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Lecture8_412 - Point Estimators MLE MOME Point Estimators...

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1 Point Estimators, MLE & MOME Point Estimators Example 1. Let X 1 , X 2 , …, be a random sample from N( ) . Please find a good point estimator for 1. 2. Solutions. 1. 2. There are the typical estimators for and Both are unbiased estimators . Property of Point Estimators Unbiased Estimators. is said to be an unbiased estimator for if E( ) . E( )= (*Please make sure you can prove this on your own. We had discussed this before.) Unbiased estimator may not be unique. Example 2. E[ ]= Let Variance of Unbiased Estimators. Methods for deriving point estimators 1. Maximum Likelihood Estimator (MLE) 2. Method of Moment Estimator (MOME) Example 1 (continued). ,i=1,2, ,n
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2 1. Derive the MLE for . 2. Derive the MOME for . Solution. 1. MLE [i] pdf: f( )= [ii] likelihood function [iii] log likelihood function [iv] Find the MLE’s R. A. Fisher ( http://en.wikipedia.org/wiki/Ronald_Fisher ) established the comprehensive theory for the maximum likelihood method.
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3 2. MOME Order Population Moment Sample Moment 1 st E(X) 2 nd k th Back to this example, we have two parameters, and we set up the equations for the first two
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