Download PowerPoint Show: Law of Large Numbers and Central Limit Theorem

Law of Large Numbers (LLN)
and
Central Limit Theorem (CLT)
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Econ 472
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.
Tobias
Econ 472
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.
Tobias
Econ 472
Consistency and LLN
• The following 5 slides present those sampling
distributions for n = 1,2,5,50 and 1,000.
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Econ 472
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Econ 472
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Econ 472
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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.
Tobias
Econ 472
Central Limit Theorem
Tobias
Econ 472
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
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Econ 472
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
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Econ 472
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.
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Econ 472
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.
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Econ 472
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:
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Econ 472
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Econ 472
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Econ 472
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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.
Tobias
Econ 472