The New sampling Procedure for Unequal Probability Sampling of Sample Size 2.

Transcription

1 . The New sampling Procedure for Unequal Probability Sampling of Sample Size. Introduction :- It is a well known fact that in simple random sampling, the probability selecting the unit at any given draw is same whereas in varying probability sampling technique the probability of drawing a specific unit differ from draw to draw. It seem in PPS sampling that such method give biased estimator as the larger unit are over represented and smaller unit are under-represented in the sample. This will occur in case of sample mean as an estimator of population mean, where all the units have equal weights. Instead of giving similar weights to all the units, if the sampling units are properly weighted at the estimation stages by taking the probability of selection in to account, it is possible to get unbiased estimators. In PPS sampling, there are two possibilities for drawing the sample, (i) (ii) With replacement Without replacement (i) Selection of unit with replacement :- In this method we returned the selected unit in the sample, therefore probability of the selected units will not change and the selection probability of a specific unit is equal at any stage. (ii) Selection of unit without replacement :- In this method we do not returned the selecting unit in the sample, therefore the probability of selected unit will changed and the probabilities are redistributed after each draw. The probability proportional 19

2 to size without replacement is more complex than the probability proportional to size sampling. The main purpose in the sampling method is to draw a sample from the population to measure the some population parameters(unknown) parameters of some interesting variables. If the sampling unit drawn from the population does not carry same probability of being selected in the sample, is termed as unequal probability sample. It is well known that the use of unequal probability sampling give much better result instead of using equal probability. The premier idea of unequal probability sampling was given by Neyman(1938) and the first mathematical frame work (with unequal probability sampling) was introduced by Hansen and Horvitz(194).They introduced the selection schemes for estimating the population total. Midzuno (1950) introduced a method of sampling.. Even though it was very easy for calculations point of view but it has some limitations also Then Narain(1951) proposed a sampling scheme which has no restrictions and the same time leads to a more efficient estimate of population value. Cochran, Hartley, Durbin, Brewer and Hanif (1953) proposed some new schemes of selecting a sample with varying probability and without replacement. Brewer and Hanif(1963) had made comprehensive discussion on the numbers of IPPS schemes. These schemes are bounded to sample of size only, because for sample size greater than,the calculation of inclusion probability become very complex. An IPPS sampling scheme must satisfy the following properties viz., (i) (ii) ( 1) (iii) ( 1) The general linear form of the Horvitz and Thompson estimator can be written as 0

3 Where is a random variate that takes the value one if unit included in the sample otherwise zero. Obviously follows binomial distribution with probability Hence E ( ) V( ) (1. ) And Cov. (, ) E( ) E( ). E( ) Where is the probability that units include in the sample. is the probability that both and units include in the sample. The estimation purposed by Horvitz and Thompson estimator is given as.. (1) Taking expectation on both sides of equation (1),we get E [ ]...() HT is an unbiased estimator of Y. 1

4 Now variance purposed by Horvitz and Thompson estimator (195) is given as ( ) + (1 ) + (1 ) +. (3) Where is the joint inclusion probability of two units in the sample. Another form of variance of equation (3) proposed by Sen. and independently by Yates Grundy is given as: (4) The estimator of equation (4), proposed by Sen. and independently by Yates Grundy is given as :-

5 (5) The variance expression given by equation (4) is more popular as compared with equation (3). Both variance expression given by equations (3) and (4) are based upon and. Suitable choice of these quantities can lead to competent reduction in variance of Horvitz and Thompson(195) estimator. Survey statisticians, from time to time, has proposed number of selection schemes which can be used with estimators given by Thompson and Horvitz(195). The purpose of these selection procedure is that the variance given by (1) is minimum and variance given by equation (5) remain positive for all possible sample from a population of size N. Shahbaz and Hanif proposed a new sampling scheme for the use with estimator given by Thompson and Horvitz(195) bounded with sample size,where the first unit is draw with probability. and second unit draw ( ) with probability. Alodat(009) has proposed a selection procedure that can be used with Thompson and Horvitz(195) estimator. and. Recently Tiwari and Chilwal (013) have suggested a new sampling scheme used with Thompson and Horvitz(195) estimator, the first unit is taken with probability and draw second unit from the remaining ( N 1) units with conditional probability / An IPPS sampling scheme must satisfies the following properties () 3

6 () ( 1) (iii) n (n 1) (). The New Selection Procedure:- In this segment we introduced a new selection scheme with Thompson and Horvitz(195) estimator of size. Suppose a population of N unit is available and a sample of size is to be selected. We propose the following sampling procedure for selection of the sample. Let us suppose that the initial probability of selection of the unit u i is p i. The inclusion probability of unit u i for include in would be given by + (/) The suggested sampling scheme consist of the following steps Step:-1 Taking the 1 st unit with probability proportion to (without replacement) Step:- Select nd unit with probability proportion to By definition the inclusion probability 1 + (/) 4

7 ( ) + ( ) h ( )(1 ) ( ) 1 + ( )(1 ) ( )(1 ) 1 ( )(1 ) + ( )(1 ) 1 ( ) (1 ) + ( )(1 ) 1 ( ) (1 ) + (6) ( )(1 ) Now the joint probability of inclusion for the i th and j th unit is given as (/) + (/) ( ) (1 ) + ( ) (1 ) 5

8 1 ( ) (1 ) + 1 ( )(1 )... (7) (3). Some desirable Properties of the Proposed Scheme: - In this section we verified the some desirable properties for the quantities and under the proposed sampling scheme. These results are very important for the validity and applicability of the selection scheme. We now verify some desirable properties for the equation (6) and (7). Proof :- Now consider equation (6) 1 ( ) (1 ) + ( )(1 ) Taking summation on both sides of above we have: 1 ( ) (1 ) + ( )(1 ) 1 + ( ) (1 ) ( )(1 ) (1 ) + ( ) (1 ) 6

9 (1 ) ( ) (1 ) [] Since n, therefore the equation (6) can be written as ( 1) Proof:- Consider the equation (7) 1 ( ) (1 ) + 1 ( )(1 ) Taking summation on both sides we have: 1 ( ) (1 ) + 1 ( )(1 ) ( ) (1 ) + ( )(1 ) ( ) (1 ) + ( )(1 ) 7

10 (1 ) ( ) (1 ) + ( )(1 ) ( )(1 ) 1 ( ) (1 ) + ( )(1 ) Comparing above equation with equation (6),we get Since n, therefore the above equation can be written as ( 1 ), ( 1) Proof:- Consider the equation (7) 1 ( ) (1 ) + 1 ( )(1 ) Applying double summation on both sides of above equation, we have, 1 ( ) (1 ) + 1 ( )(1 ) 8

11 1 ( ) (1 ) + 1 ( )(1 ), 1 ( ) (1 ) + 1 ( )(1 ), Using equation (7),we get, Since n, therefore the above equation can be re write as, ( 1) Result :-4 The sen-yates Grundy variance estimator is always gives non- negative result under this sampling scheme. Proof :- For this condition we must have 0 Using below equations 1 ( ) (1 ) + ( )(1 ) 1 ( ) (1 ) + 1 ( )(1 ) and 9

12 ( )( ) (1 ( )( ) )( )+ ( )( )( )( ) (1 )( )+ 1 ( )( ) ( )( )( )( ) ( )( )( )(... (11) The expression (11) always non-negative and so Sen- Yates -Grundy estimator is non-negative under this scheme. (4).Conclusions:- In this chapter we introduced the new sampling scheme for taking a sample of size with unequal probability sampling and we found that proposed sampling scheme satisfied all the properties of IPPS sampling. The problem considered in the chapter could be extended on for a sample size more than. 30