CORNELL ECON 3120 - Random Sampling (2 pages)

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Random Sampling

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Random Sampling


IV. Random Sampling V. Estimators VI. Large Sample Estimators

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Lecture Note
Cornell University
Econ 3120 - Applied Econometrics

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Econ 3120 1st Edition Lecture 4 Outline of Last Lecture I Covariance II Conditional Expectation III Normal Distribution Outline of Current Lecture IV Random Sampling V Estimators VI Large Sample Estimators Current Lecture Random sampling This lecture discusses how to estimate the mean from a population of interest Normally it is impractical or impossible to examine the whole population If we could do this we could simply take the mean of the population and we d be done Given that we can only examine a sample we have to use statistical inference to 1 estimate the parameters that we care about in this case it s the mean and 2 test hypotheses about these parameters To estimate the mean or any other parameter of interest we ll focus on random samples from the population Formally a random sample is a set of independent and identically distributed i i d random variables Y1 Y2 Yn that share a probability density function f y For the first part of the lecture we re going to assume that the population is distributed Normal 2 an assumption we will relax when we discuss large sample properties later in the lecture 2 Estimators Once we have our random sample we can use it to estimate the mean or any other parameter of interest An estimator is a rule or a function that uses the outcome of random sampling to assign a value to the parameter based on the sample h Y1 Y2 Yn For the mean the most obvious estimator is the sample mean or sample average Y 1 n n i 1 Yi We often write estimates by putting a hat on top of the parameter i e 1 Y Note that this is just one possible way to estimate the mean We could estimate the mean by simply looking at the first observation i e 2 Y1 In fact an estimator doesn t even need to depend on the random sample at all 3 4 is a perfectly valid estimator We ll see though that the sample mean is preferable because it is unbiased and efficient Estimators like the samples they come from have their own distributions called sampling distributions Estimators

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