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

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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