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Confidence Interval & Unbiased Estimator Review and Foreword.

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Presentation on theme: "Confidence Interval & Unbiased Estimator Review and Foreword."— Presentation transcript:

1 Confidence Interval & Unbiased Estimator Review and Foreword

2 Central limit theorem vs. the weak law of large numbers

3 Weak law vs. strong law Personal research Search on the web or the library Compare and tell me why

4 Cont.

5 Maximum Likelihood estimator Suppose the i.i.d. random variables X 1, X 2, … X n, whose joint distribution is assumed given except for an unknown parameter θ, are to be observed and constituted a random sample. f(x 1,x 2,…,x n )=f(x 1 )f(x 2 )…f(x n ), The value of likelihood function f(x 1,x 2,…,x n / θ ) will be determined by the observed sample (x 1,x 2,…,x n ) if the true value of θ could also be found. Differentiate on the θ and let the first order condition equal to zero, and then rearrange the random variables X1, X2, … Xn to obtain θ.

6 Confidence interval

7 Confidence vs. Probability Probability is used to describe the distribution of a certain random variable (interval) Confidence (trust) is used to argue how the specific sampling consequence would approach to the reality (population)

8 100(1-α)% Confidence intervals

9 100(1-α)% confidence intervals for (μ 1 - μ 2 )

10 Approximate 100(1-α)% confidence intervals for p

11 Unbiased estimators

12 Linear combination of several unbiased estimators If d 1,d 2,d 3,d 4 … d n are independent unbiased estimators If a new estimator with the form, d=λ 1 d 1 +λ 2 d 2 +λ 3 d 3 + … λ n d n and λ 1 +λ 2 + … λ n =1, it will also be an unbiased estimator. The mean square error of any estimator is equal to its variance plus the square of the bias r(d, θ)=E[(d(X)-θ)2]=E[d-E(d)2]+(E[d]-θ)2

13 The Bayes estimator

14 The value of additional information The Bayes estimator The set of observed sample revised the prior θ distribution Smaller variance of posterior θ distribution Ref. pp.274-275


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