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319lec8_2010 - Chapter 6 Parameter Estimation and...

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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 2 ) ; then f X ( x ) = f ( x; ° ) = 1 p 2 ³² 2 exp[ ° 1 2 ² 2 ( x ° ± ) 2 ] ; where ° = ( ±; ² 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 speci±c 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 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 ) ; 0 < p < 1 : Then ° = p; ^ ° = ^ p n = ° X n : Example: Suppose X n = ( X 1 ; :::; X n ) is a random sample with ( ±; ² 2 ) : Then we have ° = ² 2 ; ^ ° = S 2 n = ( n ° 1) ° 1 P n i =1 ( X i ° ° X n ) 2 : 1
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Example: Suppose X n = ( X 1 ; :::; X n ) is a random sample with ( ± 1 ; ² 2 1 ) ; and Y m = ( Y 1 ; :::; Y m ) is a random sample with ( ± 2 ; 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 2 1 ) ; and Y m = ( Y 1 ; :::; Y m ) is a random sample with ( ± 2 ; ² 2 2 ) : Then ° = ² 2 1 2 2 and ^ ° = S 2 n =S 2 m : Notations of Estimators and Estimates: parameter ( ° ) Estimator ( ^ ° ) Estimate ± ° X n ° x n ² 2 S 2 n s 2 n ² 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 2 ; 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 ² 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. Example: Let ° = ±; then the following are estimators for ° = ± : (i) ^ ° 1 = ° X n : (ii) ^ ° 2 = X 1 : Which one is a better estimator? How good is an estimator ^ ° for ° ? We need to use some appropriate criterion to measure how good the estimator is. De°nition [Mean Squared Error (MSE)]: The MSE of an estimator ^ ° is de±ned as MSE ( ^ ° ) = E ( ^ ° ° ° ) 2 : Remarks: MSE( ^ ° ) measures the variations or deviations of the estimator ^ ° from the true parameter ° . The smaller MSE( ^ ° ) ; the better the estimator ^ °: A smaller MSE means that there is smaller variation between ^ ° and °: De°nition [Relative E¢ ciency]:
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