confidence interval & unbiased estimator review and foreword
TRANSCRIPT
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Confidence Interval & Unbiased Estimator
Review and Foreword
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Central limit theorem vs. the weak law of large numbers
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Weak law vs. strong law
Personal research Search on the web or the library Compare and tell me why
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Cont.
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Maximum Likelihood estimator
Suppose the i.i.d. random variables X1, X2, …Xn, whose joint distribution is assumed given except for an unknown parameter θ, are to be observed and constituted a random sample.
f(x1,x2,…,xn)=f(x1)f(x2)…f(xn), The value of likelihood function f(x1,x2,…,xn/θ) will be determined by the observed sample (x1,x2,…,xn) if the true value of θ could also be found.
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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.
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Confidence interval
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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)
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100(1-α)% Confidence intervals
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100(1-α)% confidence intervals for (μ1 -μ2)
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Approximate 100(1-α)% confidence intervals for p
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Unbiased estimators
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Linear combination of several unbiased estimators
If d1,d2,d3,d4…dn are independent unbiased estimators If a new estimator with the form, d=λ1d1+λ2d2+λ3d3+…
λndn and λ1+λ2+…λn=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
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The Bayes estimator
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The value of additional information
The Bayes estimator The set of observed sample revised the p
rior θ distribution Smaller variance of posterior θ distributi
on Ref. pp.274-275