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1. ______________________________________________________________________

Introduction and Review of Literature ______________________________________________________________________

History of learning about population by using sampling methods could

be traced out to even very early stages of primitive life of mankind. The

development of statistical methodology used in survey sampling, to a large

extent, is aimed at estimation of the mean or the total of the population

characteristics under study with high precision or least cost.

The earlier development in the theory of estimation and statistical

inferences was based on the simples of selection procedure, namely random

sampling. It was Kiaer (1895), who introduced the concept of random sampling

to study socio-economic problems with a view to replace the usual approach of

complete count. Bowley (1906) introduced the idea of probability sampling.

Deming (1950) “Sampling is not a mere substitution of a partial

coverage for a total coverage. Sampling is the science and the art of controlling

and measuring reliability of useful statistical information the theory of

probability”.

Many strategies can be used to create a probability sample each start

with a sampling frame which can be thought of as a list of all elements in the

population of interest (e.g. names of individuals, telephone numbers, house

addresses, and census tracts). The sampling frame operationally defines the

target population from which the sample is drawn and to which the sample data

will be generalized. Simple random sampling is probably the simplest method

for obtaining a good sample. A simple random sample of say N is chosen from

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the population in such a way that every random set of n items from the

population has an equal chance of being chosen as sample. Thus simple

random sampling not only avoids bias in the choice of individual item but also

gives every possible sample an equal chance.

When the population is heterogeneous, Bowley introduced the idea of

stratification to increase the precision of the sample estimates. Bowley

introduced the idea of stratification to increase the precision of the sample

estimate. In this procedure population is divided in to a pre-assigned number of

non-overlapping sub-population or groups called “strata” before selection of

the sample. Here sample of pre-determined size are drawn independently from

strata. Stratification is more effective when there are extreme values in the

population, which can be segregated into separate within each stratum is drawn

with random sampling procedure, the design is known as stratified random

sampling.

The work of Bowley (1926) and Neyman (1934) laid the foundations of

modern sampling theory; works of R.A Fisher during twenties provided a

scientific basis for selecting a random sample. Many research papers relating to

sampling theory and methods have been written for setting up new directions

and trends of development which provide sound theoretical support to what

was in practice.

1.1 Concepts and Definitions

In this section, we present the basic concepts and definitions to be used

in this work.

An elementary units or a unit is an element or a group of elements, on

which observations can be made or from which the required statistical

information can be ascertained according to a well defined procedure. The

collection of all units of a specified type in a given region at a particular point

or period of time termed a population.

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A collection of N (given) well defined, distinct, identifiable and

observable objects

under consideration about which certain valid conclusions are to be

drawn, is called a finite population. The objects ( i=1, 2…., N) are called

sampling units or elements. The number of units in the population denoted by

N is called the size of the population and the list of all sampling units with

identification number 1,2,…….,N is called sampling frame.

A variate (character) y is a real-valued function defined on U such that

=y ( ) (i=1,2,…..,N) is the value of the character y associated with the units

of the population. Let and let denotes the

space described by which may be the N- dimensional Euclidean space or a

subspace of it. may be referred to as parametric space of y and

is called parametric vector of y.

A parameter (or parametric function) , of a character y, is a real-

valued function of .The parameters of particular interest are

population mean, population total and population variance which are

respectively defined as

Population Mean of y: =

Population Total of y: Y = N =

Population Variance of y: =

The general problem is to make some inferences about or the

parameters on the basis of the information contained in a sample (or

samples) from the population U. One may be interested in the problem of point

estimation or the problem of interval estimation or the problem of testing of

hypothesis about an unknown parameter .

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In the present work, we shall confirm ourselves to point estimation type

problems. The main problem of estimation in case of finite populations may be

specified as to obtain optimum sampling strategies that are to obtain optimum

sampling designs as well as optimum estimators for unknown parameter .

For drawing inferences about finite population survey statistician search for a

suitable sampling procedure (sampling, scheme, sampling technique) and

suitable estimation procedure (method of estimation). Since we are interested

in estimating a parameter , such as and we shall call y as the variate

under study or the study variable. When the information on a variable

is also used in estimation of , we call as an

auxiliary variable. In this case knowledge on a parameter which may be a

function of y or x, is used in estimating , we call as an auxiliary or

auxiliary parameter.

In case of drawing inference about the ‘bivariate parameter’ , the

real valued functions of (y, x). The parameter of a particular interest, in this

case, may be covariance between y and x, correlation coefficient between

between y and x, regression coefficient of y on x and the ratio R of

population mean of y to population mean of x defined respectively by

=

and R =

(z

A sample is a finite ordered sequence of units from the population U

drawn with or without replacement according to a specified probability called

sampling procedure. The totality S of samples s foe which P(s) 0 is considered

as effective sample space. The sample space or the effective sample space S

and the probability P together gives a sampling design D(S, P). The fraction of

the population selected in the sample is called the sampling fraction.

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A sample in which every unit has the equal probability of selection is

called a random sample. A sample is considered as random or probability

sample. If the selected (sampled units) are replaced in the population and

sampling is done, in that case it is called simple random sampling with

replacement (SRSWR) and if the sampled units are not replaced i.e. repetitions

are not allowed then we say simple random sampling without replacement

(SRSWOR).

When the sample is taken as every unit, it is called systematic

sampling. If the units appear in the sample with different probabilities,

sampling is said to be with unequal probabilities. When the probabilities are

based on some measure of size of the units, it is sampling with probability

proportionate to size (pps). When the population is divided into groups

(homogeneous) and a sample is selected from each group, the groups are called

strata. If the same fraction is taken into the sample from each stratum the

sampling is said to be done with proportionate allocation or with fixed

sampling fraction otherwise it is a variable sampling fraction plan.

The sample is usually selected in clusters or groups of the elementary

units since frames listing elementary units are rarely available. When clusters

are large it is difficult to enumerate them completely. We may further select a

sample from each of the selected cluster. This procedure is called sub-sampling

or two-stage sampling. Sometimes the first phase of the enquiry is limited to

the collection of auxiliary information and this information is used in the

second phase for stratification, ratio or regression estimation. This is called the

double sampling method.

A statistic is a function defined on the sample space. An estimator d(s)

is a real valued function of { }, j =1,2,…..,n(s); n(s) being the number of

units in the sample, j s, s S and an estimate the value of an estimator for a

given sample s.

An estimator d for is said to be unbiased

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iff E(d) =

where E(d) denotes the expected value of d overall S selected from

the population U. The bias of the estimator denoted by B (d) = E (d) - and

the variance of the estimator d is given by

V (d) = E (d –E (d)

The mean square error (MSE) of the estimator d for is given by

MSE = E (d – = V (d) +

An estimator d is said to be consistent estimator of if the estimator

assumes the value when the sample is taken as the population.

A sampling strategy for a parameter , is a pair of

sampling design D and an estimator d for . By the unbiasedness of

for we shall mean unbiasedness of d for and by MSE ( ), the mean

square of we shall mean MSE (d) for the given D.

The general problem of point estimation in case of finite populations,

may be described as to find optimum (optimum in some well-defined sence)

sampling strategy (or strategies) that is, to find the optimum sampling design

(or designs) as well as optimum estimator (or estimators) for an unknown

parameter .

Let and be two sampling strategies for

based on the sampling designs (or sampling procedures) and are the

estimators and both for respectively. A sampling strategy is said

to be better than

iff MSE( ) MSE( ) for all

holds with strict inequality for at least one . Let the estimators and

for be based on the same sampling design D. Then the estimator is

said to be better than

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iff MSE( ) MSE( ) for all

holds with strict inequality for at least one

The per cent relative efficiency of a sampling strategy

over is defined by 100

.

1.2 Review of Literature

In order to make effective use of available sources various sampling

technique have been developed from time to time which provide estimators of

population characteristics of interest with high precision, reduced cost and

above all will have the operational feasibility and practical applicability. In this

section a review of the work done by various authors related to work in this

thesis is presented.

1.2.1 Use of Auxiliary Information in Sample Survey

Information on variables correlated with the main variable under study

is popularly known as auxiliary information which may be fruitfully utilized

either at planning stage or at designing stage or at the information stage to

arrive at improved estimator compared to those, not utilizing auxiliary

information. The concept of multi-auxiliary information is well known in

sampling theory. Its use of paramount importance in sample surveys as it leads

to increased precision of estimators for population parameters.

The origin of utilizing auxiliary information in sample surveys can be

traced back to the origin of sampling theory itself. It has been a general view of

the survey statisticians for last five decades that the usual methods for

estimating population mean (or total) of a variable of interest, say y. It may

lead to much improvement in precision of estimation if the information on the

closely related variable (auxiliary information) is utilized judiciously in the

estimation procedure.

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It most of the surveys situations, the auxiliary information is always

available in one form or the other or can be made available by diverting for this

purpose a part of the survey resources at moderate cost. In whatever form the

auxiliary information is available, one may always utilize it to devise sampling

strategies which are better (if not uniformly then at least in a part of parametric

space) than those in which no auxiliary information is used. The method of

utilizing auxiliary information depends on the form in which it is available.

In sample surveys the auxiliary information on or more variables may be

utilized in three basic ways which has been already been discussed by Tripathi

(1970, 1976).

(1) At the pre-selection stage or the designing stage i.e. the

information may be used in stratifying the population.

(2) At the selection stage i.e. in selecting the units for sample with or

without replacement and with varying probabilities proportional to some

suitable measure of size.

(3) At the post-selection stage or at the estimation stage i.e. through

defining ratio, regression, difference and product estimators based on the

auxiliary information.

The auxiliary information may also be used in mixed ways as well as by

combining any two or the entire above situation.

Usually measurements on the auxiliary characters are available for every

unit of the population or at least the population totals or mean of auxiliary

characters are known in advance. However, when such information is not

available, it is obtained by taking a large preliminary sample in which only

those auxiliary characters are measured which lacks such information. The

purpose of this sample is to furnish a good estimate of the population mean or

total of the auxiliary characteristic or its frequency distribution. The technique

of double sampling or two phase sampling is thus used to make such

information available in surveys.

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In case , the population mean of an auxiliary character x, is known, a

large number of estimators for are available in sample survey literature.

Laplace (1820) was the first to use the auxiliary information in ratio type

estimator. The works of Bowley (1906) and Neyman (1934, 1938) can be

referred to as the initial efforts to utilize the auxiliary information in sampling

theory. However, Watson (1937) and Cochran (1940, 1942) initiated the use of

auxiliary information in dividing estimation procedures aimed at improvement

in the precision of estimation. Hansen and Hurwitz (1943) were the first to use

auxiliary information in selecting the units with varying probabilities.

The univariate ratio and regression estimators proposed by Cochran

(1940, 1942), difference estimator Hansen et al. (1953) and product estimators

by Robson (1957) , Murthy (1964) for population mean of a variable y based

on the knowledge of population mean of an auxiliary character x are quite

well known in sampling theory, and for detailed study in case of simple random

sampling without replacement (SRSWOR) and stratified sampling one may

refer to the books by Cochran (1977), Sukhatme et al. (1976), Raj (1968),

Murthy (1967), Kish (1965) and others. Das and Tripathi (1980), Das (1988),

Khare (1988) gave classes of estimators for population mean and extended the

classes of estimators, defined by Srivastava (1971, 1980), for any general

sampling design, several authors like Reddy (1974), Agarwal et al. (1980),

Gupta (1978), Ray et al. (1980), Srivenkataramana et al.(1976,1979), Kaur

(1983), Singh et al. (1983), Chaudey et al. (1984) defined estimators for

using knowledge on . Using the technique of bias correction Hartley and Ross

(1954) considered an unbiased estimator and Murthy et al. (1959) considered

almost unbiased estimators for in case of SRSWOR. Another technique well

known as Jack-Knife technique for obtaining unbiased or almost unbiased

estimators was given by Quenouille (1956) and generalized by Grey et al.

(1972). The properties of ratio, ratio-type estimators have been studied among

others, by Rao (1979) and Schucany et al. (1971). Almost unbiased ratio

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estimators (AURES) have also been considered by Rao (1966), Sahoo et al.

(1989), Pandey et al. (1989) and Singh et al. (1989).

A general class of estimators of population means using auxiliary

information has been considered by Naik and Gupta (1991). An unbiased class

of product type estimators for has been considered by Tripathi and Singh

(1988) using some transformation and estimators discussed by Gupta and

Adhavarya (1982), Kushwaha and Singh (1988) are particular members of this

class. Deng and Chikara (1991) has defined asymptotically design unbiased

estimators of population mean. Rao et al. (1990) has used auxiliary information

for estimating distribution functions and quartiles and Rao et al. (1990) have

described optimal designs for estimators. Singh (1989) has taken the problem

of estimation in case of incomplete frames. Yadav and Singh (1984) suggested

proportional allocation for simple random sampling scheme and Singh et. al.

(1985) used auxiliary information for non response. Further Chaudhary et.al.

(1989) have described the efficiency of the ratio estimators and Sampath

(1989) obtained optimal choices of unknown in ratio estimators. Further

PPSWR Tripathi (1970), Singh (1980), Bansel and Singh (1985), Gupta (1990)

have suggested estimators for population mean.

In case the required auxiliary information is not readily available the

ratio, regression, difference and product type estimators for , based on double

sampling procedure are well known in sample survey literature and have been

considered by Tripathi (1976) for general sampling designs. A class of

difference cum ratio and product estimators based on double sampling was

discussed by Ray and Singh (1979). Further Srivastava (1981), Singh et al.

(1983), Kapadia and Gupta (1984) considered the special cases of the estimator

discussed by Singh (1969). Some unbiased estimators using double sampling

and Jackknife technique have been discussed by Sengupta (1981), Singh et al.

(1985) and Shah and Gupta (1987).

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When information on p-auxiliary variables is available

and are known then Olkin (1958) was the first to deal with the

problem of estimating the mean of a survey variable when auxiliary

information are made available. He suggested the use of information on more

than one supplementary characteristic, positively correlated with study

variable, considering a linear combination of ratio estimators based on each

auxiliary variable separately. The coefficients of the linear combination were

determined so as to minimize the variance of the estimator. Analogously to

Olkin, Singh (1967 a) gave a multivariate expression of Murthy’s (1964)

product estimator, while Raj (1965) suggested a method for using multi-

auxiliary variables through a linear combination of single difference estimators.

Moreover Singh (1967b) considered the extension of the ratio cum product

estimators to multi-supplementary variables, While Rao and Mudholkar (1967)

proposed a multivariate estimator based on a weighted sum of single ratio and

product estimators. An alternative weighting system for defining weighted

ratio, regression and difference estimators has been considered by Tripathi

(1978). Srivastava and Jhajj (1983) defined the class of estimators using multi-

auxiliary information. In case where the population means are

unknown, Khan and Tripathi (1967) discussed the ratio estimator and multiple

regression estimators. Adhvaryu (1978) considered ratio-cum product estimator

in double sampling using multi-auxiliary information. Further Srivastava et al.

(1990) suggested a generalized class based on multi-auxiliary

information.Sahoo et al. (1989), Bansal and Singh (1989), Kumar and Hozel

(1988), Kothwala and Gupta (1989), Srivenkataraman and Tracy (1989) have

considered the different estimators for estimation of population mean using

auxiliary information in various sampling designs. Use of multivariate auxiliary

information for selecting units with PPSWR was considered by Maiti and

Tripathi (1976) and Agrawal et.al. (1980).

Further, an improvement over the customary estimator suggested by

Singh (1965, 67) through knowledge of . Using two auxiliary character

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and , Rao & Pereira (1968)suggested the estimator for R when the

population means of p- auxiliary characters are known. The

class of estimators of R was extended by Tripathi et al. (1979). In case in

unknown Tripathi (1970) gave a general class of estimators for R. Further

Singh (1982) and Khara (1983, 87) extended these classes to the case of p-

auxiliary character. Maiti and Tripathi (1979) also studied class of estimators

for R based on double sampling.

Further, different estimators for the estimation of population proportions

using auxiliary information were suggested by Hyett and Mckenzie (1977),

Rao (1977) and Das (1982). A ratio test for the equality of proportions has been

suggested by Chou and Owen (1991).

Many other contributions are present in sampling literature and recently,

some new estimators appeared. Deng and Chhikara (1991) have defined

asymptotically design unbiased estimators of population mean. A general class

of estimators of population means using auxiliary information has been

considered by Naik and Gupta (1991). Rabionson (1994) proposed a regression

estimator ignoring some of the assumptions usually adopted in the literature of

Srivastava. Ceccon and Diana (1996) provided a multivariate extension of

the Naik and Gupta univariate class of estimatos. Tracy et al. (1996) proposed

an alternative to Singh’s ratio-cum product estimators, when two auxiliary

variables are available. Agrawal et al. (1997) illustrated a new approach to

from a multivariate difference estimator which does not require the knowledge

of any population parameters. Abu- Dayyeh et al. (2003) introduced two

estimators which are definitely members of the class proposed by Srivastava,

while Kadilar and Cingi (2004, 2005) analyzed combinations of regression type

estimators in case of two auxiliary variables. In the same situation, Perri (2005)

proposed some new estimatiors obtain from Singh’s estimators. Pradhan (2005)

suggested a chain regression estimator for two-phase sampling using three

auxiliary variables when the population mean of one auxiliary variable is

unknown and other auxiliary population means are known.

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1.2.2 Double Sampling for Stratification

The procedure of double sampling (or two phase sampling) for

stratification was first given by Neyman (1938) and is well available in the

literature Hassen et al. (1953), Kish (1965), Raj (1968), Konijn (1973),

Cochran (1977) and Dayal (1979). In case sampling frames for strata are not

available, the usual (ordinary or prior) stratified sampling cannot be used but

PPS can be used provided strata weights are known exactly. However in

many situations, may not known exactly as they become out dated with the

passage of time and thus post stratification sampling cannot be used.

Further, the information on stratification variable x may not be readily

available but could be made available by diverting a part of the survey budget.

Under these circumstances, the technique of double sampling for stratification

(DSS) comes to our rescue as a powerful tool.

In the usual procedure of DSS, at first a preliminary sample of size is

selected and observed for x alone, which is used for stratifying the sample

giving as units falling in stratum h (

) and then a subsample with

units is selected from stratum h (h = 1,2,….L) and is observed for the main

variable y, Following Raj (1968) and Sukhatme and Sukhatme (1970) the

expression of variance of unbiased estimator of is given by Rao (1973),

Cochran (1977) in case the samples of size are random subsample from

units of the first sample in stratum h (h = 1,2,….L) as would also be a

random variable in this case. However, in the literature, the subsample of size

from stratum h (h=1,2,,…,L) used to be assumed, implicitly, non-random,

Singh and Singh (1983) pointed out that this assumption is inconsistent with

the sampling procedure. They proposed three consistent sub-sampling

procedures which treat the sub-sample size within each stratum as a random

variable:

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(i) The subsample within each stratum is selected with replacement

and all units are used in the estimator.

(ii) as in (i) but only distinct units used

(iii) sub-sampling is without replacement, the size being min ( , )

where may be predetermined.

Rao (1973) pointed out that although the procedures adopted by Singh

and Singh are free of inconsistency; procedure (i) and (ii) lead to the loss in

efficiency and further the procedure (ii ) gives rise to a variance formula which

is not suitable for the optimal determination of and for a fixed cost.

Furthermore, Rao proposed a simple procedure of double sampling for

stratification which is also free from inconsistency.

Hansen and Hurwitz (1946), Rao (1968), Srinath (1971) developed the

theory of double sampling for stratification for handling the estimation of in

the presence of non-response. Realizing the importance of analytical study of

survey data, Sedransk (1965) made empirical studies based on double sampling

for stratification.

In the discussion on DSS by various authors, the use of auxiliary

information { ) collected on the first sample is made only for

stratifying the sample. Ige and Triapthi (1987) used this information collected

on the first sample not only for stratifying the sample but also at estimation

stage and use of multi-auxiliary variable in unistage design has been proposed

by Triapthi and Bahl (1991) for improving the precision of estimation. The

multivariate auxiliary information has not been used so far and our attempt is to

use this multivariate auxiliary information at the designing as well as at the

estimation stage in two-stage design to obtain better sampling strategies.

1.2.3 Two Stage Sampling

With a view to reduce cost and/or to concentrate the field operations

around selected points and at the same time obtain precise estimates, sampling

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is sometimes carried out in stages. The procedure of first selecting large sized

units and then choosing a specified number of sub-units from the selected large

units is known as sub-sampling. The large units are called ‘first stage units’ and

the sub-units the ‘second stage units’. The procedure can be easily generalized

to three stage or multistage samples. For example, the sampling of a forest area

may be done in three stages, firstly by selecting a sample of compartments as

first stage units, secondly, by choosing a sample of topographical sections in

each selected compartment and lastly, by taking a number of sample plots of a

specified size and shape in each selected topographical section.

A sampling procedure pre-supposes the division of the population into a

finite number of distinct and identifiable units called the sampling units. The

smallest units into which the population can be divided are called the elements

of the population and groups of elements, the clusters. When a list of elements

is not available using an element as the sampling unit is clearly not feasible. In

that case the method of cluster sampling is used. A necessary condition for the

validity of the procedure is that every unit of the population under study must

correspond to one and only one unit of the cluster so that the total number of

sampling units in the list(frame) will cover all the units of the population under

study with no omission on duplication otherwise biases are introduced.

Mahalanobis (1940, 1942, 1944) has considered in detail the question of

determining the optimum cluster size in case of crop surveys. Further Smith

(1938), Hasen and Hurwitz (1942), Jessen (1942) Sukhatme (1947, 1950) and

Seng (1951) have also studied the question of sampling efficiency of cluster

sampling.

In cluster sampling all the elements of the selected cluster are

enumerated. The large the cluster, the less efficiency it is usually relative to the

elements the sampling units. It is, therefore, logical to expect that for a given

number of elements, greater precision will be attained by distributing them over

large number of clusters than by taking a small number of clusters and

sampling a large number of elements from each of them or completely

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enumerating them. The procedure of first selecting the clusters and then

choosing a specified number of elements from each of selected cluster is

known as sub-sampling or two-stage sampling. The clusters that form the units

of sampling at the first stage are called first stage units or primary units and the

elements or groups of elements within clusters which form the units of

sampling at the second stage are called sub-units or second stage units. Hence

the procedure is generalized to three or more stages and is then termed multi-

stage sampling.

Use of multistage sampling using various sampling procedures is well

known in literature.

Raj (1968), Durbin (1967), Hansen and Hurwitz (1943), Horvitz and

Thompson (1952), Hartley and Rao (1962), Murthy and Sethi (1959,1961), Rao

(1962), Rao, Harttley and Cochran (1962), Rao (1975), Sukhatme and Koshal

(1959), Sukhatme (1962,1950). Mahalanobis (1940) used this sampling

procedure in crop surveys. Ganguli (1941), Cochran (1939) and Hansen and

Hurwitz (1943) have considered the use of this procedure in agricultural and

population surveys respectively. Lahiri (1954) has discussed the use of

multistage sampling in the Indian National Sample Survey, Rao (1957) and

Singh (1958) have considered the estimation of variance components for this

sampling scheme.

Various authors Singh and Srivastava (1973), Sahoo (1987),) and Arnab

(1991) so far have made use of auxiliary information on one variable whether it

is available or collected through the preliminary sample in multistage designs

for estimation of population mean, population total, population ratio and

proportion. Mahajan and Singh (1996) proposed an estimator of population

total in two stage sampling. Ye, A Zhong (1997) extended the multistage

sampling with unequal probability. Ye, A Zhong (1998) proposed an allocation

of the sample sizes in three or four stage sampling. Goswami and Sukhatama

(1965) extended the result of Sukhatame and Koshal (1959) to several auxiliary

variables with unknown mean for a three stage design and these results can be

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extended to design with any number of stages. Chatterjee (1968, 1972)

discussed the multivariate stratified surveys and optimum allocation in

multivariate stratified surveys. Garg and Pillai (1975) has developed two ratio-

type estimator of the population mean in the case of two stage sampling when

the auxiliary information in not available for all the units in the populations.

Using cost function, the optimum allocations of sample units for attaining a

given precision, the total cost of the survey being fixed, have been worked out

when two phase sampling in multistage adopted. Jain (1981) discussed a

rotation scheme for a stratified multistage sampling to satisfy the condition (i)

there is a constraints on the number of units that can be replaced in each round

and (ii) it is relatively inexpensive to increase the sample size gradually. He

derived estimators of the population proportion of elements of specified

characteristics. Ruiz Espezo (1991) proposed a minimum allocation in stratified

sampling with highly correlated variable of interest and auxiliary variable. Yi

Neng (1996) extended the mean precision of a ratio type estimator in two stage

sampling. Okafor (1996) proposed a double sampling for stratification with sub

sampling for the non-respondents. Our aim in the present work is to devise

methods of using multivariate auxiliary information, is collected through the

preliminary sample, for the estimation of population parameters in the two-

stage designs.

1.2.4 Probability Proportional to Size Sampling

In many instances, the sampling units vary considerably in size and

simple random sampling may not be effective in such cases as it does not take

into account the possible importance of the larger units in the population. In

such cases, it has been found that auxiliary information about the size of the

units can be gainfully utilized in selecting the sample so as to get a more

efficient estimator of the population parameters. One such method is to assign

unequal probabilities for selection to different units of the population. For

example, villages with larger geographical area are likely to have larger area

under food crops and in estimating the production; it would be desirable to

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adopt a sampling scheme in which villages are selected with probability

proportional to geographical area. When units vary in their size and the

variable under study is directly related with the size of the unit, the

probabilities may be assigned proportional to the size of the unit. This type of

sampling where the probability of selection is proportion to the size of the unit

is known as ‘PPS Sampling’.

In sampling from a finite population often the values of some auxiliary

character x closely related to main character y of interest are available for all

the units of the population. The variable x suitably taken as a measure of size of

unit. For example in socio economic surveys, the data on the size of population

which may be available from some previous census may measure the size of

villages, in a survey of industries, x may be number of workers, in agriculture

survey for estimating the yield of crops the data on area under the crop if

available, may provide the size of the farm. In such cases instead of sampling

the units with equal probability with replacement or without replacements one

may sample the units with probability proportional to size measure x (pps) with

replacement or without replacement.

Since a unit with value of x is expected to contribute more to the

population total of y than those with smaller size, one may expect that a

selection procedure which gives higher selection probability to bigger units

than to smaller units should be more efficient than SRS.

The technique of PPS sampling was first put forward by Mahalanobis

(1938) wild sampling plots for a crop survey and its details worked out by

Hansen and Hurwitz (1943, 1949) initiated the use of auxiliary information in

selecting the units with probabilities proportional to size (PPS).They introduced

the method of selecting units of finite population with probability proportional

to a given size measure (pps) and demonstrated its efficiency over the simple

random sampling method and discussed the problem of choosing selection

probability optimally. The Cumulative total method for selection of units which

requires data on the size variable for all the units before-hand, was found to be

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unsuitable when the population are large as well as when the sizes of some

units are missing or not available.

The method of pps selection was familiar to Mahalanobis even as early

as 1937. He realized that, in agricultural surveys it would be necessary to select

plots using the cumulative totals of their areas since the vary considerably

(Mahalanobis (1938)).

Under the initial guidance of Prof. Mahalanobis, using the 1941 census

list of villages as frame, National Sample Survey (NSS) in their first three

rounds, selected the first stage units (fsu’s)using a pps with replacement (wr)

method, size being the village population (where available) or village area

(where population is not available). With the availability of 1951 census,

tehsils as fsu’s and villages as second stage units (ssu’s) were selecting using

ppswr from fourth round onwards. The use of pps selection continued in the

later rounds as well except for a few rounds where equal probability and

circular systematic sampling were resorted to, for economy in conducting the

surveys. To eliminate repetitions, pps systematic sampling was also widely

used in the rural as well as urban design. While Hansen and Hurwitz developed

the theory of pps sampling based on one fsu per stratum Midzuno (1950, 1952)

considered selection of a combination of n elements with probability of

selection proportional to the aggregate size measure.

In addition to obtaining better estimators the use of PPS sampling has

also been made to obtain unbiased ratio estimators. Lahiri (1951) showed that

the ratio estimator = ( ) , which is biased in equal probability sampling,

would became unbiased for if the probability of selecting a samples from U

is made proportionate to its mean or total size. Horvitz and Thompson (1952)

generalized the theory to pps sampling without replacement (wor) and defined

three classes of linear estimators noting the ‘identifiable’ nature of the finite

population. It may be noted that Narain (1951) independently discussed

varying probability sampling technique and also gave comparison between wor

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and wr methods. Raj (1954 a) considered variance and an unbiased variance

estimator of the ratio estimator in case of a multistage design where the sample

of first stage units is selected with PPS. Raj (1954 b) investigated about the

superiority of PPS sampling over the equal probability sampling.

For ppswor Raj (1956) developed a simple estimator for the population

total depends on the order of selection of units while Murthy (1957) obtained a

summarized version of this estimator. Fellegi (1963) developed a sampling

design for rotating and non rotating samples which is practically suitable if the

units in sample have to be rotated as in the case of repetitive surveys. An

alternative estimator in PPS sampling for multiple characteristic was developed

by Rao (1966) and Durbin (1967) describes an estimator for estimating the

sampling error in multistage sampling survey. Rao and Bayless (1969) put

forward an empirical study of the stabilities of the estimates and variance

estimators in unequal probability sampling of two units per stratum.

Ramakrishnan (1971) generalized the Yates and Grundy estimates.

Mukhopadhya (1972,80), Sinha (1973), Haezel (1986) considered

constructions of sampling designs which realize pre-assigned sets of inclusion

probabilities of first two orders. Recently, Srivenkataramana and Tracy

(1986) reviewed transformations which can be used after the sample is selected

for ratio, product methods and for pps sampling. Use of transformations for

reduction in variance in sampling with ppswr and wor was discussed in detail

by Stuart (1986).

Use of multiple auxiliary variables for obtaining a suitable composite

size measure was also made by Singh, Kumar and Chandak (1983) and Tripathi

and Chaudey (1990). In large scale sample surveys, where one is interested in

estimating parameters relating to several characteristics, it is sometimes

observed that some of the study variable poorly correlated with selection

probabilities while pps sampling adopted. J.N.K. Rao (1966) has suggested

alternative estimator with ppswr scheme which is shown to be better than the

conventional unbiased estimator, though biased. Singh and Horn (1998)

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showed empirically that their estimator becomes more efficient than

conventional estimator proposed by Kumar and Agarwal (1997) for varying

probability sampling scheme. Arnab (2003) uses the auxiliary information for

the finite population related to a study variable plays eminent role for selection

of sample with varying probabilities to get the efficient estimator.

1.2.5 Predictive Estimators

In sample surveys, supplementary population information is often used

at the estimation stage to increase the precision of estimators of a population

mean or total. It is common practice to use auxiliary information on a character

x in the estimation of the finite population mean or total of a character under

study. A variety of approaches are available to construct more efficient

estimators for the population mean and total, including design based and model

based methods. The model-based approach is based on super population

models, which assume that the population under study is a realization of super-

population random variables having a super population model ξ. This super

population model ξ formalizes our prior knowledge about the population and is

used to predict the nonsampled values of the population, and hence finite

population quantities Y or total Y. Some advantages of this approach are as

follows:

1. Prediction theory for sampling surveys (or model- based theory)

can be considered as a general framework for statistical inferences on the

character of finite population. Well-known estimators of population totals

encounter in the classical theory, as expansion, ratio, regression, another

estimators, can be predictors is a general prediction theory, under some special

model.

2. This approach is aligned with mainline statistics approachs in

other application areas.

3. In large samples and with certain distribution, results can parallel

those from design-based inference.

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4. Model-based estimators often have a smaller variance than their

design-based competitors.

In a predictive approach a model is specified for the population values

and is used to predict the non sampled values.

Predictive approach advocated by Basu (1971) is adopted for estimating

the mean of a finite population; it is observed that the use of mean per unit

estimator, regression estimator and ratio estimator as a predictor for the mean

of unobserved units in the population results in the corresponding customary

estimators of the mean of the whole population. Royall (1970) extended the

predictive approach to the case where information on auxiliary characters is not

available, this approach is essentially model based. Srivastava (1983)

suggested, if the product estimator is used as a predictor for the mean of

unobserved units in the population, the resulting estimator of the mean of the

whole population is different from the customary product estimator. Srivastava

et al. (1988) proposed a double sampling based approach. Sahoo et al. (1995)

proposed two unbiased ratio estimators of the population mean and study their

efficiencies under a linear model. Hossian et al. (2001) suggested a class of

predictive estimator for a two stage sampling with unequal first stage units

using unequal first stage units using auxiliary information, ratio, regression and

product estimators were proposed, minimum mean square of these estimators

are obtained. Ahmed et al. (2003) gave a class of predictive estimators in

multistage sampling using auxiliary information. Further Ahmed (2004)

proposed some estimators for a finite population mean under two stage

sampling using multivariate auxiliary information. Sud et al. (2007) proposed a

estimating population mean square through predictive approach when auxiliary

character is estimated. Sahoo et al. (2009) introduced a new class of estimators

for the finite population mean availing information on two auxiliary variables

in two stage sampling.

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1.3 Thesis at Glance

It was during the first half of this century that a majority of the basic

sampling techniques, now in vogue, were developed. During the subsequent

period practical use, on a large scale, was made of these techniques in a variety

of fields like agriculture, socio-economic industry, medicine etc. These

applications threw up a number of problems, which required basic research, the

efforts on which resulted in more recent development enriching the sampling

theory. The present thesis is an effort in that direction.

The research work included in this thesis consists of investigation in

multistage design using double sampling technique and auxiliary information.

The present work included in this thesis has been divided into six chapters.

In the first chapter, we have discussed the necessary background and

introduced the problems considered in this thesis. This chapter gives a general

sampling concepts and its development over time. A review of whole work

done in auxiliary information, double sampling for stratification, two stage

sampling, PPS sampling and predictive estimators are mentioned in this

chapter.

In Chapter – II we propose difference and ratio type estimators for

estimating the population mean of the study variable when the auxiliary

information is not available in two stage design, but is collected through a large

preliminary sample and we use this information at selection stage as well as

estimation stage. The estimators have been proposed in two different ways:

a. When the information is collected at fsu level and after the

collection of auxiliary information further sample of fsu’s is selected with

PPSWR sampling.

b. When the information is collected at ssu level then after the

collection of auxiliary information sample of ssu’s is selected with PPSWR

sampling.

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The mean, variance, bias and mean square error of these (proposed)

estimators have been obtained. These estimators are compared for their

precision with usual two stage design using on auxiliary information and the

estimators using auxiliary information are found to be more efficient as

compare usual two stage design. An empirical comparison of proposed

estimators based on census data is made to observe the relative behavior of

sampling scheme proposed by us compared to usual two stage design when

there is no use of auxiliary information.

In Chapter – III we developed difference and ratio estimators for

estimating the population mean in two stage design using multi-auxiliary

information which is collected for fsu and ssu level separately. We use this

information at selection stage as well as estimation stage for the estimation of

population mean. The estimators have been proposed in two different ways:

a. When the information is collected at fsu level and after the

collection of auxiliary information further sample of fsu’s is selected with

PPSWR sampling.

b. When the information is collected at ssu level then after the

collection of auxiliary information sample of ssu’s is selected with PPSWR

sampling.

We study their general properties and also find optimum estimators for

both levels separately. We compare the estimators using auxiliary information

at selection stage as well as estimation stage with respect to the estimator using

auxiliary information at selection stage only and after comparison we conclude

that the former estimators are found to be more efficient than latter. An

empirical study is made to compare the relative performance of the proposed

estimators.

In Chapter- IV we propose multivariate difference and ratio estimators

based on Double Sampling for Stratification (DSS) using multi-auxiliary

information, at FSU level, for constructing strata as well as constructing

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estimators for population mean . We study their general properties and

obtain optimum estimators. Their comparison with corresponding estimator

based on Unstratified Double Sampling is made and under the moderate

conditions the estimators using DSS are more efficient with respect to USDS.

An empirical study is made to observe the efficiency of the proposed

estimators.

In Chapter- V we propose multivariate difference and ratio estimators

based on Double Sampling for Stratification (DSS) using multi-auxiliary

information, at SSU level for constructing strata as well as constructing

estimators for population mean . We study their general properties and

obtain optimum estimators. Their comparison with corresponding estimators in

Unstratified Double Sampling is made and these estimators are better than the

estimators based on Unstratified Double Sampling in two stage design. An

empirical study, using census data is made to compare the relative performance

of the proposed estimator.

In Chapter- VI we propose a class of predictive estimators based on a

two-stage design for the estimation of population parameter. The proposed

class consists of two different types of estimators namely ratio and regression.

The Mean square error (MSE) and minimum mean square error of this class

have been derived. We compare the efficiency of predictive estimators in two-

stage design with respect to two stage design using double sampling. An

empirical study is made to compare the relative performance of the proposed

estimators

The results presented in this thesis are mainly theoretical. Applications

of the results obtained are demonstrated through empirical studies.

In the end we propose the possible areas for further extension of the

research work contained in this thesis.

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