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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
Search on the web or the library
Compare and tell me why
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
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
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
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
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
The Bayes estimator
The set of observed sample revised the
prior θ distribution
Smaller variance of posterior θ distribution
Ref. pp.274-275