Ratio Estimator In Simple Random Sample

 

                                                     RATIO ESTIMATORS IN SIMPLE RANDOM SAMPLE

-Faiza Rahman

2048121

 

Let’s see when does ratio method of estimation is required.

In many surveys the information on auxiliary variate which is highly correlated to the variable under study is readily available and it can be used to improve the sampling design. This auxiliary information can be used  to construct various sampling deigns such as probability proportional to size with replacement (PPSWR), probability proportional to size without replacement (PPSWOR),stratified sampling etc. to yield efficient estimators for population parameters (e.g., total, mean, and proportions). In these schemes the information on auxiliary variate on individual sample units is available prior to preparation of the sampling design. However these sampling schemes cannot be used when data on auxiliary variate for individual sampling units are not available but only the aggregated value for all units of auxiliary variate is available, that is when ratio method of estimation comes into picture.

An Introduction to ratio method of estimation.

When the information on auxiliary variate which is highly correlated with our characteristic of interest is available then we go for ratio method of estimation. More precisely, the situations wherein the aggregated data on auxiliary variate is available which can be used in the time of estimation of parameters and the data on auxiliary variate for sampled units can be easily obtained while recording the values of the study variate then we go for either of the two methods i.e.

1.     Ratio method of estimation

2.      Regression method of estimation

But, here we are only interested in ratio method of estimation.

What is a ratio estimator?

The ratio estimator is a statistical parameter and is defined to be the ratio of means of two random variables. Ratio estimates are biased and preferred over other unbiased estimators since they are comparatively more efficient. They are used in order to obtain higher precision without changing the cost of the survey or sampling design.

Ratio estimator in simple random sampling.

In ratio method of estimation in SRS a sample of size n are drawn from a population of size N using simple random sample using without replacement scheme which more preferable because it is estimates obtained using without replacement are more efficient than estimates obtained using SRS with replacement technique and then the parameters are estimated using ratio estimators.

Notations and Terminologies

Let us denote

yi is the value of the characteristic under study for the ith unit of the population.

xi is the value of the auxiliary characteristic on the same unit.

Y is the total of y characteristic of the population.

X is the total of x characteristic of the population.

R is the ratio of the population totals or means of character y and x.

ρ is the correlation coefficient between x and y in the population.






The ratio estimator of population ratio is 


 

The ratio estimator of population total is

   

The ratio estimator of population mean is

 

and

Where y and x are the sample totals for y and x, respectively. 

Important Formulae

·         Bias of ratio estimators





               Approximate variance and mean square error of ratio estimate

         In a simple random sampling of size n (n large )

     

 

Where f=n/N is the sampling fraction.

·         First order approximation of the variance of ratio estimates.



Where,

ρ is the population correlation coefficient between X and Y.

Sx  is the standard deviation of X

Sy  is the standard deviation of Y.

Note: If we are interested in the sample estimate of variance we replace standard deviation, correlation coefficient by their sample estimates.

Applications and examples of Ratio estimators in simple random sample.

Ratio method of estimation is basically applied to the data wherein information on auxiliary variate is given and auxiliary variable is highly correlated to the variable of interest.

For example,

1.      The data consisting of sex and height of a person. Here height is our variable of interest and sex is the auxiliary variate.

2.      Amount of milk produced and a particular breed of cow.

3.      Amount of yield of wheat crop and a particular variety of wheat.

4.      No. of plants and production of apples in each orchards. This example is used to illustrate how to estimate the parameter using ratio estimators in simple random sample. The analysis (in R) of the same is given below,

AIM:

Suppose a sample of 15 orchards was selected from 125 orchards of a certain locality by SRSWOR method. We want to

(i)        Estimate the average yield of apple per plant and its standard error.

(ii)      Estimate the mean yield of apple per orchard and obtain 90% confidence interval of the mean yield when it is known that the average number of plants per orchard is 40.

(iii)   Estimate total production of apples and it’s standard error.

Here,

X denotes number of plants in each orchard.

Y denote production of apples in each orchard.

ANALYSIS:









Hence, we could easily find all the estimates using ratio estimators in R.

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