chap07PRN - Chapter 7.1 Properties of Point Estimators Let...

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Unformatted text preview: Chapter 7.1 Properties of Point Estimators Let the random sample,, . . . , be a set of random variables that are independently and identically distributed. 1 X 2 X n X Population characteristics are summarized by parameters the true values are typically unknown. For example, the population mean is denoted by . An estimation rule can be specified for a parameter of interest. This estimation rule is called a point estimator . For example, a point estimator for the population mean is: P n 1i i X n 1 X An estimator is a random variable that is a function of the sample information. An estimator has a probability distribution called the sampling distribution. Econ 325 Chapter 7 1 An applied study works with a data set. The numeric observations are:,, . . . , . 1 x 2 x n x The estimation rule given by the estimator X can be used to calculate a point estimate of the population mean : n 1i i x n 1 x An important distinction is made between an estimator and an estimate. A point estimator is a random variable. A point estimate is a numeric outcome. Different samples of data will have different numeric observations and, therefore, will result in different point estimates of the population parameter. Econ 325 Chapter 7 2 Properties of Point Estimators Denote (the Greek letter theta) as a population parameter to be estimated (as a special case this may be the population mean ). T P Let T (theta-hat) be a point estimator of. T is a function of the sample information: )X,,X,X(f n 2 1 5 T This estimator is a random variable with a sampling distribution. T is said to be an unbiased estimator of if: T T ) (E The bias of an estimator T is defined as: T0T T ) (E) ( Bias It follows that the bias of an unbiased estimator is zero. Econ 325 Chapter 7 3 Example: Let, , be a random sample from a population with mean 1 X 2 X 3 X . That is, )X(E)X(E)X(E 3 2 1 Consider two alternative point estimators of : )X X X( 3 1 X 3 2 1 . . and )X X4 X( 6 1 X 3 2 1 W . . The second estimator is a weighted average of the sample information. To compare these estimators consider: P ,P . . 3( 3 1 )] X(E)X(E)X(E[ 3 1 )X(E 3 2 1 and P ,P.P.P . . 4 ( 6 1 )] X(E)X(E4)X(E[ 6 1 ) X(E 3 2 1 W Therefore, both estimators are unbiased estimators of the population mean P . Econ 325 Chapter 7 4 The above example demonstrated that there may be several unbiased estimators of a population parameter of interest. A problem that arises is: how can an estimator be selected from among a number of competing unbiased estimators ? A suggestion is to choose the estimator with minimum variance. Let 1 T and 2 T be two unbiased estimators of the population parameter T . That is, T T ) (E 1 and T T ) (E 2 1 T is said to be more efficient than 2 T if: ) ( Var ) ( Var 2 1 T ? T The relative efficiency of one estimator with respect to another is the variance ratio: ) ( Var ) ( Var 1 2 T T If T is an unbiased estimator of , and no other unbiased...
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This note was uploaded on 01/28/2011 for the course ECON 325 taught by Professor Whistler during the Spring '10 term at The University of British Columbia.

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chap07PRN - Chapter 7.1 Properties of Point Estimators Let...

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