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Unformatted text preview: Chapter 7 Lecture # 29 Estimation • To estimate numerical value of a population parameter using a sample of some size n. • We must device an appropriate statistic (on random sample of size n) so that we only need to consider its value on the observed sample. • It is desirable that the statistic satisfies certain properties. We may want to have such a statistic. Estimator and estimate • A statistic (which is a function on a random sample, and hence a random variable) used to estimate the population parameter θ is called a point estimator for θ and is denoted by • The value of the point estimator on a particular sample of that size is called a point estimate for θ. θ ˆ U nbiased estim ator : Let θ be the param eter of interest and θ ˆ be a statistic. Then a statistics θ ˆ is said to be an unbiased estim ator , or its value an unbiased estim ate , if and only if the m ean of the sam pling distribution of the estim ator equals θ , w hatever the value of θ , viz. E[ θ ˆ ]= θ . Theorem : The sample mean X of a random sample of size n from population X is an unbiased estimator of the population mean μ . [ ] . ) ( 1 ] [ ... ] [ 1 ] ... [ 1 ... ] [ : 1 1 1 μ μ = = + + = + + = + + = n n X E X E n X X E n n X X E X E n n n Proof More efficient unbiased estimator : A statistics 1 ˆ θ is said to be a more efficient unbiased estimator of the parameter θ than the statistics 2 ˆ θ if 1. 1 ˆ θ and 2 ˆ θ are both unbiased estimators of θ ; 2. The variance of the sampling distribution of the first estimator is not larger than that of the second and is smaller for at least one value of θ . Since $ 1 θ has a smaller variance than $ 2 θ , the estimator $ 1 θ is more likely to produce an estimate close to the true value θ . n X...
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This note was uploaded on 05/14/2010 for the course MATHEMATIC mathe taught by Professor Xyz during the Spring '10 term at Birla Institute of Technology & Science.
 Spring '10
 XYZ
 Math

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