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Chapter 7

# Chapter 7 - Ch 7 Introduction to Estimation Characteristics...

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1 Ch 7: Introduction to Estimation Characteristics of a Good Estimator Methods of Estimation: Methods of Moments, MLE Sampling Distribution for X-bar Confidence Intervals for the Mean

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Estimation In an estimation problem, there is at least one parameter θ whose value is to be estimated on the basis of a sample. The approximation is done by using an appropriate statistic, denoted as Point Estimate: A estimate of a population parameter given by a single value calculated from sample data. 2 θ
3 How good is this estimator? There are two qualities that we look for in an estimator: Unbiasedness Relative Efficiency (small variance for larger sample sizes)

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4 Unbiasedness Unbiased Statistic: A statistic is said to be unbiased if the mean of the sampling distribution is equal to the population parameter that it estimates. In other words, an estimator, , is an unbiased estimator for a population parameter, θ, if and only if E( )= θ. There can be more than one unbiased estimator for a population parameter. θ θ
5 Unbiased Estimators of Population Mean Let X 1 , X 2 , …, X n be a random sample of size n from a distribution with mean μ . The sample mean, , is an unbiased estimator for μ . E( ) = μ Var( ) = σ 2 /n x x x

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The sample median is also an unbiased estimator for μ . E(sample median) = μ Var(sample median) = How do we decide which unbiased estimator to use? 6 n 2 2 σ Π
7 Relative Efficiency An unbiased estimator is considered to be more relatively efficient if it has a smaller variance than other unbiased estimators of the same population parameter. Variance of sample mean: Variance of sample median: n 2 σ n 2 2 σ Π

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8 Frequently Used Estimators
9 Methods of Estimation Classical Method: Using only sample data to make inferences about a population parameter Bayesian Method: Utilizing prior knowledge about the probability distribution of the unknown parameter(s) along with sample data – WE WILL COME BACK TO THIS LATER THIS SEMESTER

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7.2 Classical Methods of Deriving Estimators
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Chapter 7 - Ch 7 Introduction to Estimation Characteristics...

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