Ratio Estimators in Stratified Sampling


 

                   Ratio Estimators in Stratified Sampling

                               

                                - Amala Johnson(2048112)

Introduction:

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 corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical and symmetrical tests such as the t test should not be used to generate confidence intervals.

The ratio estimator was observed to be more precise than the usual sample mean estimator under different conditions for estimating the population mean of the study character. Several researchers diverted their attention in the direction of using prior value of certain population parameters to find the estimates that are more precise. Searls (1964) used coefficient of variation of study character at estimation stage. In practice, coefficient of variation is seldom known. Motivated by Searls (1964) work, various authors including Sen (1978), Sesodiya and Dewivedi (1981) Singh et al (1991) and Upadhyaya and Singh (1984) used the known coefficient of variation of auxiliary character for estimating population mean of the study character in ratio method of estimation. Singh et al (1973) first made the use of prior value of coefficient of kurtosis in estimating the population variance of study character. Later used by Searls and Interapanich (1990). Recently Singh and Tailor (2003) proposed a modified ratio estimator by using the known value of correlation coefficient. Taking into consideration the point biserial correlation coefficient between auxiliary attribute and study variable, Jhajj et al (2006) and Singh, et al (2008) defined ratio estimators of population mean when the priori information on auxiliary variable possessing some attribute is available.

Some ratio-type estimators have been proposed in stratified random sampling using auxiliary attribute. The expressions for the bias and mean square errors of the proposed estimators have been derived up to first order of approximation. Comparisons have been made with traditional combined ratio estimator and it is shown that the proposed estimators are more efficient than combined ratio estimator under certain condition.

When using ratio estimation with stratified random sampling, there are two different ways to produce estimates. One way is to perform ratio estimation separately in each stratum, and then combine them. This gives a separate ratio estimator. The second way is to compute estimators for µy and µx using estimators for stratified random sampling, and then use y(bar)st/x(bar)st as a ratio estimator of µy/µx . This gives a combined ratio estimator.

v  Separate ratio-type estimators for population mean with their properties are considered. Some separate ratio-type estimators for population mean using known parameters of auxiliary variate are proposed. The bias and mean squared error of the proposed estimators are obtained up to the first degree of approximation. It is shown that the proposed estimators are more efficient than unbiased estimators in stratified random sampling and usual separate ratio estimators under certain obtained conditions.

v  Combined ratio estimator:It is assumed, in case of seprate estimator that nm ‘s were large in each stratum.However,it may not hold good always in practice.

Generally, the concern with the separate ratio estimator is that with small sample sizes per stratum, the individual stratum variance estimates will be biased, and that bias is added across strata. Thus it is recommended to use the separate ratio estimator unless the stratum sizes are small, say (ni < 20), or if the within-stratum ratios are approximately equal. Estimates of population totals are obtained by multiplication by the population size N, giving ˆτyRS = NµˆyRS or ˆτyRC = NµˆyRC .

Important formulas

Estimating the Ratio

                              


                  

In a situation with two strata (labeled A and B), the expression for estimating a mean using the separate ratio estimator is

            


        

with estimated variance:

                       


In a situation with two strata (labeled A and B), the expression for estimating a mean using the combined ratio estimator is:

                     


with estimated variance:

                    


 

Advantages and Disadvantages of Ratio Estimator in stratified sampling:

Advantages

Ø  Helps in forecasting and planning by performing trend analysis.

Ø  Helps in estimating budget for the firm by analyzing previous trends.

Ø  It helps in determining how efficiently a firm or an organization is operating.

Ø  It provides significant information to users of accounting information regarding the performance of the business.

Ø  It helps in comparison of two or more firms.

Disadvantages:

Ø  Financial statements seem to be complicated.

 

Ø  Several organizations’ work in various enterprises each possessing different environmental positions such as market structure, regulation, etc., such factors are important that a comparison of 2 organizations from varied industries might be ambiguous.

 

Application:

Suppose we want to estimate the average number of trees per acre on a 1000-acre plantation. The investigator samples 10 one-acre plots by simple random sampling and counts the number of trees (y) on each plot. She also has aerial photographs of the plantation from which she can estimate the number of trees (x) on each plot of the entire plantation. Hence, she knows μx = 19.7 and since the two counts are approximately proportional through the origin, she uses a ratio estimate to estimate μy.


                                         

                                         

                                           

                            

Conclusion:

We have developed some ratio-type estimators in stratified random sampling for estimating population mean by using information on auxiliary attributes. By comparison, it is found that the proposed estimators are more efficient and less bias than the traditional combined ratio estimator is. These theoretical conditions are also satisfied by the results of an application with original data. For practical purposes, the choice of the estimator depends upon the availability of the population parameters.










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