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Section 7.1-7.2

# Section 7.1-7.2 - Chapter 7.1 Point estimate Properties of...

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Chapter 7.1 : Point Estimation Point estimate Properties of Estimators The primary aim of statistical inference is to draw some conclusions about the unknown parameter θ , say, mean or variance. Point estimation deals with methods to come up with a single value, based on the observed data. Let the rv X follow ) | ( or ) | ( θ θ x f x p where θ is the unknown parameter. Definition : A point estimate of a population parameter θ is a single number calculated from sample data. The statistic used to calculate an estimate is called an estimator and denoted by θ ˆ . Some examples are given in the table below: characteristic Population Parameter θ Sample Estimator ˆ θ Proportion π p Mean μ x Standard Deviation σ S 1

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An estimator is unbiased if in repeated random samples the numerical values are grouped around θ ; that is, if the mean value of . ) ˆ ( ˆ θ θ μ θ = = E The difference between the mean of ˆ θ and θ is called the bias. That is, ˆ bias of θ μ θ θ - = . - ) ˆ ( θ θ E = Example 1 : Sample mean - x is an unbiased estimator of the population mean μ , because x μ μ = Sample proportion p is an unbiased estimator of the population proportion π , because µ p = π . Definition . An estimator is consistent if ] - ˆ [| ε θ θ P 0, as n . Result : If n as 0, ) ˆ ( θ SE , then the estimator is consistent. Example 2 : 2
(i) Note x is consistent, because its SE 0 ) / ( = n x σ σ , as n .

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Section 7.1-7.2 - Chapter 7.1 Point estimate Properties of...

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