MIDZUNO SAMPLING SCHEME

 

INTRODUCTION

 A finite population is nothing but a set containing finite number of distinguishable elements. The number of elements in a finite population is called population size and it is denoted by N and it is always a known finite number .A sample is nothing but a subset of the population S. Usually it is denoted by s. Choosing a subset of the population according to a probability sampling design is called probability sampling. probability sampling can be done in equal and unequal ways. For equal probability sampling the samples taken from population have equal probability of selection and for unequal probability sampling If the selection probabilities are unequal, the sample mean is not unbiased for population mean and sample total is not unbiased for population total. Under this unequal probability sampling Midzuno sampling falls under. its also known as Midzuno Zen Sampling.

                                  The famous Midzuno Sampling was proposed by the statistician Midzuno H and A.R. Zen in 1952. This sampling design is more elaborated in the research paper "On the sampling system with probability proportional to sum of sizes ".Ann. Inst. Statist. Maths.,3, 99–107. The Midzuno Sen scheme of sampling describes the selection of the first unit with probability proportional to a given size measure (x) and remaining (n-1) units of the sample by srswor. that is it's a sampling algorithm to select a sample with unequal probabilities, while estimating unbiasedly a ratio. It may be of interest with a moderate sample size if the small sample bias can be appreciable. Midzuno sampling has been recently considered to an appllication in Escobar and Berger (2013) and Hidiroglou (2016) studies.

METHODOLGY

 

                                 

The above sampling design can be implemented by using the following sampling method. To draw a sample of size n. one unit is drawn by probability proportional to size method and from the remaining (N -1) units a simple random sample of size (n -1) will be drawn.

DERIVATION OF INCLUSION PROBABILITES UNDER MS SAMPLING

1)Under Midzuno sampling design the first order inclusion probabilities are

The following theorem gives the second order inclusion probabilities under Midzuno scheme.

2)Under Midzuno sampling design, the second order inclusion probabilities are 

Thus we have derived the first and second order inclusion probabilities under Midzuno sampling scheme. These expressions can be used in the Horvitz Thompson estimator to estimate the population total and derive the variance of the estimator. Midzuno sampling design is one in which the Yates-Grundy estimator of variance is non-negative.

APPLICATION OF MIDZUNO SAMPLING

1) In adaptive cluster sampling with unequal probability sample without replacement MS sampling           design employed for having accurate results.

2) To derive that Yates-Grundy estimator is non-negative under Midzuno sampling design.

    Proof 

FRANK YATES

    A set of sufficient conditions for the nonnegativity of Yates-Grundy estimator are          given by

     Using the earlier expressions we have 

    Since the right hand side of the above expression is non-negative, we conclude that the Yates Grundy      estimator is always nonnegative

MIDZUNO SAMPLING SCHEME IN R PROGRAMING

• INTRODUCTION

To use the MS in R environment we can  go by command Pmidzuno. This command uses  Midzuno method to select a sample of units (unequal probabilities, without replacement, fixed sample size). Its usage is given by UPmidzuno(pik,eps=1e-6) where pik is the vector of the inclusion probabilities and  eps the control value, by default equal to 1e-6. This value Returns a vector (with elements 0 and 1) of size N, the population size. Each element k of this vector indicates the status of unit k (1, unit k is selected in the sample; 0, otherwise). The value ’eps’ is used to control pik (pik>eps & pik < 1-eps).

       Lets Consider a dataset of of 22 lime growing villages in one of the tehsils of Bangalore district. There are 2 variables in the data that is the number of bearing lime trees and area reported under limes. Let us estimate the average number of bearing lime tress using exact variance estimate- Horvitz-Thompson by performing a Midzuno Sampling Scheme and then we conclude the findings.

• ANALYSIS OF R CODES 

• CONCLUSION

The estimated average number of bearing lime trees is 1476.636 tress and the standard error of it is 60.62787. The population mean using the Horvitz-Thompson estimator is 4503.097, $y_{HT}=$1476.636 and Std Error=60.62787.

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