Lect15_2700_s09 - Standard Errors Methods: mome Methods:...

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Unformatted text preview: Standard Errors Methods: mome Methods: MLE Confidence Intervals Home Page Title Page JJ II J I Page 1 of 26 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 15: Point Estimation: Methods David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA dm484@cornell.edu March 23, 2009 Standard Errors Methods: mome Methods: MLE Confidence Intervals Home Page Title Page JJ II J I Page 2 of 26 Go Back Full Screen Close Quit 1. More on standard errors. Consider the following examples. Example 1. Suppose X 1 , . . . , X n iid N ( , (3 . 5) 2 ) . Estimate with = X with SE = q Var ( X ) = r 1 n Var( X 1 ) = 3 . 5 n . For instance, if n = 10, then SE = 3 . 5 10 = 1 . 11 . On the other hand, if X 1 , . . . , X n iid N ( , 2 ) , Standard Errors Methods: mome Methods: MLE Confidence Intervals Home Page Title Page JJ II J I Page 3 of 26 Go Back Full Screen Close Quit where 2 is unknown, then Var( X ) = 2 n and we must estimate so that SE = n where = v u u t 1 n- 1 n X i =1 ( X i- X ) 2 . Standard Errors Methods: mome Methods: MLE Confidence Intervals Home Page Title Page JJ II J I Page 4 of 26 Go Back Full Screen Close Quit Example 2. Suppose now X 1 , . . . , X n iid Bernoulli( p ) . Then with p = X we have Var( p ) = Var( X ) = 1 n Var( X 1 ) = pq n . Since we do not know p we estimate it to get the SE: SE p = s X (1- X ) n . Standard Errors Methods: mome Methods: MLE Confidence Intervals Home Page Title Page JJ II J I Page 5 of 26 Go Back Full Screen Close Quit 2. Two Methods for Getting Point Estimators. How do we generate sensible estimators for model parame- ters? The two simplest and common methods are: Method of moments estimators (momes), Maximum likelihood estimators (mles). 2.1. Momes. General philosophy: If is d-dimensional, equate d differ- ent population characteristics with the corresponding sam- ple characteristics. Solve for = ( 1 , . . . , d ) . For example: If d = 1, equate E ( X 1 ) = Z xf ( x ) dx = Z x d F n ( x ) = X, and solve for . Standard Errors Methods: mome Methods: MLE Confidence Intervals Home Page Title Page JJ II J I Page 6 of 26 Go Back Full Screen Close Quit For general d : equate E ( X r 1 ) = Z x r f ( x ) dx = Z x r d F n ( x ) = 1 n n X i =1 X r i for r = 1 , . . . , d and solve the d equations for = ( 1 , . . . , d ) . Features: Provides a method and a way to proceed if other meth- ods complex. Resulting estimators may or may not be optimal. Simple. Occasionally the method gives dumb estimators. (This is true of most methods.) Ambiguity: If, for example, d = 1, why not solve E ( X 17 1 ) = 1 n n X i =1 X 17 i Standard Errors Methods: mome Methods: MLE Confidence Intervals Home Page Title Page JJ II J I Page 7 of 26 Go Back Full Screen Close Quit...
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Lect15_2700_s09 - Standard Errors Methods: mome Methods:...

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