COMPARISION OF RATIO AND REGRESSION ESTIMATOR UNDER SRSWOR -2148139

 ESTIMATING BOLLYWOOD BOX OFFICE REVENUE USING RATIO AND  REGRESSION ESTIMATOR 

LAXIA VIOLA PEREIRA 

        2148139      

India is the largest producer of films in the world .The Hindi Film Industry Popularly known as the Bollywood is one of the biggest industries in India. We have used a data set on revenue of 190 Bollywood films and their gross and budget .In order to know what makes movies successful and what factors allow filmmaker to use their resources to create as much as revenue possible we use methods of estimation. Using Ratio and Regression Estimation we will determine the factors that are corelated with Box Office Returns. We also Find best estimator to see whether there exists Linear Relationship Between Gross and Budget.

     1.   INTRODUCTION

Ratio and regression estimators produce more precise estimates than the ordinary mean by invoking an auxiliary variable x in the estimation process. These estimators provide more efficient estimates among those that utilize auxiliary variables. However, these estimators rely heavily on certain assumptions, e.g., linear dependence of the target variable y on the auxiliary variable x. This study aims to compare the two estimators, in terms of bias and coefficient of variation, in estimating the population mean. The comparison will provide insights on scenarios where one estimator outperforms the other. We will use both the estimators and check the better one for our Research.

      2. DATA DESCRIPTION

This study uses a cross-section of data from the top 190 Bollywood films from 2013-2017 in INDIA .This data set is taken from the Source http://www.bollymoviereviewz.com. We have 3 variables in this data set Movie Gross and Budget. Firstly 190 box office hit Bollywood movie names are taken for this project. The dependent variable (y) is gross domestic box office revenue, is measured in crores and was pulled from the Internet Movie Database. This is solely revenue from a film’s first run in theaters. Here the independent variable (x) is the production budget for films was also found on the Internet Movie Database. It is measured in crores and it is expected to have a positive coefficient. Increasing the budget is expected to increase gross revenues but at a diminishing rate. Movie production budgets make for an interesting variable because they include actors’ wages, costs of special effects, and other factors designed to increase the value of the movie. The gross revenue of a preceding movie was found using the Internet Movie Database. If a movie was a sequel, then the preceding movie’s gross box office performance was recorded. This is expected to have a positive coefficient because past success may indicate future success.

    3. METHOD OF ESTIMATION

RATIO ESTIMATION

Ratio estimator uses auxiliary information about the population to estimate the unknown parameter of interest. The linear relationship between the auxiliary variable and the variable of interest increases the precision of the estimate. The ratio estimate  of the population mean Y bar in simple random sampling is Given as  

where y is the sample mean of Y , and x and X are the sample and the population

means of X, respectively. Its variance is defined as

Cx is the coefficient of variation of xi

Cy is the coefficient of variation of yi

ρ is the sample correlation value between x and y

y is the sample mean

Ratio Estimation in Stratified Random Sampling Consists of two different methods to construct estimators of a ratio in stratified sampling.

• Separate Ratio Estimator: Estimate the ratio of µy to µx within each stratum and then form a weighted average of the separated estimates.

• Combined Ratio Estimator: Compute the usual yst and xst, then use their quotient as an estimator of 

• If the stratum sample sizes are large (more than 20) it is better to use separate ratio estimators. Otherwise, if the sample sizes are small or the within-stratum ratios are approximately equal, it is better to use combined ratio estimators.

                                                                                   REGRESSION ESTIMATION

If the study variate (y) is approximately a constant and a multiple of the auxiliary variate, it is more precise to estimate the population mean or total by fitting a linear regression. Such an estimator is called a regression estimator. Like the ratio estimator, the regression estimator is not unbiased for the  population mean or total .Regressions analysis uses data, specifically two or more variables, to provide some idea of where future data points will be. The benefit of regression analysis is that this type of statistical calculation gives businesses a way to see into the future.

Regression Estimator is the linear regression estimator is one of the

most commonly used estimators of Y given an auxiliary variable. The linear

regression estimator is defined by

where

is the sample regression coefficient of y on x. The variance is best approximated by

 

where 

      4. ANALYSIS OF  DATA USING R 

Here We select a random sample using the SRSWOR procedure of 190 Bollywood movies which is collected from data set of 300 movies. Here we are considering Production budget as independent and Gross Revenue as Independent .We Find best method of estimation using different packages in R.

OBTAING CORRELATION BETWEEN PRODUCTION BUDGET AND GROSS REVENUE

The correlation between Gross Revenue and Production budget  is 0.6821581. We can conclude High Degree correlation.

          1. RATIO ESTIMATION

OBTAINING SAMPLE SURVEY ANALYSIS IN RATIO ESTIMATION    

We got the ratio estimators of budget/gross using ratio estimation as 0.8475796 and standard error as 0.05865716.

OBTAINING ESTIMATED AVERAGE OF GROSS REVENUE AND STANDARD ERROR USING RATIO ESTIMATION

The estimated mean gross value is 34.32697 crores and The standard error using ratio estimation is 2.654391 crores.

             2. REGRESSION ESTIMATION

OBTAINING ESTIMATED AVERAGE OF GROSS REVENUE USING REGRESSION ESTIMATOR

The estimated average gross revenue is 47.20725 crores

OBTAINING VARIANCE AND STANDARD ERROR USING REGRESSION ESTIMATOR

The Variance of Gross Revenue is 4.9148 using regression estimator and  standard error through regression estimation is 2.216937

OBTAINING REGRESSION COEFFICIENT

We found the value of the regression coefficient as 1.301 crores.

   5 .CONCLUSION

(1) The regression coefficient 1.301 crores shows that there is a positive correlation between Gross revenue of Bollywood movies and their budget. This infers that the coefficient value signifies how much the mean of the Gross Revenue changes given a one-unit shift in the Production Budget while holding other variables in the model constant. 

(2) The estimated average gross revenue is 47.20725 crores using regression estimation .The estimated average gross revenue is 34.32697 crores using ratio estimation.

(3) The standard error through regression estimation is 2.216937 and the The standard error through ratio estimation is 2.654391.

 

The mean value of Gross revenue of films 

(y) with ratio estimation is  

34.32697 crores 

.

With the given value of 

the estimator of the mean 40.5 crores, we can conclude that the 

population mean lies within these values. We can see that 

standard error in regression estimation is smaller than that in 

ratio estimation, So regression estimation gives greater 

precise 

value than ratio estimation. Hence Regression estimator is the best and efficient estimator for evaluating box office revenue .