Confidence Interval & Unbiased Estimator

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

Cont.

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 couldalso befound. 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 (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 • 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 • The Bayes estimator • The set of observed sample revised the prior θ distribution • Smaller variance of posterior θ distribution • Ref. pp.274-275

Confidence Interval & Unbiased Estimator