319lec8_2010 - Chapter 6: Parameter Estimation and...

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Unformatted text preview: Chapter 6: Parameter Estimation and Evaluation References: Chapter 9, Sections 9.1&9.13 (only for references) 6.1 Point estimation Suppose f X i g n i =1 is an i.i.d. random sample from some population f X ( x ) : [Recall the popu- lation distribution f X ( x ) is the distribution of each X i : ] Often, one assumes f X ( x ) = f ( x;& ) ; where f ( x;& ) is unknown in functional form, but & is unknown. For example, if f X i g is i.i.d. N ( ; 2 ) ; then f X ( x ) = f ( x;& ) = 1 p 2 2 exp[ & 1 2 2 ( x & ) 2 ] ; where & = ( ; 2 ) : Suppose the random sample X n = f X 1 ;X 2 ;::;X n g has a realization x n = f x 1 ;x 2 ;:::;x n g : Then the realization x n is called a data set. A random sample X n can generate many di/erent data sets. Estimator versus Estimate De&nition [Estimator and Estimate]: Suppose & is a parameter of a population, and is unknown. An estimator ^ & = ^ & ( X 1 ;:::;X n ) of & is a statistic that depends on the sample information and whose realizations provide an approximation to &: A specic realization of that r.v. is called an estimate. De&nition [Point Estimator]: A point estimator ^ & is an estimator for & that only gives a single number for each sample realization x n = ( x 1 ;x 2 ;:::;x n ) . Example: Suppose X n = ( X 1 ;:::;X n ) is a random sample with ( ; 2 ) : Then we have & = ; ^ & = & X n = n & 1 P n i =1 X i : Example: Suppose X n = ( X 1 ;:::;X n ) is a random sample where X i Bernoulli ( p ) ; < p < 1 : Then & = p; ^ & = ^ p n = & X n : Example: Suppose X n = ( X 1 ;:::;X n ) is a random sample with ( ; 2 ) : Then we have & = 2 ; ^ & = S 2 n = ( n & 1) & 1 P n i =1 ( X i & & X n ) 2 : 1 Example: Suppose X n = ( X 1 ;:::;X n ) is a random sample with ( & 1 ; 2 1 ) ; and Y m = ( Y 1 ;:::;Y m ) is a random sample with ( & 2 ; 2 2 ) : Then we have = & 1 & & 2 ; and ^ = & X n & & Y m : Example: Suppose X n = ( X 1 ;:::;X n ) is a random sample with ( & 1 ; 2 1 ) ; and Y m = ( Y 1 ;:::;Y m ) is a random sample with ( & 2 ; 2 2 ) : Then = 2 1 = 2 2 and ^ = S 2 n =S 2 m : Notations of Estimators and Estimates: parameter ( ) Estimator ( ^ ) Estimate & & X n & x n 2 S 2 n s 2 n S n s n p ^ p n p n : Example: Price-earnings ratios for a random sample of ten stocks traded in NYSE on a partic- ular date is 10 ; 16 ; 5 ; 10 ; 12 ; 8 ; 4 ; 6 ; 5 ; 4 : Find the point estimates of the population mean, variance and standard deviation, and the proportion of stocks in the population for which the price-earnings ratio exceeded 8.5. ANS: &; 2 ; and p = P ( X > 8 : 5) : & x n = 80 = 10 = 8 : s 2 x = ( n & 1) & 1 ( P n i =1 x 2 & n & x 2 n ) = (782 & 10 8 2 ) = 9 = 15 : 78 : s x = p 15 : 78 = 3 : 97 : ^ p x = 4 = 10 = 0 : 4 : Questions: For any , ther are more than one estimators....
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319lec8_2010 - Chapter 6: Parameter Estimation and...

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