For illustration, in forest studies, it is hard to go to remote country to do on land finding. However, aerial exposure of wood are comparatively cheap, which can be used to make up one’s mind about forest type ; a strongly correlated variable with land finding.

## 1.1.1 Notation of Multiphase Sampling

Let a population of N units is designated as, is the value of variable of involvement associated with, and be information of subsidiary variables associated with. Let, and be population mean of X, W and Y severally. Further Lashkar-e-Taiba, and are matching discrepancies. Besides and are population correlativity coefficients between X & A ; Y, X & A ; W and Y & A ; W severally. Let a first stage sample of units is drawn from the population and information on subsidiary variables is recoded ; farther allow a sub-sample of units is drawn from the first stage sample and information of subsidiary variables aboard variable of involvement.

The sample agencies of subsidiary variables based on units are denoted byand etc and sample agencies of 2nd stage are denoted by and. We will besides utilize and such that. For notational intent it will be assumed that the mean of estimand and subsidiary variables can be approximated from their population means so that where is sample mean of subsidiary variable Ten at h-th stage ; h = 1 and 2. Similar notation will be used for other quantitative subsidiary variables. For qualitative subsidiary variables we will utilize will be used and for variable of involvement we will utilize the notation with usual premises.

Following outlooks for deducing the average square mistake of calculators which are based upon quantitative subsidiary variables will be used:

( 1.1.1.1 )

In instance of several subsidiary variables ; state Q ; the sample mean of i-th subsidiary at h-th stage will be denoted by. The vector notations in instance of multiple subsidiary variables and sample average vector of subsidiary variables at h-th stage will be denoted by with relation. Following extra outlooks are besides utile:

Similar outlooks for qualitative subsidiary variables are:

( 1.1.1.2 )

For multivariate and Zelner calculator we will utilize undermentioned notations:

( 1.1.1.3 )

where is covariance matrix of variables of involvement, the notation of will be used for new calculators under two stage sampling.

## 1.2 Some Popular Univarate Calculators in Multiphase Sampling based on Quantitative Forecasters

In this subdivision some well-known ratio and arrested development calculators for gauging population mean along with their average square mistakes for two-phase trying utilizing one and two quantitative subsidiary variables are discussed.

The traditional ratio calculator for unknown population mean suggested by Cochran ( 1977 ) is

( 1.2.1 )

and its average square mistake is

, ( 1.2.2 )

where is coefficient of fluctuation of ten and is correlation coefficient between x and Y.

Cochran ( 1977 ) suggested following simple arrested development calculator for unknown as under:

, ( 1.2.3 )

where is based on 2nd stage sample and look for average square mistake is

. ( 1.2.4 )

Another simple arrested development calculator of ; suggested by Cochran ( 1977 ) when is known:

, ( 1.2.5 )

With average square mistake:

. ( 1.2.6 )

It can be instantly seen that.

Mohanty ( 1967 ) suggested the undermentioned regression-cum-ratio calculator by uniting the arrested development and ratio method when information on is non available:

, ( 1.2.7 )

where is calculated from 2nd stage sample and the look for average square mistake of

. ( 1.2.8 )

Another regression-cum-ratio calculator suggested by Mohanty ( 1967 ) when information on both subsidiary variables is unavailable:

, ( 1.2.9 )

where is calculated from 2nd stage sample and look for average square mistake is

. ( 1.2.10 )

The concatenation ratio-type calculator proposed by Chand ( 1975 ) for two-phase trying utilizing two-auxiliary variables ; when population mean is known:

( 1.2.11 )

and its average square mistake is:

. ( 1.2.12 )

Another concatenation ratio-type calculator suggested by Chand ( 1975 ) when information on subsidiary variable Z is unavailable for population is

( 1.2.13 )

and its average square mistake is

. ( 1.2.14 )

The ratio-to-regression calculator suggested by Kiregyera ( 1980 ) is:

( 1.2.15 )

where is calculated from the first stage sample. The average square mistake of can be written as:

. ( 1.2.16 )

The ratio-in-regression calculator developed by Kiregyera ( 1984 ) is

( 1.2.17 )

where computed from 2nd stage sample. The average square mistake of ( 1.2.17 ) can be written as:

. ( 1.2.18 )

Another regression-in-regression calculator suggested by Kiregyera ( 1984 ) is:

( 1.2.19 )

where is based on 2nd stage sample while is based on first stage sample and average square mistake of is:

. ( 1.2.20 )

Following the building of regression-in-regression calculator by Kiregyera ( 1984 ) , Mukerjee, Rao, & A ; Vijayan ( 1987 ) developed the calculator when information on both subsidiary variables is unavailable. The arrested development calculator utilizing two subsidiary variables is:

( 1.2.21 )

where and are based on 2nd stage sample. The average square mistake of ( 1.2.21 ) can be written as:

. ( 1.2.22 )

Mukerjee, et Al. ( 1987 ) besides proposed following calculator when population information on subsidiary variable Z is available:

( 1.2.23 )

where and are based on 2nd stage sample. The average square mistake:

. ( 1.2.24 )

Mukerjee, et Al. ( 1987 ) developed the 3rd calculator when information on subsidiary variable Z is available for population as:

( 1.2.25 )

where and are based on 2nd stage sample while is based on first stage sample. The average square mistake for this calculator can be written as:

( 1.2.26 )

J. Sahoo, Sahoo, & A ; Mohanty ( 1993 ) suggested the undermentioned calculator

( 1.2.27 )

where and are based on 2nd stage sample. The average square mistake for this calculator can be written as:

. ( 1.2.28 )

J. Sahoo & A ; Sahoo ( 1994 ) proposed three arrested development type calculators utilizing information of two subsidiary variables. The first calculator proposed by J. Sahoo & A ; Sahoo ( 1994 ) when information on subsidiary variable “ omega ” is available for population is

( 1.2.29 )

where and are based on 2nd stage sample. The average square mistake for this calculator is:

. ( 1.2.30 )

The 2nd calculator proposed by J. Sahoo & A ; Sahoo ( 1994 ) when information on subsidiary variable “ omega ” is available for population is:

( 1.2.31 )

where and are based on 2nd stage sample while is based on the first stage sample. The average square mistake for this calculator is:

( 1.2.32 )

Third calculator proposed by J. Sahoo & A ; Sahoo ( 1994 ) when information on subsidiary Z is available for population, is:

( 1.2.33 )

where and are based on the 2nd stage sample while is based on the first stage sample. The average square mistake for this calculator is:

( 1.2.34 )

S. K. Srivastava ( 1970 ) suggested the undermentioned general ratio calculator utilizing individual subsidiary variable as:

( 1.2.35 )

where is unknown changeless and the value of for which the mean square mistake of is minimal is and the average square mistake of is

( 1.2.36 )

Another general ratio calculator suggested by S. R. Srivastava, Khare, & A ; Srivastava ( 1990 ) utilizing information of two subsidiary variables is:

( 1.2.37 )

The values of and for which the mean square mistake of is minimal are and severally. The average square mistake of can be written as:

( 1.2.38 )

Roy ( 2003 ) suggested the undermentioned general arrested development calculator for two stage trying when information on Z is available:

. ( 1.2.39 )

The optimal values of unknown invariables are

, and

and the look for average square mistake is:

. ( 1.2.40 )

H. P. Singh, Upadhyaya, & A ; Chandra ( 2004 ) proposed following generalized calculator when information on subsidiary variable Z is available:

( 1.2.41 )

The optimal values of unknown invariables are

## ,

Mean square mistake of is:

. ( 1.2.42 )

Further, H. P. Singh, et Al. ( 2004 ) investigated that for different values of, the average per unit calculator, the usual two-phase trying ratio calculator, the usual two-phase trying merchandise calculator, S. K. Srivastava ( 1971 ) calculator, Chand ( 1975 ) ratio-type calculator, S. R. Srivastava, et Al. ( 1990 ) calculator, G. N. Singh & A ; Upadhyaya ( 1995 ) calculator, Upadhyay and Singh ( 2001 ) calculators are particular instances of their calculator.

Samiuddin & A ; Hanif ( 2007 ) has proposed different calculators by sing following state of affairs in two stage sampling:

In add-on to the sample, the population means of both subsidiary variables are known. They called it the “ Full Information Case ” .

In add-on to the sample, is given merely, ( being unknown ) . They called it the “ Partial Information Case ” .

When and are unknown, they called it the “ No Information Case ” .

The arrested development calculator suggested by Samiuddin & A ; Hanif ( 2007 ) for Full information Case is:

( 1.2.43 )

The optimal values of unknown invariables are

and

and intend square mistake of is

( 1.2.44 )

where is the partial correlativity coefficient of “ Y ” and combined effects of “ ten ” and “ omega ”

The undermentioned arrested development calculator has been suggested by Samiuddin & A ; Hanif ( 2007 ) for Partial Information Case.

. ( 1.2.45 )

The optimal values for unknown invariables are

## ,

and.

The average square mistake of ( 1.2.45 ) is:

. ( 1.2.46 )

Samiuddin & A ; Hanif ( 2007 ) proposed following arrested development calculator for No Information Case

( 1.2.47 )

The optimal values of unknown invariables are

and

The minimal average square mistake is:

. ( 1.2.48 )

The ratio calculator suggested by Samiuddin & A ; Hanif ( 2007 ) for Full information Case is:

. ( 1.2.49 )

The optimal values of unknown invariables are

and

The average square mistake of is:

. ( 1.2.50 )

The ratio calculator suggested by Samiuddin & A ; Hanif ( 2007 ) for Partial information Case is:

, ( 1.2.51 )

The optimal values of unknown invariables are

## ,

and

The average square mistake of is:

. ( 1.2.52 )

Ratio calculator proposed by Samiuddin & A ; Hanif ( 2007 ) for No information Case is:

. ( 1.2.53 )

The optimal values of unknown invariables are

and

average square mistake of ( 1.2.53 ) is:

. ( 1.2.54 )

Z. Ahmed, Hanif, & A ; Ahmad ( 2009 ) suggested three categories of regression-cum-ratio calculators for gauging population mean of variable of involvement for two-phase trying utilizing multi-auxiliary variables for full, partial and no information instances.

The proposed calculator by Z. Ahmed, Hanif, & A ; Ahmad ( 2009 ) are: ( 1.2.55 )

Regressions-Cum-Ratio Estimator for Partial Information Case is:

( 1.2.56 )

Regressions-Cum-Ratio Estimator for No Information Case is:

( 1.2.57 )

The average square mistakes of above calculators are:

( 1.2.58 )

( 1.2.59 )

( 1.2.60 )

## 1.3 Some Popular Univariate Calculators in Multiphase Sampling based on Qualitative Forecasters

In this subdivision some calculators in multiphase sampling have been discussed which used information on subsidiary properties. The pioneering work in multiphase trying based on subsidiary properties has been the work of Naik & A ; Gupta ( 1996 ) .

The household of calculators for two-phase sampling for no information instance by Jhajj, et Al. ( 2006 ) under same regularity conditions is

, ( 1.3.1 )

where, and

The followerss are some maps ( calculators ) of ( 1.3.11 ) .

## ,

## ,

## ,

## ,

where I± is unknown changeless. Many other maps ( calculators ) may be constructed.

The average square mistake of each calculator to the footings of order of this household is,

. ( 1.3.2 )

Shabbir and Gupta ( 2007 ) proposed an calculator which utilize the property subsidiary information:

( 1.3.3 )

where and are unknown invariables.

The average square mistake of ( 1.3.13 ) to the footings of order is,

. ( 1.3.4 )

Hanif, Haq, & A ; Shahbaz ( 2009 ) proposed a generalised household of calculators based on the information of “ K ” subsidiary properties and discussed the calculator for full, partial and no information instances. Hanif, Haq, et Al. ( 2009 ) showed that the proposed household has smaller average square mistake than given by Jhajj, et al. ( 2006 ) . The proposed calculator for Partial Information Case is:

( 1.3.5 )

The average square mistake of ( 1.3.17 ) is:

( 1.3.6 )

The proposed calculator for No Information Case is:

( 1.3.7 )

The average square mistake of ( 1.3.19 ) is:

( 1.3.8 )

Hanif, Haq, & A ; Shahbaz ( 2010 ) proposed some ratio calculators for individual stage and two stage sampling by utilizing information on multiple subsidiary properties. The proposed calculators are generalisation of the calculator proposed by Naik & A ; Gupta ( 1996 ) . Hanif, et Al. ( 2010 ) besides drive the shrinking version of the proposed calculators by utilizing the method given Shahbaz & A ; Hanif ( 2009 ) . The calculator for two stage sampling is:

( 1.3.9 )

The average square mistake of ( 1.3.23 ) up to first order estimate is:

( 1.3.10 )

## 1.4 Multivariate Calculators

Hanif, Ahmed, & A ; Ahmed ( 2009 ) proposed a figure of generalised multivariate ratio calculators for two-phase and multi-phase sampling in the presence of multi-auxiliary variables for gauging population mean for a individual variable and a vector of variables of involvement. Hanif, Ahmed, et Al. ( 2009 ) proposed more general ratio calculator when information on all subsidiary variables are non available for population ( No Information Situation ) , the calculator is:

( 1.4.1 )

The variance-covariance matrix of the calculator is of the undermentioned signifier:

( 1.4.2 )

Where is covariance matrix of.

Z. Ahmed, et Al. ( 2010 ) besides following multivariate arrested development calculator by utilizing information of multiple subsidiary variables:

( 1.4.3 )

The variance-covariance matrix of the calculator is of the undermentioned signifier:

( 1.4.4 )

## 1.5 Introduction to Zellner Models

Apparently unrelated arrested development equations ( SURE ) theoretical account, proposed by Zellner ( 1962 ) , is a generalisation of a additive arrested development theoretical account that consists of several arrested development equations, each holding its ain dependant variable and potentially different sets of independent variables. Each equation is a additive arrested development theoretical account in its ain and can be estimated individually, that ‘s why the system is called apparently unrelated arrested development theoretical accounts Greene ( 2003 ) .

The theoretical account can be estimated equation by equation utilizing ordinary least squares ( OLS ) method. Such estimations are consistent, nevertheless by and large non every bit efficient as calculators obtained by SUR method, which amounts to feasible generalised least squares with a specific construction of the variance-covariance matrix. Two state of affairss when SUR is tantamount to OLS, are: either when the mistake footings are uncorrelated between the equations ( genuinely unrelated ) , or when each theoretical account contains precisely the same set of forecasters on the right-hand-side.

## 1.5.1 The SURE Model

Suppose there are thousand arrested development equations

Where represents the equation figure, and, is the observations index. The figure of observations is assumed to be big plenty, such that in the analysis we take, whereas the figure of theoretical accounts remains same.

Each equation has a individual dependant variable, and a -dimensional vector of forecasters. If we stack observations matching to the equation into -dimensional vectors and matrices, so the arrested development theoretical account can be written in vector signifier as:

where Lolo and Iµi are TA-1 vectors, Xi is a TA-ki matrix, and I?i is a kiA-1 vector.

Finally, if we stack these Ks vector equations on top of each other, the system will take signifier Zellner ( 1962 )

( 1.5.1.1 )

The theoretical account ( 1.5.1.1 ) can be jointly estimated by utilizing Feasible Generalized Least Square ( FGLS )

## 1.6 The Shrinking Calculator

Shrinking calculator is an calculator that, either explicitly or implicitly, incorporates the effects of shrinking. In simple words this means that a natural estimation is improved by uniting it with other information. One general consequence is that many standard calculators can be improved, in footings of mean squared mistake ( MSE ) , by shriveling them towards nothing. Assume that the expected value of the natural estimation is non zero and see other calculators obtained by multiplying the natural estimation by a certain parametric quantity. A value for this parametric quantity can be specified as that understating the MSE of the new estimation. For this value of the parametric quantity, the new estimation will hold a smaller MSE than the natural 1. Thus it has been improved. An consequence here may be to change over an indifferent natural estimation to an improved biased one. A well-known illustration arises in the appraisal of the population discrepancy based on a simple sample ; for a sample size of N, the usage of a factor N a?’ 1 in the usual expression gives an indifferent calculator while a factor of n + 1 gives one which has the minimal average square mistake.

## 1.6.2 General Shrinkage Estimator Shahbaz & A ; Hanif ( 2009 )

Let a population parametric quantity can be estimated by utilizing an calculator whit average square mistake. Shahbaz & A ; Hanif ( 2009 ) has defined a general shrinking calculator as where vitamin D is a changeless to be determined such that mean square mistake of is minimized.

( 1.6.2.1 )

The look for average square mistake given in ( 1.6.2.1 ) can be used to obtain the average square mistake of shrinkage version of any calculator. The calculator proposed by Searl ( 1964 ) turned out to be particular instance of shrinking calculator proposed by Shahbaz and Hanif ( 2009 ) by utilizing.

## Chapter 2: Literature Reappraisal

Neyman ( 1938 ) was the first 1 who gave the construct of two-phase sampling as:

“ A more accurate estimation of the original character may be obtained for the same entire outgo by set uping the sampling of population in two stairss. The first measure is to procure informations, for the 2nd character merely, from a comparatively big random sample of the population in order to obtain an accurate estimation of the distribution of this character.

The 2nd measure is to split this sample, as in graded trying into categories or strata harmonizing to the value of the 2nd character and to pull at random from each of the strata, a little sample for the dearly-won intensive questioning necessary to procure informations sing the first character.

An estimation of the first character based on these samples may be more accurate than based on an every bit expensive sample drawn at random without stratification. The inquiry is to find for a given outgo, the sizes of the initial sample and the subsequent samples which yield the most accurate estimation of the first character ” .

Cochran ( 1940 ) developed Ratio calculator for gauging population sum by using the subsidiary information and discussed the comparative efficiency of the calculator. The ratio calculator is an efficient calculator of population entire if there exist strong additive relationship between variable of involvement and subsidiary variable. The arrested development calculator is ever more efficient than the ratio calculator if population arrested development coefficient is used as a edifice block of the calculator. Both calculators are every bit precise if the arrested development line passes through beginning. Use of subsidiary variable are good studied in literature of study trying as discussed in the standard books on study sampling by assorted writers including Hartley & A ; Ross ( 1954 ) , Yates ( 1960 ) , Kish ( 1965 ) , Murthy ( 1967 ) , Raj ( 1968 ) , Cochran ( 1977 ) and P. V. Sukhatme, Sukhatme, Sukhatme, & A ; Ashok ( 1984 ) .

Hartley & A ; Ross ( 1954 ) developed exact ratio calculator. Rao & A ; Rao ( 1971 ) studied public presentation of the ratio calculator based on little samples. B. V. Sukhatme ( 1962 ) developed a general ratio-type calculator in two-phase sampling. Mohanty ( 1967 ) discussed that the preciseness in gauging the population mean may be increased by utilizing another aide variable which was correlated with variable of involvement. Swain ( 2000 ) constructed concatenation arrested development calculator in which the aide variable with known population mean was used to gauge the unknown population mean of another subsidiary variable say “ ten ” so this estimated mean of “ x ” was used to gauge the population mean of survey variable “ Y ” . Chand ( 1975 ) developed two concatenation ratio-type calculators by utilizing the information of two subsidiary variables for gauging finite population mean. Kiregyera ( 1980 ) constructed a concatenation ratio-to-regression type calculator by utilizing two subsidiary variables and discussed the comparative efficiency with Chand ( 1975 ) concatenation ratio-type calculator.

S. K. Srivastava ( 1970 ) suggested a general household of ratio-type calculators for gauging mean of a finite population by utilizing individual subsidiary variable. Kiregyera ( 1984 ) developed two calculators, one is ratio-in-regression and other is regression-in-regression calculator ; both use two subsidiary variables. The efficiency of calculators was investigated through empirical observation every bit good as under super-population theoretical account, both constructed calculators performed better than arrested development calculator utilizing one subsidiary variable for two-phase sampling. The regression-in-regression calculator performed better than ratio-in-regression calculator and their public presentation was better than Kiregyera ( 1980 ) calculator. Mukerjee, et Al. ( 1987 ) developed three calculators following the method of Kiregyera ( 1984 ) . Mukerjee, et Al. ( 1987 ) besides extended their consequences to the instance when multi-auxiliary information was utilized.

H. P. Singh ( 1987 ) proposed a arrested development calculator for gauging population mean in two-phase sampling by utilizing anterior cognition of correlativity coefficient between variable of involvement and subsidiary variable. H. P. Singh ( 1987 ) proposed his calculator and demonstrated that the proposed calculator is more efficient than usual arrested development calculator in two-phase sampling. Tripathi, Singh, & A ; Upadhyaya ( 1988 ) provided a general model for gauging a general map of parametric quantities with the aid of a general map of auxiliary parametric quantity, for bivariate population, discrepancy of survey variable was estimated through general consequences derived from gauging general map of parametric quantities. An asymptotically optimal subclass of the wider category was besides identified in it. H. P. Singh & A ; Namjoshi ( 1988 ) suggested a category of multivariate arrested development calculators of population mean of survey variable in two-phase sampling. H. P. Singh & A ; Namjoshi ( 1988 ) provided exact look of average square mistake and optimal calculator of the proposed category. H. P. Singh, Tripathi, & A ; Upadhyaya ( 1989 ) proposed a general category of calculators for population mean and discussed that usual ratio, arrested development and merchandise calculators in two-phase sampling may ever be improved under moderate conditions. H. P. Singh, et Al. ( 1989 ) besides provided a general status under which two-phase sampling calculators were preferred over usual indifferent calculator for individual sample for a additive cost construction.

Tripathi & A ; Khattree ( 1989 ) discussed the appraisal of agencies of several variables of involvement, utilizing multi-auxiliary variables, under simple random sampling. Further Tripathi ( 1989 ) extends the consequences to the instance of two occasions. Tripathi & A ; Chaubey ( 1993 ) have considered the job of obtaining optimal chances of choice, based on multi-auxiliary variables, in unequal chance trying for gauging the finite population mean.

S. R. Srivastava, et Al. ( 1990 ) developed a general household of concatenation ratio-type calculators for gauging population mean by utilizing two subsidiary variables.

H. P. Singh, Upadhyaya, & A ; Iachan ( 1990 ) proposed a category of calculators based on general sampling designs for population parametric quantity using subsidiary information of some other parametric quantities. They besides discussed the belongingss of the suggested category and happen the asymptotic lower edge to the average square mistake of the calculators belonging to the category. H. P. Singh, et Al. ( 1990 ) besides proposed several indifferent ratio and merchandise calculators with their looks of asymptotic discrepancies utilizing Jackknife technique in two-phase sampling.

H. P. Singh, Singh, & A ; Kushwaha ( 1992 ) suggested a category of concatenation ratio-to-regression calculators in two-phase sampling for finite population mean of variable of involvement. Optimum calculator was identified from this category. The public presentation of optimal calculator is investigated theoretically every bit good as through empirical observation.

L. N. Upadhyaya, Dubey, & A ; Singh ( 1992 ) suggested a category of ratio-in-regression calculators for population mean of the survey variable utilizing two subsidiary variables in two-phase sampling and investigated its asymptotic belongingss.

J. Sahoo & A ; Sahoo ( 1993 ) gave a general frame work of appraisal of population mean of variable of involvement by utilizing an extra aide variable for two-phase sampling when the population mean of the chief subsidiary variable was unknown. Chand ( 1975 ) and Kiregyera ( 1980, 1984 ) calculators can be seen as the particular instances of J. Sahoo & A ; Sahoo ( 1993 ) category of calculators.

H. P. Singh ( 1993 ) developed a category of concatenation ratio-cum-difference calculator for mean of a finite population utilizing two subsidiary variables with asymptotic looks for its prejudice and average square mistake in two-phase sampling. H. P. Singh ( 1993 ) besides theoretically and through empirical observation proved that the constructed category of calculator was more efficient than Chand ( 1975 ) and S. R. Srivastava, et Al. ( 1990 ) calculators.

J. Sahoo, et Al. ( 1993 ) suggested a regression-type calculator based upon the information on 2nd subsidiary variable when population mean of the chief subsidiary variable was unknown. H. P. Singh & A ; Biradar ( 1994 ) developed general category of indifferent ratio-type calculators in two stage sampling and derived look of its asymptotic discrepancy.

J. Sahoo & A ; Sahoo ( 1994 ) discussed comparative efficiency of four chain-type calculators in two-phase trying under super-population theoretical account. J. Sahoo, Sahoo, & A ; Mohanty ( 1994a ) provided a arrested development attack for appraisal utilizing two subsidiary variables for two-phase sampling. J. Sahoo, Sahoo, & A ; Mohanty ( 1994b ) considered an alternate attack for gauging mean in two-phase trying utilizing two subsidiary variables. V. K. Singh & A ; Singh ( 1994 ) proposed a category of calculators for gauging ratio and merchandise of agencies of two finite populations in two-phase sampling. V. K. Singh & A ; Singh ( 1994 ) obtained the asymptotic look for prejudice and average square mistake.

G. N. Singh & A ; Upadhyaya ( 1995 ) developed a generalised calculator for gauging the population mean in two-phase trying utilizing two-auxiliary variables. H. P. Singh & A ; Gangele ( 1995 ) suggested an calculator utilizing information of coefficient of fluctuation and information on two-auxiliary variables for population mean in two stage trying. Their proposed calculator was efficient than Chand ( 1975 ) , Chand ( 1975 ; Kiregyera ( 1980, 1984 ) and J. Sahoo & A ; Sahoo ( 1993 ) calculators.

H. P. Singh, Katyar, & A ; Gangwar ( 1996 ) discussed a category of about indifferent arrested development type calculators in two-phase sampling by utilizing Quenouille ( 1956 ) and Jack-Knife technique. Naik & A ; Gupta ( 1996 ) proposed ratio, merchandise and arrested development calculators for the population mean when subsidiary property information is available.

Hidiroglou & A ; Sarndal ( 1998 ) discussed that two-phase sampling is cost effectual and preciseness of ratio and arrested development estimations under two-phase sampling additions if there exist high correlativity between the subsidiary variable and variable under survey.

M. S. Ahmed ( 1998 ) interpreted the arrested development coefficients right for the calculators suggested by Mukerjee, et al. ( 1987 ) . M. S. Ahmed ( 1998 ) mentioned that the corrected mean square mistakes of Kiregyera ( 1984 ) calculators are computed presuming that the arrested development coefficient and are ordinary non partial arrested development coefficient. Furthermore he proved that Kiregyera ( 1984 ) calculators were better than Mukerjee, et Al. ( 1987 ) calculators and besides showed that calculator suggested by Tripathi & A ; Ahmed ( 1995 ) was more efficient than Kiregyera ( 1984 ) calculators. H. P. Singh & A ; Gangele ( 1999 ) suggested about indifferent ratio-type and product-type calculators for population mean in two-phase sampling. The public presentation of suggested calculators was through empirical observation evaluated. Tracy & A ; Singh ( 1999 ) proposed a category of concatenation arrested development calculators with asymptotic look of prejudice and mean squared mistake for gauging the population mean of variable of involvement in two-phase sampling by utilizing two-auxiliary variables.

Tracy & A ; Singh ( 1999 ) besides derived asymptotic optimum indifferent ratio-type calculator with its discrepancy in two-phase sampling and besides in consecutive sampling with the usage of two subsidiary variables. Tracy & A ; Singh ( 1999 ) proved that proposed calculator is better than Olkin ( 1958 ) and Sen ( 1971 ) calculators.

J. Sahoo & A ; Sahoo ( 1999a ) developed a category of calculators by utilizing two stage sampling and J. Sahoo & A ; Sahoo ( 1999b ) conducted a comparative survey of the calculators considered by Chand ( 1975 ) , Kiregyera ( 1980, 1984 ) , Mukerjee, et Al. ( 1987 ) , J. Sahoo, et Al. ( 1993 ) and J. Sahoo, et Al. ( 1994a ) under the ace population theoretical account utilizing two subsidiary variables.

H. P. Singh & A ; Tailor ( 2000 ) suggested some ratio-type calculators of population mean of survey variable utilizing two subsidiary variables in two-phase sampling with coefficient of fluctuation of the 2nd subsidiary variable was known. H. P. Singh & A ; Tailor ( 2000 ) obtained the conditions in which proposed calculators were more efficient than usual two-phase trying ratio-estimator, Chand ( 1975 ) calculator, and G. N. Singh & A ; Upadhyaya ( 1995 ) .

A. K. Singh, Singh, & A ; Upadhyaya ( 2001 ) proposed two categories of concatenation ratio-type calculators and besides derived looks of prejudice and average square mistakes in two-phase sampling by utilizing two-auxiliary variables. L. N. Sahoo & A ; Sahoo ( 2001 ) proposed calculators of finite population mean by utilizing prognostic attack in two-phase trying utilizing two subsidiary variables. A. K. Singh, et Al. ( 2001 ) considered a generalised concatenation calculator for finite population mean utilizing two subsidiary variables in two stage sampling.

Radhey, Singh, & A ; Singh ( 2002 ) provided a modified ratio calculator with approximative looks for its prejudice and average square mistake in two-phase sampling for population mean of variable of involvement by utilizing two-auxiliary variables. Radhey, et Al. ( 2002 ) investigated through empirical observation that asymptotic optimal calculators performed better than conventional indifferent ratio, traditional ratio, Chand ( 1975 ) , Kiregyera ( 1980 ) and L. Upadhyaya, Kushwaha, & A ; Singh ( 1990 ) calculators. H. P. Singh & A ; Singh ( 2002 ) estimated the population coefficient of fluctuation of survey variable with concatenation ratio-type calculator utilizing two subsidiary variables in two-phase sampling and besides derived looks for the prejudice and mean squared mistake.

Chandra & A ; Singh ( 2003 ) discussed a category of indifferent calculators with its belongingss for the population mean of survey variable in two-phase trying utilizing two-auxiliary variables when information for the mean of chief subsidiary variable was non available. The indifferent calculators suggested by Chand ( 1975 ) and Dalbehera & A ; Sahoo ( 2000 ) found to be the particular instances of proposed category. Diana & A ; Tommasi ( 2003 ) proposed a general category of calculators for finite population mean in two-phase sampling. Diana & A ; Tommasi ( 2003 ) category of calculators was based on the sample agencies and discrepancies of two subsidiary variables. Diana & A ; Tommasi ( 2003 ) besides provide the minimal discrepancy edge for any member of the category.

H. P. Singh & A ; Espejo ( 2003 ) proposed a category of ratio-product calculators for gauging a finite population mean in two-phase sampling and identified an asymptotically optimal calculator in their category along with its approximative mean-square mistake by utilizing the anterior cognition of the parametric quantity. H. P. Singh & A ; Espejo ( 2003 ) besides found that the calculators are every bit efficient for known value of C every bit good as for consistent calculator of C.

R. Singh & A ; Singh ( 2003 ) proposed a regression-type calculator in

two-phase trying for population mean when information on 2nd variable was known and discrepancy of chief subsidiary variable was non known. The proposed calculator was more efficient than Chand ( 1975 ) , Kiregyera ( 1980, 1984 ) and usual ratio, arrested development calculators.

Roy ( 2003 ) constructed a regression-type calculator of population mean of the chief variable in the presence of available information on 2nd subsidiary variable, when the population mean of the first subsidiary variable was non known. Roy ( 2003 ) calculator was more efficient than Mohanty ( 1967 ) , Chand ( 1975 ) , Kiregyera ( 1980, 1984 ) and J. Sahoo, et Al. ( 1993 )

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M. S. Ahmed ( 2003 ) proposed concatenation based general calculators for finite population mean utilizing multivariate subsidiary information under multiphase sampling. M. S. Ahmed ( 2003 ) considered a figure of subsidiary variables in each stage under a general sampling design and studied the belongingss of these calculators and presented the consequences for simple random trying without replacing strategies. M. S. Ahmed ( 2003 ) besides derived the optimal sample sizes utilizing a modified cost.

H. P. Singh, et Al. ( 2004 ) proposed a household of calculators, which is more efficient than those considered by S. K. Srivastava ( 1970 ) , Chand ( 1975 ) , S. R. Srivastava, et Al. ( 1990 ) , H. P. Singh & A ; Biradar ( 1994 ) , H. P. Singh & A ; Gangele ( 1995 ) and A. K. Singh, et al. ( 2001 ) . H. P. Singh & A ; Vishwakarma ( 2005-2006 ) suggested a modified version of Sahai ( 1979 ) calculator in two-phase sampling and discussed its belongingss. L. N. Upadhyaya, Singh, & A ; Tailor ( 2006 ) proposed a household of concatenation ratio-type calculators for population mean by using information of mean for first subsidiary variable and coefficient of fluctuation for 2nd subsidiary variable.

H. P. Singh, Singh, & A ; Kim ( 2006 ) considered concatenation ratio and arrested development type calculators of average and provided looks for its discrepancy. The optimal sample sizes were besides obtained for first stage and 2nd stage utilizing fixed cost of the study. Comparison was made with calculators suggested by A. K. Singh & A ; Singh ( 2001 ) . Jhajj, et Al. ( 2006 ) has proposed a household of calculators in individual and two stage trying utilizing information on a individual subsidiary properties, the proposed household is based upon a general map. Shabbir & A ; Gupta ( 2007 ) have besides proposed an calculator for population mean in individual stage trying utilizing information of individual subsidiary property.

H. P. Singh & A ; Espejo ( 2007 ) suggested a category of ratio-product calculators in two-phase sampling for population mean in the presence of two-auxiliary variables and besides discussed their belongingss. H. P. Singh & A ; Espejo ( 2007 ) besides identified asymptotically optimal calculators with their discrepancies and compared their efficiency with two-phase ratio, merchandise and average per unit calculator under some conditions. Shabbir & A ; Gupta ( 2007 ) have besides proposed an calculator for population mean in individual stage trying utilizing information of individual subsidiary property.

Samiuddin & A ; Hanif ( 2007 ) introduced ratio and arrested development appraisal processs for gauging population mean in two-phase sampling for different three state of affairss depending upon the handiness of information on two subsidiary variables for population. Samiuddin & A ; Hanif ( 2007 ) considered three state of affairss, foremost when information on both subsidiary variables was non available, 2nd when information on one subsidiary variable was available and 3rd, when information was available on both subsidiary variables. Samiuddin & A ; Hanif ( 2007 ) calculators developed in 2nd state of affairs were found to be every bit efficient as H. P. Singh, et Al. ( 2004 ) and Roy ( 2003 ) . But the calculators developed in 3rd state of affairs were more efficient so H. P. Singh, et Al. ( 2004 ) and Roy ( 2003 ) every bit good as their ain calculators developed in first two state of affairss.

Z. Ahmed, et Al. ( 2009 ) proposed generalized regression-cum-ratio calculators for two-phase trying utilizing multi-auxiliary variables. Z. Ahmed, et Al. ( 2009 ) suggested three categories of regression-cum-ratio calculators for gauging population mean of variable of involvement for two-phase sampling based on multi-auxiliary variables for full information, partial information and no information instances. Hanif, Ahmed, et Al. ( 2009 ) proposed a figure of generalised multivariate ratio calculators for two-phase and multi-phase sampling in the utilizing multi-auxiliary variables for gauging population mean for a individual variable and a vector of variables of involvement ( s ) . Hanif, Ahmed, et Al. ( 2009 ) besides made theoretical and empirical to look into the efficiencies of the calculators.

Hanif, Haq, et Al. ( 2009 ) proposed general household of calculators and derived general look of average square mistake of calculators proposed by Jhajj, et al. ( 2006 ) . The household has been proposed for single-phase sampling in instance of full information and for two-phase sampling in instance of partial and no information instances. Hanif, Haq, et Al. ( 2009 ) discussed that the proposed household has smaller average square mistake than given by Jhajj, et Al. ( 2006 ) .

Z. Ahmed, et Al. ( 2010 ) suggested a figure of generalised multivariate arrested development calculators for two-phase and multi-phase sampling in the presence of multi-auxiliary variables for gauging population mean for a individual variable and a vector of variables. Hanif, et Al. ( 2010 ) proposed some ratio calculators for individual stage and two stage trying utilizing information on multiple subsidiary properties. The proposed calculators are generalisation of the calculator proposed by Naik & A ; Gupta ( 1996 ) . Hanif, et Al. ( 2010 ) proposed some ratio calculators for individual stage and two stage sampling by utilizing information on multiple subsidiary properties. The proposed calculators are generalisation of the calculator proposed by Naik & A ; Gupta ( 1996 ) . Hanif, et Al. ( 2010 ) besides drive the shrinking version of the proposed calculators by utilizing the method given Shahbaz & A ; Hanif ( 2009 ) .