RATIO METHOD OF ESTIMATION IN SAMPLE SURVEY

RATIO METHOD OF ESTIMATION IN SAMPLE SURVEY

POURAB BHATTACHARJEE

2148109

 An important goal of every statistical estimate technique is to get more precise estimators of parameters of interest. It is also well established that include additional data in the estimate technique results in better estimators, assuming the data is genuine and correct. To generate a better estimator of the population mean, the ratio technique of estimation is used to incorporate such auxiliary data. Auxiliary information on a variable, which is linearly related to the variable under investigation and is used to estimate the population mean, is provided in the ratio method of estimation.

Let Y be the research variable, and X be an auxiliary variable that is associated with Y. For each sampling unit, the observations x_i on X and y­_i on Y are obtained. It is necessary to know the population mean X̄ of X (or, equivalently, the population total, X_tot). x_i' s, for example, may be the values of y_i' s from-

- some earlier completed census,

- some earlier surveys,

- some characteristic on which it is easy to obtain information etc.

 

If y_i is the quantity of fruits produced in the i ^th plot, then x_i can be the area of the i ^th plot or the previous year's fruit production in the same plot.

Let (x₁, y₁), (x₂, y₂),...,(x_n, y_n) be a random sample of size n on the paired variable (X, Y) chosen from a population of size N, ideally using SRSWOR. The ratio estimate of  population mean Y is

 

Where the population mean X̄ is assumed to be known. The ratio estimator of population total,

Where X_tot is the population total which is assumed to be known,

are the sample totals of Y and X respectively.

The ratio technique estimates the relative change X_tot /Y_tot that occurred after (x_i,y_i) were observed, as seen in the structure of ratio estimators. It is obvious that if the fluctuation between the values of y_i/x_i is roughly the same for all i = 1,2,...,n, then the values of x_tot/y_tot fluctuate little from sample to sample, and the ratio estimate will be accurate.

BIAS AND MEAN SQUARED ERROR OF RATIO ESTIMATOR:

Let us assume that, by SRSWOR, a random sample (x_i, y_i),i=1,2,...,n is drawn and that the population mean X is known. Then

 

Moreover, it is difficult to find the exact expression for

So they are approximated and we proceed as follows

Let,

Since we are following SRSWOR here, so

where,

and  ,   

 is the coefficient of variation related to Y.

Similarly,

where C_x= S_x/ X̄ is the coefficient of variation related to X and r is the population correlation coefficient between X and Y.

We write ,

in terms of  e 's,

If  |e₁| is assumed, the term (1+e₁)^-1 can be stretched into an infinite series that is convergent. According to this assumption,

i.e., a potential estimate x̄ of the population mean X̄, is between 0 and 2 X̄. If the fluctuation in x̄ isn't too significant, this is likely to hold. Assume that the sample size n is quite large to ensure that the variation in x̄ is minimized. With this assumption,

Therefore , the error of ratio estimation of population mean is ,

When the sample size is big, e₀ and e₁are likely to be small quantities, and the terms involving the second and higher powers of e₀ and e₁would be negligible. In that case,

And

So, up to the first order of approximation, the ratio estimator is an unbiased estimator of the population mean.

If we assume that only terms of e₀ and e₁ involving powers of more than two are negligibly small (which is more reasonable than supposing that powers of more than one are negligibly small), we may approximate the estimation error of

as follows.

The bias of  the ratio estimator of population mean is given by,

upto the second order of approximation. The bias generally decreases as the sample size increases.

Now, the bias is zero ,i.e.,

When the regression line of Y on X crosses through the origin, this condition is satisfied.

Now consider the following to determine the mean squared error:

upto the second order of approximation.

Upper limit of ratio estimator:

Let us consider,

Thus,

Thus,

where C_X is the coefficient of variation of X. If C_X < 0.1, then the bias in  may be safely regarded as negligible in relation to the standard error of .

Confidence interval of ratio estimator:

If the sample size is large enough to use the normal approximation, the 100(1-α) percent confidence intervals for Ȳ and  are

is approximately N(0,1).

Conditions under which the ratio estimate is optimum:

The best linear unbiased estimator of Ȳ  is the ratio estimate

when,

1.    the relationship between y_i and x_i is linear passing through the origin, i.e. y_i= 𝛃x_i + e_i, where e_i's are independent with E(e_i/x_i)=0 and 𝛃is the slope parameter, and

2.   this line is proportional to x_i, i.e. Var (y_i/x_i)= E(e_i²)=Cx_i, where C is a constant.

APPLICATION

Now that we have seen the theory of ratio estimation, let us look at one of its applications.

We will be using R-programming language codes to show and explain a practical example of ration estimation.

OBJECTIVE: To estimate the average real estate farm loans assuming that the average nonreal estate farm loans in the country is known and is equal to $878.16. Also using the ratio estimator to give the estimates with 95% confidence interval for this data set and discuss the results. Notation X - Nonreal estate farm loans Y - Real estate farm loans Data

DATA DESCRIPTION: Given below is a random sample of 21 states from a population of 50 states of a country using SRSWOR.

 

        

         

        

        

            

n

Conclusion: The data shows a moderately high correlation between the variables. We obtain the ratio estimate as 0.6699792 with a standard error of 0.1380009. The estimate of the mean is $588.3489 with the standard error estimate of 121.1869. The 95% confidence interval is (335.5576, 841.1403). With the given value of the estimator of the mean, we can conclude that the population mean lies within these values