M312_2 - Probability and Statistics II M-312 Lecture 2 2...

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Probability and Statistics II M-312 Lecture 2 2. Point Estimation 2.1. General Concepts Definition 2.1. A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of θ. The symbol θ ˆ (“theta hat”) is customarily used to denote both the estimator of θ and the point estimate resulting from a given sample. Thus X = μ ˆ is read as “the point estimate of μ is the sample mean X ”. The statement “the point estimate of μ is 5.77” can be written concisely as 77 . 5 ˆ = μ . Example 2.1. Consider the accompanying 20 observations on dielectric breakdown voltage for pieces of epoxy resin. 24.46 25.61 26.25 26.42 26.66 27.15 27.31 27.54 27.74 27.94 27.98 28.04 28.28 28.49 28.50 28.87 29.11 29.13 29.50 30.88 The pattern in the normal probability plot is quite straight, so we assume that the distribution of breakdown voltage is normal with mean value μ. The given observations are then assumed to be the result of a random sample 20 2 1 , , , X X X from this normal distribution. Consider the following estimators for μ: a . Estimator = X , estimate = 793 . 27 20 86 . 555 = = = n x x i b . Estimator = X ~ , estimate = 960 . 27 2 98 . 27 94 . 27 ~ = + = x c . Estimator = ( 29 ( 29 2 max min i i X X + , estimate = ( 29 ( 29 670 . 27 2 88 . 30 46 . 24 2 max min = + = + i i x x d . Estimator = ) 10 ( tr X , the 10% trimmed mean (discard the smallest and largest 10% of the sample and then average),
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estimate = 838 . 27 16 88 . 30 50 . 29 62 . 25 46 . 24 86 . 555 ) 10 ( = - - - - = tr x Each one of the estimators (a) – (d) uses a different measure of the center of the sample to estimate μ. Which of the estimates is closest to the true value? We cannot answer without knowing the true value. A question that can be answered is, “Which estimator,
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