History Of Sampling Has Been Discussed Biology Essay

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Estimation problem has always been vital in all the domains of life. Effective planning depends upon preciseness of the estimates that's why researchers are always in process of developing methods that can produce more precise estimates. Several methods are available in literature that can be used for efficient estimation of the characteristic under study, these methods are collectively called sampling methods. The scientific development in the field of survey sampling has long history but the groundbreaking work in this field was done by Neyman(1934). The work of Neyman(1934) guided the number of statisticians for significant development in various areas of survey sampling. The historical work done by Hansen & Hurwitz(1943) and by Horvitz & Thompson(1952) in the development of unequal probability sampling is also based upon the ideas given by Neyman(1934).

History of sampling has been discussed by many survey statisticians, some notable references are Chang(1976), Dalenius(1962), Duncan & Shelton(1978), Hansen(1987), Kruskal & Mosteller(1980), Seng(1951) and Stephan(1948). {Kiaer, 1895 #164} in a meeting of International Statistical Institute (ISI) put forward the idea that a partial investigation could provide useful information. Detailed discussions of {Kiaer, 1895 #164} work and its impact on sampling methodology may be found in Seng(1951) and Kruskal & Mosteller(1980). The initial reaction to {Kiaer, 1895 #164} work was negative and generally not receptive; however in 1901 and 1903 Kiaer was supported by C. D. Wright and later by A. L. Bowley. Kiaer(1897) mentions the possibility of randomization, in his words a sample 'selected through the drawing of lots', but does not develop the idea further in his writings.

Like Kiaer, Bowley actively promoted his ideas on sampling and randomization specially. Bowley(1906) paper containing an empirical verification to simple random sampling, at this point Bowley has probably equated random sampling to any sampling scheme in which the inclusion probabilities are the same for every sampling units. Bowley(1913) used a systematic sample of buildings of "Reading" from street listing in the local directory of residential buildings. Bowley(1913) also checked the representativeness of his samples by comparing his sample results to known population counts. For two cases in which Bowley(1913) found a discrepancy between his sample and official statistics, on further checking it was discovered that the official statistics contained error. This work is discussed in details by Seng(1951) and Kruskal & Mosteller(1980). Also Bowley(1926) provided a theoretical monograph summarizing the known results in random and purposive selection.

The work of Neyman(1934) paper has been recognized as an important contribution to the field of survey sampling. Kruskal & Mosteller(1980) have discussed the work as "the Neyman watershed" and Hansen, Dalenius, & Tepping(1985) have commented that the "paper played a paramount role in promoting theoretical research, developments, and application of what is now known as probability sampling". The work of Neyman(1934) is considered as a classic work on two grounds; Firstly Neyman was able to provide valid reasons, both theoretically and with practical examples that why randomization gave a much more reasonable solution than purposive selection. Secondly, the paper provides a paradigm in the history of sampling is that the theory of point and interval estimation is provided under randomization. Neyman(1938) introduced the use of cost function into survey sampling in connection with two-phase sampling.

In early 1940's Hansen & Hurwitz(1943) made some fundamental contribution to theory of sampling, they took an important step forward by extending the idea of sampling with unequal inclusion probabilities for units in different strata. This allowed the development of very complex multi-stage designs that are the backbone of large scale social and economic survey research.

Introduction to Multiphase Sampling

Information related to the variable of interest is termed as auxiliary information, which can be utilized to improve the efficiency of the estimators. In Multiphase sampling certain items of information are drawn from the whole sampling units and certain other items of information are taken from the subsample. Multiphase sampling is used when it is expensive to collect data on the variable of interest but it is relatively inexpensive to collect data on variables that are correlated with the variables of interest. For example, in forest surveys, it is difficult to travel to remote area to make on ground determination. However, aerial photographs of forest are relatively inexpensive, which can be used to decide about forest type; a strongly correlated variable with ground determination.

1.1.1 Notation of Multiphase Sampling

Let a population of N units is designated as , is the value of variable of interest associated with , and be information of auxiliary variables associated with . Let , and be population mean of X, W and Y respectively. Further let , and are corresponding variances. Also and are population correlation coefficients between X & Y, X & W and Y & W respectively. Let a first phase sample of units is drawn from the population and information on auxiliary variables is recoded; further let a sub-sample of units is drawn from the first phase sample and information of auxiliary variables alongside variable of interest.

The sample means of auxiliary variables based on units are denoted byand etc and sample means of second phase are denoted by and . We will also use and such that . For notational purpose it will be assumed that the mean of estimand and auxiliary variables can be approximated from their population means so that where is sample mean of auxiliary variable X at h-th phase; h = 1 and 2. Similar notation will be used for other quantitative auxiliary variables. For qualitative auxiliary variables we will use will be used and for variable of interest we will use the notation with usual assumptions.

Following expectations for deriving the mean square error of estimators which are based upon quantitative auxiliary variables will be used:

(1.1.1.1)

In case of several auxiliary variables; say q; the sample mean of i-th auxiliary at h-th phase will be denoted by . The vector notations in case of multiple auxiliary variables and sample mean vector of auxiliary variables at h-th phase will be denoted by with relation . Following additional expectations are also useful:

Similar expectations for qualitative auxiliary variables are:

(1.1.1.2)

For multivariate and Zelner estimator we will use following notations:

(1.1.1.3)

where is covariance matrix of variables of interest, the notation of will be used for new estimators under two phase sampling.

1.2 Some Popular Univarate Estimators in Multiphase Sampling based on Quantitative Predictors

In this section some well-known ratio and regression estimators for estimating population mean along with their mean square errors for two-phase sampling using one and two quantitative auxiliary variables are discussed.

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

(1.2.1)

and its mean square error is

, (1.2.2)

where is coefficient of variation of x and is correlation coefficient between x and y.

Cochran(1977) suggested following simple regression estimator for unknown as under:

, (1.2.3)

where is based on second phase sample and expression for mean square error is

. (1.2.4)

Another simple regression estimator of ; suggested by Cochran(1977) when is known:

, (1.2.5)

With mean square error:

. (1.2.6)

It can be immediately seen that .

Mohanty(1967) suggested the following regression-cum-ratio estimator by combining the regression and ratio method when information on is not available:

, (1.2.7)

where is calculated from second phase sample and the expression for mean square error of

. (1.2.8)

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

, (1.2.9)

where is calculated from second phase sample and expression for mean square error is

. (1.2.10)

The chain ratio-type estimator proposed by Chand(1975) for two-phase sampling using two-auxiliary variables; when population mean is known:

(1.2.11)

and its mean square error is:

. (1.2.12)

Another chain ratio-type estimator suggested by Chand(1975) when information on auxiliary variable Z is unavailable for population is

(1.2.13)

and its mean square error is

. (1.2.14)

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

(1.2.15)

where is calculated from the first phase sample. The mean square error of can be written as:

. (1.2.16)

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

(1.2.17)

where computed from second phase sample. The mean square error of (1.2.17) can be written as:

. (1.2.18)

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

(1.2.19)

where is based on second phase sample while is based on first phase sample and mean square error of is:

. (1.2.20)

Following the construction of regression-in-regression estimator by Kiregyera(1984), Mukerjee, Rao, & Vijayan(1987) developed the estimator when information on both auxiliary variables is unavailable . The regression estimator using two auxiliary variables is:

(1.2.21)

where and are based on second phase sample. The mean square error of (1.2.21) can be written as:

. (1.2.22)

Mukerjee, et al.(1987) also proposed following estimator when population information on auxiliary variable Z is available:

(1.2.23)

where and are based on second phase sample. The mean square error :

. (1.2.24)

Mukerjee, et al.(1987) developed the third estimator when information on auxiliary variable Z is available for population as:

(1.2.25)

where and are based on second phase sample while is based on first phase sample. The mean square error for this estimator can be written as:

(1.2.26)

J. Sahoo, Sahoo, & Mohanty(1993) suggested the following estimator

(1.2.27)

where and are based on second phase sample. The mean square error for this estimator can be written as:

. (1.2.28)

J. Sahoo & Sahoo(1994) proposed three regression type estimators using information of two auxiliary variables. The first estimator proposed by J. Sahoo & Sahoo(1994) when information on auxiliary variable "z" is available for population is

(1.2.29)

where and are based on second phase sample. The mean square error for this estimator is:

. (1.2.30)

The second estimator proposed by J. Sahoo & Sahoo(1994) when information on auxiliary variable "z" is available for population is:

(1.2.31)

where and are based on second phase sample while is based on the first phase sample. The mean square error for this estimator is:

(1.2.32)

Third estimator proposed by J. Sahoo & Sahoo(1994) when information on auxiliary Z is available for population, is:

(1.2.33)

where and are based on the second phase sample while is based on the first phase sample. The mean square error for this estimator is:

(1.2.34)

S. K. Srivastava(1970) suggested the following general ratio estimator using single auxiliary variable as:

(1.2.35)

where is unknown constant and the value of for which the mean square error of is minimum is and the mean square error of is

(1.2.36)

Another general ratio estimator suggested by S. R. Srivastava, Khare, & Srivastava(1990) using information of two auxiliary variables is:

(1.2.37)

The values of and for which the mean square error of is minimum are and respectively. The mean square error of can be written as:

(1.2.38)

Roy(2003) suggested the following general regression estimator for two phase sampling when information on Z is available:

. (1.2.39)

The optimum values of unknown constants are

, and

and the expression for mean square error is:

. (1.2.40)

H. P. Singh, Upadhyaya, & Chandra(2004) proposed following generalized estimator when information on auxiliary variable Z is available:

(1.2.41)

The optimum values of unknown constants are

,

Mean square error of is:

. (1.2.42)

Further, H. P. Singh, et al.(2004) investigated that for different values of, the mean per unit estimator, the usual two-phase sampling ratio estimator, the usual two-phase sampling product estimator, S. K. Srivastava(1971) estimator, Chand(1975) ratio-type estimator, S. R. Srivastava, et al.(1990) estimator, G. N. Singh & Upadhyaya(1995) estimator, Upadhyay and Singh (2001) estimators are special cases of their estimator.

Samiuddin & Hanif(2007) has proposed different estimators by considering following situation in two phase sampling:

In addition to the sample, the population means of both auxiliary variables are known. They called it the "Full Information Case".

In addition to the sample, is given only, ( being unknown). They called it the "Partial Information Case".

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

The regression estimator suggested by Samiuddin & Hanif(2007) for Full information Case is:

(1.2.43)

The optimum values of unknown constants are

and

and mean square error of is

(1.2.44)

where is the partial correlation coefficient of "y" and combined effects of "x" and "z"

The following regression estimator has been suggested by Samiuddin & Hanif(2007) for Partial Information Case.

. (1.2.45)

The optimum values for unknown constants are

,

and .

The mean square error of (1.2.45) is:

. (1.2.46)

Samiuddin & Hanif(2007) proposed following regression estimator for No Information Case

(1.2.47)

The optimum values of unknown constants are

and

The minimum mean square error is:

. (1.2.48)

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

. (1.2.49)

The optimum values of unknown constants are

and

The mean square error of is:

. (1.2.50)

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

, (1.2.51)

The optimum values of unknown constants are

,

and

The mean square error of is:

. (1.2.52)

Ratio estimator proposed by Samiuddin & Hanif(2007) for No information Case is:

. (1.2.53)

The optimum values of unknown constants are

and

mean square error of (1.2.53) is:

. (1.2.54)

Z. Ahmed, Hanif, & Ahmad(2009) suggested three classes of regression-cum-ratio estimators for estimating population mean of variable of interest for two-phase sampling using multi-auxiliary variables for full, partial and no information cases.

The proposed estimator by Z. Ahmed, Hanif, & 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 mean square errors of above estimators are:

(1.2.58)

(1.2.59)

(1.2.60)

1.3 Some Popular Univariate Estimators in Multiphase Sampling based on Qualitative Predictors

In this section some estimators in multiphase sampling have been discussed which used information on auxiliary attributes. The pioneering work in multiphase sampling based on auxiliary attributes has been the work of Naik & Gupta(1996).

The family of estimators for two-phase sampling for no information case by Jhajj, et al.(2006) under same regularity conditions is

, (1.3.1)

where , and

The followings are some functions (estimators) of (1.3.11).

,

,

,

,

where α is unknown constant. Many other functions (estimators) may be constructed.

The mean square error of each estimator to the terms of order of this family is,

. (1.3.2)

Shabbir and Gupta (2007) proposed an estimator which utilize the attribute auxiliary information:

(1.3.3)

where and are unknown constants.

The mean square error of (1.3.13) to the terms of order is,

. (1.3.4)

Hanif, Haq, & Shahbaz(2009) proposed a generalized family of estimators based on the information of "k" auxiliary attributes and discussed the estimator for full, partial and no information cases. Hanif, Haq, et al.(2009) showed that the proposed family has smaller mean square error than given by Jhajj, et al.(2006). The proposed estimator for Partial Information Case is:

(1.3.5)

The mean square error of (1.3.17) is:

(1.3.6)

The proposed estimator for No Information Case is:

(1.3.7)

The mean square error of (1.3.19) is:

(1.3.8)

Hanif, Haq, & Shahbaz(2010) proposed some ratio estimators for single phase and two phase sampling by using information on multiple auxiliary attributes. The proposed estimators are generalization of the estimator proposed by Naik & Gupta(1996). Hanif, et al.(2010) also drive the shrinkage version of the proposed estimators by using the method given Shahbaz & Hanif(2009). The estimator for two phase sampling is:

(1.3.9)

The mean square error of (1.3.23) up to first order approximation is:

(1.3.10)

1.4 Multivariate Estimators

Hanif, Ahmed, & Ahmed(2009) proposed a number of generalized multivariate ratio estimators for two-phase and multi-phase sampling in the presence of multi-auxiliary variables for estimating population mean for a single variable and a vector of variables of interest. Hanif, Ahmed, et al.(2009) proposed more general ratio estimator when information on all auxiliary variables are not available for population (No Information Situation), the estimator is:

(1.4.1)

The variance-covariance matrix of the estimator is of the following form:

(1.4.2)

Where is covariance matrix of .

Z. Ahmed, et al.(2010) also following multivariate regression estimator by using information of multiple auxiliary variables:

(1.4.3)

The variance-covariance matrix of the estimator is of the following form:

(1.4.4)

1.5 Introduction to Zellner Models

Seemingly unrelated regression equations (SURE) model, proposed by Zellner(1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially different sets of independent variables. Each equation is a linear regression model in its own and can be estimated separately, that's why the system is called seemingly unrelated regression models Greene(2003).

The model can be estimated equation by equation using ordinary least squares (OLS) method. Such estimates are consistent, however generally not as efficient as estimators obtained by SUR method, which amounts to feasible generalized least squares with a specific structure of the variance-covariance matrix. Two situations when SUR is equivalent to OLS, are: either when the error terms are uncorrelated between the equations (truly unrelated), or when each model contains exactly the same set of predictors on the right-hand-side.

1.5.1 The SURE Model

Suppose there are k regression equations

Where represents the equation number, and , is the observations index. The number of observations is assumed to be large enough, such that in the analysis we take , whereas the number of models remains same.

Each equation has a single dependent variable , and a -dimensional vector of predictors . If we stack observations corresponding to the equation into -dimensional vectors and matrices, then the regression model can be written in vector form as:

where yi and εi are TÃ-1 vectors, Xi is a TÃ-ki matrix, and βi is a kiÃ-1 vector.

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

(1.5.1.1)

The model (1.5.1.1) can be collectively estimated by using Feasible Generalized Least Square(FGLS)

1.6 The Shrinkage Estimator

Shrinkage estimator is an estimator that, either explicitly or implicitly, incorporates the effects of shrinkage. In simple words this means that a raw estimate is improved by combining it with other information. One general result is that many standard estimators can be improved, in terms of mean squared error (MSE), by shrinking them towards zero. Assume that the expected value of the raw estimate is not zero and consider other estimators obtained by multiplying the raw estimate by a certain parameter. A value for this parameter can be specified as that minimising the MSE of the new estimate. For this value of the parameter, the new estimate will have a smaller MSE than the raw one. Thus it has been improved. An effect here may be to convert an unbiased raw estimate to an improved biased one. A well-known example arises in the estimation of the population variance based on a simple sample; for a sample size of n, the use of a divisor n − 1 in the usual formula gives an unbiased estimator while a divisor of n + 1 gives one which has the minimum mean square error.

1.6.2 General Shrinkage Estimator Shahbaz & Hanif(2009)

Let a population parameter can be estimated by using an estimator whit mean square error . Shahbaz & Hanif(2009) has defined a general shrinkage estimator as where d is a constant to be determined such that mean square error of is minimized.

(1.6.2.1)

The expression for mean square error given in (1.6.2.1) can be used to obtain the mean square error of shrinkage version of any estimator. The estimator proposed by Searl (1964) turned out to be special case of shrinkage estimator proposed by Shahbaz and Hanif (2009) by using .

Chapter 2: Literature Review

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

"A more accurate estimate of the original character may be obtained for the same total expenditure by arranging the sampling of population in two steps. The first step is to secure data, for the second character only, from a relatively large random sample of the population in order to obtain an accurate estimate of the distribution of this character.

The second step is to divide this sample, as in stratified sampling into classes or strata according to the value of the second character and to draw at random from each of the strata, a small sample for the costly intensive interviewing necessary to secure data regarding the first character.

An estimate of the first character based on these samples may be more accurate than based on an equally expensive sample drawn at random without stratification. The question is to determine for a given expenditure, the sizes of the initial sample and the subsequent samples which yield the most accurate estimate of the first character".

Cochran(1940) developed Ratio estimator for estimating population total by utilizing the auxiliary information and discussed the relative efficiency of the estimator. The ratio estimator is an efficient estimator of population total if there exist strong linear relationship between variable of interest and auxiliary variable. The regression estimator is always more efficient than the ratio estimator if population regression coefficient is used as a building block of the estimator. Both estimators are equally precise if the regression line passes through origin. Use of auxiliary variable are well studied in literature of survey sampling as discussed in the standard books on survey sampling by various authors including Hartley & Ross(1954) , Yates(1960), Kish(1965), Murthy(1967), Raj(1968), Cochran(1977) and P. V. Sukhatme, Sukhatme, Sukhatme, & Ashok(1984).

Hartley & Ross(1954) developed exact ratio estimator. Rao & Rao(1971) studied performance of the ratio estimator based on small samples. B. V. Sukhatme(1962) developed a general ratio-type estimator in two-phase sampling. Mohanty(1967) discussed that the precision in estimating the population mean may be increased by using another auxiliary variable which was correlated with variable of interest. Swain(2000) constructed chain regression estimator in which the auxiliary variable with known population mean was used to estimate the unknown population mean of another auxiliary variable say "x" then this estimated mean of "x" was used to estimate the population mean of study variable "y". Chand(1975) developed two chain ratio-type estimators by using the information of two auxiliary variables for estimating finite population mean. Kiregyera(1980) constructed a chain ratio-to-regression type estimator by using two auxiliary variables and discussed the relative efficiency with Chand(1975) chain ratio-type estimator.

S. K. Srivastava(1970) suggested a general family of ratio-type estimators for estimating mean of a finite population by using single auxiliary variable. Kiregyera(1984) developed two estimators, one is ratio-in-regression and other is regression-in-regression estimator; both use two auxiliary variables. The efficiency of estimators was investigated empirically as well as under super-population model, both constructed estimators performed better than regression estimator using one auxiliary variable for two-phase sampling. The regression-in-regression estimator performed better than ratio-in-regression estimator and their performance was better than Kiregyera(1980) estimator. Mukerjee, et al.(1987) developed three estimators following the method of Kiregyera(1984). Mukerjee, et al.(1987) also extended their results to the case when multi-auxiliary information was utilized.

H. P. Singh(1987) proposed a regression estimator for estimating population mean in two-phase sampling by using prior knowledge of correlation coefficient between variable of interest and auxiliary variable. H. P. Singh(1987) proposed his estimator and demonstrated that the proposed estimator is more efficient than usual regression estimator in two-phase sampling. Tripathi, Singh, & Upadhyaya(1988) provided a general framework for estimating a general function of parameters with the help of a general function of supplementary parameter, for bivariate population, variance of study variable was estimated through general results derived from estimating general function of parameters. An asymptotically optimum subclass of the wider class was also identified in it. H. P. Singh & Namjoshi(1988) suggested a class of multivariate regression estimators of population mean of study variable in two-phase sampling. H. P. Singh & Namjoshi(1988) provided exact expression of mean square error and optimum estimator of the proposed class. H. P. Singh, Tripathi, & Upadhyaya(1989) proposed a general class of estimators for population mean and discussed that usual ratio, regression and product estimators in two-phase sampling may always be improved under moderate conditions. H. P. Singh, et al.(1989) also provided a general condition under which two-phase sampling estimators were preferable over usual unbiased estimator for single sample for a linear cost structure.

Tripathi & Khattree(1989) discussed the estimation of means of several variables of interest, using multi-auxiliary variables, under simple random sampling. Further Tripathi(1989) extends the results to the case of two occasions. Tripathi & Chaubey(1993) have considered the problem of obtaining optimum probabilities of selection, based on multi-auxiliary variables, in unequal probability sampling for estimating the finite population mean.

S. R. Srivastava, et al.(1990) developed a general family of chain ratio-type estimators for estimating population mean by using two auxiliary variables.

H. P. Singh, Upadhyaya, & Iachan(1990) proposed a class of estimators based on general sampling designs for population parameter utilizing auxiliary information of some other parameters. They also discussed the properties of the suggested class and find the asymptotic lower bound to the mean square error of the estimators belonging to the class. H. P. Singh, et al.(1990) also proposed several unbiased ratio and product estimators with their expressions of asymptotic variances using Jackknife technique in two-phase sampling.

H. P. Singh, Singh, & Kushwaha(1992) suggested a class of chain ratio-to-regression estimators in two-phase sampling for finite population mean of variable of interest. Optimum estimator was identified from this class. The performance of optimum estimator is investigated theoretically as well as empirically.

L. N. Upadhyaya, Dubey, & Singh(1992) suggested a class of ratio-in-regression estimators for population mean of the study variable using two auxiliary variables in two-phase sampling and investigated its asymptotic properties.

J. Sahoo & Sahoo(1993) gave a general frame work of estimation of population mean of variable of interest by using an additional auxiliary variable for two-phase sampling when the population mean of the main auxiliary variable was unknown. Chand(1975) and Kiregyera(1980, 1984) estimators can be seen as the special cases of J. Sahoo & Sahoo(1993) class of estimators.

H. P. Singh(1993) developed a class of chain ratio-cum-difference estimator for mean of a finite population using two auxiliary variables with asymptotic expressions for its bias and mean square error in two-phase sampling. H. P. Singh(1993) also theoretically and empirically proved that the constructed class of estimator was more efficient than Chand(1975) and S. R. Srivastava, et al.(1990) estimators.

J. Sahoo, et al.(1993) suggested a regression-type estimator based upon the information on second auxiliary variable when population mean of the main auxiliary variable was unknown. H. P. Singh & Biradar(1994) developed general class of unbiased ratio-type estimators in two phase sampling and derived expression of its asymptotic variance.

J. Sahoo & Sahoo(1994) discussed relative efficiency of four chain-type estimators in two-phase sampling under super-population model. J. Sahoo, Sahoo, & Mohanty(1994a) provided a regression approach for estimation using two auxiliary variables for two-phase sampling. J. Sahoo, Sahoo, & Mohanty(1994b) considered an alternative approach for estimating mean in two-phase sampling using two auxiliary variables. V. K. Singh & Singh(1994) proposed a class of estimators for estimating ratio and product of means of two finite populations in two-phase sampling. V. K. Singh & Singh(1994) obtained the asymptotic expression for bias and mean square error.

G. N. Singh & Upadhyaya(1995) developed a generalized estimator for estimating the population mean in two-phase sampling using two-auxiliary variables. H. P. Singh & Gangele(1995) suggested an estimator using information of coefficient of variation and information on two-auxiliary variables for population mean in two phase sampling. Their proposed estimator was efficient than Chand(1975), Chand(1975; Kiregyera(1980, 1984) and J. Sahoo & Sahoo(1993) estimators.

H. P. Singh, Katyar, & Gangwar(1996) discussed a class of almost unbiased regression type estimators in two-phase sampling by using Quenouille(1956) and Jack-Knife technique. Naik & Gupta(1996) proposed ratio, product and regression estimators for the population mean when auxiliary attribute information is available.

Hidiroglou & Särndal(1998) discussed that two-phase sampling is cost effective and precision of ratio and regression estimates under two-phase sampling increases if there exist high correlation between the auxiliary variable and variable under study.

M. S. Ahmed(1998) interpreted the regression coefficients correctly for the estimators suggested by Mukerjee, et al.(1987). M. S. Ahmed(1998) mentioned that the corrected mean square errors of Kiregyera(1984) estimators are computed assuming that the regression coefficient and are ordinary not partial regression coefficient. Furthermore he proved that Kiregyera(1984) estimators were better than Mukerjee, et al.(1987) estimators and also showed that estimator suggested by Tripathi & Ahmed(1995) was more efficient than Kiregyera(1984) estimators. H. P. Singh & Gangele(1999) suggested almost unbiased ratio-type and product-type estimators for population mean in two-phase sampling. The performance of suggested estimators was empirically evaluated. Tracy & Singh(1999) proposed a class of chain regression estimators with asymptotic expression of bias and mean squared error for estimating the population mean of variable of interest in two-phase sampling by using two-auxiliary variables.

Tracy & Singh(1999) also derived asymptotic optimum unbiased ratio-type estimator with its variance in two-phase sampling and also in successive sampling with the use of two auxiliary variables. Tracy & Singh(1999) proved that proposed estimator is better than Olkin(1958) and Sen(1971) estimators.

J. Sahoo & Sahoo(1999a) developed a class of estimators by using two phase sampling and J. Sahoo & Sahoo(1999b) conducted a comparative study of the estimators considered by Chand(1975), Kiregyera(1980, 1984), Mukerjee, et al.(1987), J. Sahoo, et al.(1993) and J. Sahoo, et al.(1994a) under the super population model using two auxiliary variables.

H. P. Singh & Tailor(2000) suggested some ratio-type estimators of population mean of study variable using two auxiliary variables in two-phase sampling with coefficient of variation of the second auxiliary variable was known. H. P. Singh & Tailor(2000) obtained the conditions in which proposed estimators were more efficient than usual two-phase sampling ratio-estimator, Chand(1975) estimator, and G. N. Singh & Upadhyaya(1995).

A. K. Singh, Singh, & Upadhyaya(2001) proposed two classes of chain ratio-type estimators and also derived expressions of bias and mean square errors in two-phase sampling by using two-auxiliary variables. L. N. Sahoo & Sahoo(2001) proposed estimators of finite population mean by using predictive approach in two-phase sampling using two auxiliary variables. A. K. Singh, et al.(2001) considered a generalized chain estimator for finite population mean using two auxiliary variables in two phase sampling.

Radhey, Singh, & Singh(2002) provided a modified ratio estimator with approximate expressions for its bias and mean square error in two-phase sampling for population mean of variable of interest by using two-auxiliary variables. Radhey, et al.(2002) investigated empirically that asymptotic optimum estimators performed better than conventional unbiased ratio, traditional ratio, Chand(1975), Kiregyera(1980) and L. Upadhyaya, Kushwaha, & Singh(1990) estimators. H. P. Singh & Singh(2002) estimated the population coefficient of variation of study variable with chain ratio-type estimator using two auxiliary variables in two-phase sampling and also derived expressions for the bias and mean squared error.

Chandra & Singh(2003) discussed a class of unbiased estimators with its properties for the population mean of study variable in two-phase sampling using two-auxiliary variables when information for the mean of main auxiliary variable was not available. The unbiased estimators suggested by Chand(1975) and Dalbehera & Sahoo(2000) found to be the special cases of proposed class. Diana & Tommasi(2003) proposed a general class of estimators for finite population mean in two-phase sampling. Diana & Tommasi(2003) class of estimators was based on the sample means and variances of two auxiliary variables. Diana & Tommasi(2003) also provide the minimum variance bound for any member of the class.

H. P. Singh & Espejo(2003) proposed a class of ratio-product estimators for estimating a finite population mean in two-phase sampling and identified an asymptotically optimum estimator in their class along with its approximate mean-square error by using the prior knowledge of the parameter. H. P. Singh & Espejo(2003) also found that the estimators are equally efficient for known value of C as well as for consistent estimator of C.

R. Singh & Singh(2003) proposed a regression-type estimator in

two-phase sampling for population mean when information on second variable was known and variance of main auxiliary variable was not known. The proposed estimator was more efficient than Chand(1975), Kiregyera(1980, 1984) and usual ratio, regression estimators.

Roy(2003) constructed a regression-type estimator of population mean of the main variable in the presence of available information on second auxiliary variable, when the population mean of the first auxiliary variable was not known. Roy(2003) estimator was more efficient than Mohanty(1967), Chand(1975), Kiregyera(1980, 1984) and J. Sahoo, et al.(1993)

.

M. S. Ahmed(2003) proposed chain based general estimators for finite population mean using multivariate auxiliary information under multiphase sampling. M. S. Ahmed(2003) considered a number of auxiliary variables in each phase under a general sampling design and studied the properties of these estimators and presented the results for simple random sampling without replacement schemes. M. S. Ahmed(2003) also derived the optimum sample sizes using a modified cost.

H. P. Singh, et al.(2004) proposed a family of estimators, which is more efficient than those considered by S. K. Srivastava(1970), Chand(1975), S. R. Srivastava, et al.(1990), H. P. Singh & Biradar(1994), H. P. Singh & Gangele(1995) and A. K. Singh, et al.(2001). H. P. Singh & Vishwakarma(2005-2006) suggested a modified version of Sahai(1979) estimator in two-phase sampling and discussed its properties. L. N. Upadhyaya, Singh, & Tailor(2006) proposed a family of chain ratio-type estimators for population mean by utilizing information of mean for first auxiliary variable and coefficient of variation for second auxiliary variable.

H. P. Singh, Singh, & Kim(2006) considered chain ratio and regression type estimators of median and provided expressions for its variance. The optimum sample sizes were also obtained for first phase and second phase using fixed cost of the survey. Comparison was made with estimators suggested by A. K. Singh & Singh(2001) . Jhajj, et al.(2006) has proposed a family of estimators in single and two phase sampling using information on a single auxiliary attributes, the proposed family is based upon a general function. Shabbir & Gupta(2007) have also proposed an estimator for population mean in single phase sampling using information of single auxiliary attribute.

H. P. Singh & Espejo(2007) suggested a class of ratio-product estimators in two-phase sampling for population mean in the presence of two-auxiliary variables and also discussed their properties. H. P. Singh & Espejo(2007) also identified asymptotically optimum estimators with their variances and compared their efficiency with two-phase ratio, product and mean per unit estimator under some conditions. Shabbir & Gupta(2007) have also proposed an estimator for population mean in single phase sampling using information of single auxiliary attribute.

Samiuddin & Hanif(2007) introduced ratio and regression estimation procedures for estimating population mean in two-phase sampling for different three situations depending upon the availability of information on two auxiliary variables for population. Samiuddin & Hanif(2007) considered three situations, first when information on both auxiliary variables was not available, second when information on one auxiliary variable was available and third, when information was available on both auxiliary variables. Samiuddin & Hanif(2007) estimators developed in second situation were found to be as efficient as H. P. Singh, et al.(2004) and Roy(2003). But the estimators developed in third situation were more efficient then H. P. Singh, et al.(2004) and Roy(2003) as well as their own estimators developed in first two situations.

Z. Ahmed, et al.(2009) proposed generalized regression-cum-ratio estimators for two-phase sampling using multi-auxiliary variables. Z. Ahmed, et al.(2009)suggested three classes of regression-cum-ratio estimators for estimating population mean of variable of interest for two-phase sampling based on multi-auxiliary variables for full information, partial information and no information cases. Hanif, Ahmed, et al.(2009) proposed a number of generalized multivariate ratio estimators for two-phase and multi-phase sampling in the utilizing multi-auxiliary variables for estimating population mean for a single variable and a vector of variables of interest(s). Hanif, Ahmed, et al.(2009) also made theoretical and empirical to check the efficiencies of the estimators.

Hanif, Haq, et al.(2009) proposed general family of estimators and derived general expression of mean square error of estimators proposed by Jhajj, et al.(2006). The family has been proposed for single-phase sampling in case of full information and for two-phase sampling in case of partial and no information cases. Hanif, Haq, et al.(2009) discussed that the proposed family has smaller mean square error than given by Jhajj, et al.(2006).

Z. Ahmed, et al.(2010) suggested a number of generalized multivariate regression estimators for two-phase and multi-phase sampling in the presence of multi-auxiliary variables for estimating population mean for a single variable and a vector of variables. Hanif, et al.(2010) proposed some ratio estimators for single phase and two phase sampling using information on multiple auxiliary attributes. The proposed estimators are generalization of the estimator proposed by Naik & Gupta(1996). Hanif, et al.(2010) proposed some ratio estimators for single phase and two phase sampling by using information on multiple auxiliary attributes. The proposed estimators are generalization of the estimator proposed by Naik & Gupta(1996). Hanif, et al.(2010) also drive the shrinkage version of the proposed estimators by using the method given Shahbaz & Hanif(2009).

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