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Confidence Interval &
Unbiased Estimator
Review and Foreword
Central limit theorem vs. the
weak law of large numbers
Weak law vs. strong law

Personal research

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Search on the web or the library
Compare and tell me why
Cont.
Maximum Likelihood estimator
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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.
^
the maximum likelihood estimator of  , denoted by  , would maximize
the probabilit y of likelihood function of observed values
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.
Confidence interval
Confidence vs. Probability
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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)
100(1-α)% Confidence intervals
100(1-α)% confidence
intervals for (μ1 -μ2)
Approximate 100(1-α)%
confidence intervals for p
Unbiased estimators
Linear combination of several
unbiased estimators

If d1,d2,d3,d4…dn are independent unbiased
estimators
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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
The Bayes estimator
The value of additional
information
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The Bayes estimator
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The set of observed sample revised the
prior θ distribution
Smaller variance of posterior θ distribution
Ref. pp.274-275