Chapter 7 - CHAPTER-7 Estimation Point Estim ation : D...

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Unformatted text preview: CHAPTER-7 Estimation Point Estim ation : D efinition : Point estim ation is a choice of statistics, i.e. a single num ber calculated from sam ple data for w hich w e have som e expectation, or assurance that it is reasonably close to the param eter it is supposed to estim ate. Desirable Properties of a point estimator: 1. to be unbiased for . 2. to have a small variance for large sample sizes. Definition (Unbiased): An estimator is an unbiased estimator for parameter if and only if E[ ] = . Theorem : Let be a random sample of size n from a distribution with mean . The sample mean, , is an unbiased estimator for . n X X ,......, 1 X Proof: E[ ] = E[1/n( X 1 + X 2 + X 3 +.+ X n )] By the rules of expectations, X E[ ] = 1/n(E[X 1 ] +E[X 2 ] +E[X 3 ] ++ E[X n ]) Since X 1 , X 2 , X 3 ,X n constitutes a random sample of size n from a distribution with mean , each of these random variables have mean . Hence E[ ] = 1/n( + + + + ) = 1/n(n) = n terms X X Theorem : Let be the sample mean based on a random sample of size n from a distribution with mean and variance . Then Var = /n. Var[ ] = Var[(X 1 + X 2 ++ X n ) /n] = 1/n [Var(X 1 ) + Var(X 2 ) + + Var(X n )] = 1/n [ + + + ] = n /n = /n. X 2 2 X 2 2 2 2 2 X Definition (Standard error of the mean) : Let denote the sample of size n drawn from a distribution with standard deviation . The standard deviation of is given by / n and is called the standard error of the mean....
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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.

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Chapter 7 - CHAPTER-7 Estimation Point Estim ation : D...

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