Supplement 2 Comparing the two estimators of population

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Supplement 2: Comparing the two estimators of population variance by simulations Ka-fu Wong ©

Supplement 2: Comparing the two estimators of population variance by simulations Ka-fu Wong © 2007 ECON 1003: Analysis of Economic Data Lesson 0 -1

Population Variance n Average of squared deviations of values from the mean n Population

Population Variance n Average of squared deviations of values from the mean n Population variance: Where = population mean N = population size xi = ith value of the variable x Ka-fu Wong © 2007 ECON 1003: Analysis of Economic Data 2

Sample analogue of the population variance n Average of squared deviations of values from

Sample analogue of the population variance n Average of squared deviations of values from the mean n Sample variance #1: Where = arithmetic mean n = sample size Xi = ith value of the variable X Ka-fu Wong © 2007 ECON 1003: Analysis of Economic Data 3

Sample Variance n Average (approximately) of squared deviations of values from the mean n

Sample Variance n Average (approximately) of squared deviations of values from the mean n Sample variance #2: Where = arithmetic mean n = sample size Xi = ith value of the variable X Ka-fu Wong © 2007 ECON 1003: Analysis of Economic Data 4

Sample variance #1 vs. #2 n Claim: Sample variance #1 is a biased estimator

Sample variance #1 vs. #2 n Claim: Sample variance #1 is a biased estimator of population variance. Sample variance #2 is a unbiased estimator of population variance. An estimator is unbiased of the population variance only if on average, the estimator correctly estimates the population variance. Ka-fu Wong © 2007 ECON 1003: Analysis of Economic Data 5

Using Monte Carlo simulation to compare the properties of the two estimators n Create

Using Monte Carlo simulation to compare the properties of the two estimators n Create a population of 100 population values from 1 to 100 with equal chance of occurrence in a worksheet called “population. ” n Generate 1000 sample of size 10 in a worksheet called “samples” based on random sample with replacement from the population according to the probability assigned. [Tools > Data Analysis > Random Number Generation > Discrete …. ] n For each of the sample compute the variance #1 and variance #2. n Produce average of the 1000 sample variance #1 and variance #2 in a worksheet called “summary”. Result: variance. xls Ka-fu Wong © 2007 ECON 1003: Analysis of Economic Data 6

Supplement 2: Comparing the two estimators of population variance by simulations - END -

Supplement 2: Comparing the two estimators of population variance by simulations - END - Ka-fu Wong © 2007 ECON 1003: Analysis of Economic Data Lesson 0 -7