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### Transcript of Lecture 27+. The Bootstrap Method: Simulation zuev/teaching/2013Spring...آ  Math 408 -...

• Math 408 - Mathematical Statistics

Lecture 27+. The Bootstrap Method: Simulation Results

April 8, 2013

Konstantin Zuev (USC) Math 408, Lecture 27+ April 8, 2013 1 / 7

• Example: Gaussian Model Suppose that:

X1, . . . ,Xn ∼ N (µ, σ2), true values: µ = 1 and σ = 2 Exact Confidence Intervals:

µ : µ̂MLE ± 1√ n − 1

σ̂MLEtn−1(α/2) σ 2 :

( nσ̂2MLE χ2n−1(

α 2 ) ,

nσ̂2MLE χ2n−1(1− α2 )

)

Konstantin Zuev (USC) Math 408, Lecture 27+ April 8, 2013 2 / 7

• Bootstrap

Konstantin Zuev (USC) Math 408, Lecture 27+ April 8, 2013 3 / 7

• Confidence Intervals for µ, n = 100

Konstantin Zuev (USC) Math 408, Lecture 27+ April 8, 2013 4 / 7

• Confidence Intervals for σ, n = 100

Konstantin Zuev (USC) Math 408, Lecture 27+ April 8, 2013 5 / 7

• Confidence Intervals for µ, n = 1000

Konstantin Zuev (USC) Math 408, Lecture 27+ April 8, 2013 6 / 7

• Confidence Intervals for σ, n = 1000

Konstantin Zuev (USC) Math 408, Lecture 27+ April 8, 2013 7 / 7