Sampling Distribution of Some Special Price Index Numbers

Transcription

1 International Journal of Management and Social Sciences Research (IJMSSR) ISSN: Volume 3,, January Sampling Distribution of Some Special Price Index Numbers Jayant Dubey, Dept. of Business Studies, Babulal Tarabai Institute of Research & Technology, Sironja, Sagar, (M.P.), India Diwakar Shukla, Professor, Department of Mathematics & Statistics, Dr. H. Gour University, Sagar (M.P.), India. ABSTRACT Index numbers are the indicator of the relative changes in the level of certain phenomenon in any given period. They give the idea of the economic pressure over the market situations and due to this they are treated as barometers of economic activity. The sampling distribution is helpful in explaining the estimates with the variability factor due to sample. This paper presents an examination of sampling distribution of Marshal-Edgeworth and Dorbish-Bowley s price index numbers. The content has graphical representation of these price indices showing the concentration of index estimate around the expected value. Some Special weighted aggregate price indices are discussed in detail in the form of sampling distributions using numerical support. The useful conclusions related to moments of the price indices are also drawn. Keywords: Index Number, Sampling Distribution, Prices,, SRSWOR, Raw and Central Moments Skewness and Kurtosis.. INTRODUCTION: Index numbers indicate the average change in a group of related variables over varying time, space or situations. They are referred to as a measure of change, a device to measure change or a series representing the process of change. A price index is a measure or function that summarizes the change in the prices of many products from one situation (a time period or place) to another situation. More specifically, for most practical purposes, a price index can be regarded as a weighted mean of the change in the relative prices of the products under consideration in the two situations. Most of the time they are helpful in providing framework for decision making and to forecast future of economy. According to Croxton and Cowden (967) Index numbers can be classified into following three categories: (a) Price Index Number (b) Index Number and (c) Value Index Number. More elaborative properties and applications about indices are described in Bradley, R. (2), Dubois (964). Gupta et. al. (24), Dasgupta (24), Fisher and Kotwal and Shinde (25) have taken into consideration the probability distributions with their relative properties. A limitation of price indices is that their calculation, for all practical purposes, are based on single sample data. If this data varies, the index computation changes accordingly. Therefore, a motivation arises to think over for obtaining databased sampling distribution of these price indices. This paper gives emphasis only on price index number with different formulae and database sampling distribution is derived to examine the moment properties. 2. SAMPLE BASED SIMULATION SCHEME: Following Ahmed et. al. (26) a scheme for sampling distribution of some special price index number is describe as under; Step - I: Draw a random sample of size n by Simple Random Sampling without Replacement (SRSWOR.) Step -II: Calculate price index numbers using different methods like Laspayre s, Paasche s, Fisher, Marshal- Edgeworth, Dorbish-Bowley s, Walsh, Kelley s etc. on these samples. Step- III: Compare these values over a large number of samples. Step-IV : Tabulate the value using class interval and frequencies. Convert frequencies into probabilities f f i i by p. i Step- V: Draw the curve between the mid values of class intervals with their respective frequencies. Step VI: Compute moments, skewness and Kurtosis and observe the tendency of the curve. Step VII: Interpret the result from the above analysis. 3.: PRICE INDEX FORMULAE USED: As per Croxton & Cowden (967) and Dubois (964) different price indices are Laspayre s, Paasche s, Fisher, Marshal-Edgeworth, Dorbish-Bowley s, Walsh, Kelley s Price Index Number. Out of which some of them are described in this section and are numerically supported.

2 International Journal of Management and Social Sciences Research (IJMSSR) ISSN: Volume 3,, January Marshal - Edge worth s Price Index Number: Let P be the price of the base year, P the price of the current year, Q quantity of the base year, Q the quantity of the current year, then Marshall and Edgeworth defined the Price Index Number as follows; P ME OR P {( Q + Q ) 2} {( Q Q ) 2} P + o X P ME P {( Q + Q )} {( Q Q )} P + o X 3.2 Dorbish - Bowley s Price Index Number: Similarly P be the price of the base year, P the price of the current year, Q the quantity of the base year, Q the quantity of the current year, then Dorbish - Bowley s defined the Price Index Number as the mean of Laspayre s and Paasche s Price Index Number and is given by the formula as; P D 2 PQ P Q + PQ P Q o X 4. EMPIRICAL STUDY: The data of 5 commodities are taken, the prices and quantities of base and current year are given and the respective values for different formulas of Marshal- Edgeworth and Dorbish - Bowley s are calculated. A sample of 54 units with 7 commodities together is drawn for convenience four entries were not considered for calculation. The details are provided in the table 4. and 4.2. The frequency distribution is prepared and is appended in table 4.3. To find the tendency of the data set the Moment about any point, Central Moments, Skewness and Kurtosis are also calculated. Graphical representation of the data is presented by table 6. and 6.2 and the graphs are drawn in section 6.. Price of Base Price of Table 4.: List of 5 commodities is taken and the calculated values are as under: of Base of Base Yrs Base Yr Yr Yr Price & Base Yr Yrs Mean of quantities of Base & Product of Base Yr Mean Quantities P Product of Yr Mean Quantities P P Q Q P Q P Q P Q P Q (Q +Q )/2 (Q +Q )/2 P (Q +Q )/ TABLE 4.2 Total 54 Samples drawn each of 7 commodities from the list of 5 Commodities: Sample P Q P Q P Q P Q P (Q +Q )/2 P (Q +Q )/2 M & E D & B to to to ,2,4,7,9,, to 6 and

3 International Journal of Management and Social Sciences Research (IJMSSR) ISSN: Volume 3,, January 24 3 Sample P Q P Q P Q P Q P (Q +Q )/2 P (Q +Q )/2 M & E D & B 6 to 6 and to 6 and to 6 and to 6 and to 6 and to 6 and to 6 and to 7 & to 7 & to 7 & to 7 & to 7 & to 7 & to 7 and & , , , , , to to 9, , , , , to , , , , to toand to, to, to To 2, To 2, to to 3, to & & 6,7,9, & 6,7, & 6,7, & 6,7, & 6,7, & 6,7, & 6,7,5, Note: Four Entries serial number 42, and 45 are not considered for calculation.

4 International Journal of Management and Social Sciences Research (IJMSSR) ISSN: Volume 3,, January 24 3 Table 4.3: Frequency and probability Distribution of price Index numbers of 5 samples considered for calculation purpose; Class Interval Frequency of Marshal & Edgeworth Index Prob. Freq/Total Freq Cumulative Frequency Frequency of Dorbish & Bowley s Index Prob. Freq /Total Freq Cumulative Frequency TOTAL SAMPLING DISTRIBUTION OF PRICE INDEX NUMBERS: With the help of Table.3 we can find out the values of Moments, Skewness and Kurtosis as under; 5. Moments of Index Numbers. After calculating the values we find the value of moments. 5..: Moments for Marshal-Edgeworth s Price Index Numbers. µ.2, µ , 3 µ , µ , µ µ 3.488, µ β.36; β ; γ.66; γ : Moments for Dorbish-Bowley s Price Index Numbers. µ.6, µ , µ , µ µ 2 2.2, µ , µ β.53; β ; γ.23; γ GRAPHICAL PRESENTATION OF THE GIVEN DATA SET: According to the scheme discussed in section 2. of step V the graphical presentation of the given data set is as under: 6. Marshal-Edgeworth Price Index Number: The graph is drawn between the mid value of class interval and the frequencies Graph 6. and 6.2. Table 6. Class Interval Mid Value Freq of Marshal & Edgeworth TOTAL 5 Graph 6.: Graph showing Marshal-Edgeworth Price Index Number Frequencies Marshal and Edgeworth Price Index Numbers Freq M& E Mid Value of Class Interval

5 International Journal of Management and Social Sciences Research (IJMSSR) ISSN: Volume 3,, January Table 6.2: Class Interval Mid Value TOTAL 5 Freq of Dorbish & Bowley Graph 6.2: Graph showing Dorbish-Bowley s Price Index Number Frequencies Dorwish and Bowley Price Index Numbers Freq DB Mid Value of Class Interval 7. CONCLUSION: For the empirical study 5 commodities are considered out of which different combinations of seven commodities are taken at a time for calculation. A sample of 54 units are drawn each having seven commodities together. Table 4. and 4.2 gives the detail of the commodities and sample respectively. With the help of table 4.2 the frequencies and probability distribution of price index number for 5 samples are prepared in table 4.3. With the help of table 4.3 Moments About Any Points, Central Moments, Skewness And Kurtosis are calculated to find out the nature of the curve. Mean of Marshal-Edgeworth is.2, Variance equals 9.84 and the mean of Dorbish-Bowley is.6 with the variability of 2.2. The variability of Marshal-Edgeworth is less and is considered to be more consistent whereas the mean of Dorbish-Bowley is less than that of Marshal- Edgeworth price Index Number. The value of Skewness of Marshal-Edgeworth is.36 and that of Dorbish-Bowley is.53 which is much lesser than Marshal-Edgeworth Price Index Number and is highly skewed as compared to Dorbish-Bowley. The value of kurtosis in both the cases is approximately same. Table 6. and 6.2 illustrates the tendency of graphs by drawing it between the mid value of the class interval and the frequencies. The graph 6. and 6.2 shows that they are Unimodal having model value 6 and 7 respectively. By analyzing the problem one can choose Marshal- Edgeworth Price Index Number as compared to Dorbish- Bowley since the variance of the formal is less than the variance of later and is considered to be more consistent. REFERENCES. Ahmad, M., Al-Tiyi, O., Al- Rawi Z. and Abu- Dayyeh, W. (26): Estimation of population mean using different imputation methods, Statistics in Transition, Vol.7, No-6, pp Bradley, R. (2): Finite Sample Effects in the Construction of Price Index, Jour. of Official Statistics, Vol. 7, 3,. 3. Croxton, F.E. and Cowden, D.J.: Applied General Statistics, Prentice-hill, 967, and Prentice-hill of India, Dasgupta, Ratan (24): Distribution of parallelism, Cal. Stat. Asso. Bull., 55, Sept-Dec., 29-22, pp Dubois, E.N.: Essential Method in Business Statistics, McGraw- Hill Fisher, Irving, The making of Index Number, Houghton, Miffin Co., Boston. 7. Gupta, R.P., Jayakumar, K. and Thomas, M. (24): On Logistic and generalized logistic distributions, Cal. Stat. Asso. Bull., 55, Sept-Dec., 29-22, pp Kotwal, K. and Shinde, R.L. (25): Markov Binomial and trinomial distributions, Jour. Ind. Stat. Asso., Vol 43, -, pp Dubey, Jayant and Shukla, Diwakar (28): Simulated Sampling Distribution of Price Index Numbers, Inter-Stat Journal, Statistics on the Internet, Vol 5(5), pp-8.