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Law of Large Numbers (LLN) and Central Limit Theorem (CLT)

Law of Large Numbers (LLN) and Central Limit Theorem (CLT). Consistency and LLN. As shown in class, a law of large numbers is a powerful theorem that can be used to establish the consistency of an estimator.

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Law of Large Numbers (LLN) and Central Limit Theorem (CLT)

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  1. Law of Large Numbers (LLN)andCentral Limit Theorem (CLT) Econ 472

  2. Consistency and LLN • As shown in class, a law of large numbers is a powerful theorem that can be used to establish the consistency of an estimator. • We illustrate what we mean by consistency by showing what happens to the sampling distributions of sample averages as the sample size tends toward infinity. Econ 472

  3. Consistency and LLN • We again consider the case of random (iid) sampling from a uniform distribution. • We obtain 5,000 random samples of sizes n = 1,2,5,50 and 1,000. • For each experiment, we calculate the sample average of the drawn values. Doing this 5,000 different times (for each sample size n) enables us to characterize the sampling distributions of the estimators. Econ 472

  4. Consistency and LLN • The following 5 slides present those sampling distributions for n = 1,2,5,50 and 1,000. Econ 472

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  10. Results • As we can see from the progression of these slides, the sampling distribution collapses around the population average, (i.e., .5), as n approaches infinity. This is what we mean by the consistency of the sample average under iid sampling. • We also see that the Normal approximation to the sample average appears to work well for moderate to large n, but not so well for very small n. Econ 472

  11. Central Limit Theorem Econ 472

  12. CLT • As suggested by the last point, the central limit theorem is a powerful statistical tool that can be used to establish that the sampling distribution of the standardized sample average converges to a standard Normal distribution as the sample size n !1 Econ 472

  13. CLT, continued • By standardized sample average, we mean taking the sample average, subtracting off its mean, and then dividing through by its standard deviation. • Since Econ 472

  14. CLT, continued • the CLT can be used to establish that: • Where “!” means “converges to as n approaches 1” and N(0,1) denotes a standard Normal distribution. Econ 472

  15. CLT, continued • To demonstrate this convergence, we again illustrate with random sampling from a uniform distribution. • We obtain random samples of sizes n=1,2,5 and 1,000 from the uniform distribution, and calculate the sample average and the standardized sample average. • We do this 5,000 times for each sample size. Econ 472

  16. CLT, continued • We then characterize the sampling distributions of the standardized sample averages and compare them to the standard Normal distribution. • The results of this exercise are found on the following 4 slides: Econ 472

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  21. CLT results • As you can see, for very small n, the Normal approximation is not very accurate. • For this exercise, the normal approximation is reasonable even for n=5. Econ 472

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