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ECON206_1011_01_Handout_05[1]

# ECON206_1011_01_Handout_05[1] - ECON 206 METU Department of...

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ECON 206 November 22, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 1 LECTURE 05 ESTIMATION - BIAS - MEAN SQUARE ERROR (MSE) Outline of today’s lecture: I. Introduction . ....................................................................................................................... 1 II. Bias and Mean Square Error of Point Estimators . ............................................................ 2 III. Evaluating the Goodness of a Point Estimator . ............................................................... 9 I. Introduction Purpose of statistics is to permit the user to make an inference about a population based on information contained in a sample. An estimate may be given in two distinct forms: 1. Point Estimate 2. Interval Estimate Definition 8.1 An estimator is a rule that tells how to calculate the value of an estimate based on the measurements contained in a sample.

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ECON 206 November 22, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 2 II. Bias and Mean Square Error of Point Estimators Definition 8.2 Let ˆ θ be a point estimator for a parameter . Then ˆ is an unbiased estimator if ˆ () E = . Otherwise ˆ is said to be biased . Note that ˆ E equals to a constant, it is not a random variable. We want a point estimator to be unbiased. In other words, we would like the mean or expected value of the distribution of estimates to equal the parameter estimated. Definition 8.3 The bias of a point estimator ˆ is given by ˆˆ BE θθ =− Note that, ˆ B is not a random variable since ˆ E is not a random variable Figure 1 Sampling Distribution for a positively biased estimator
ECON 206 November 22, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 3 Figure 1 Sampling Distributions for two unbiased estimators: (a) estimator with large variation; (b) estimator with small variation We would prefer that our estimator have the type of distribution indicated in Figure 8.3(b) because the smaller variance guarantees that, in repeated sampling, a higher fraction of values of 2 ˆ θ will be "close" to . Thus, in addition to preferring unbiasedness , we want the variance of the distribution of the estimator ˆ () V to be as small as possible. o Given two unbiased estimators of parameter , and all other things being equal, we would select the estimator with the smaller variance . o To characterize the goodness of a point estimator, we might employ the expected value of 2 ˆ , the average of the square of the distance between the estimator and its target parameter.

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ECON 206 November 22, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 4 Definition 8.4 The mean square error of a point estimator ˆ θ is the expected value of 2 ˆ () : 2 ˆˆ ( ) MSE E θθ =− o Hence, the mean square error of an estimator ˆ , ˆ MSE is a function of both its variance and its bias .
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ECON206_1011_01_Handout_05[1] - ECON 206 METU Department of...

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