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Unformatted text preview: § 6 Confidence Intervals § 6.1 Introduction 6.1.1 Point estimator ˆ θ — a single fixed number (or vector) thought to be close to the unknown value of the parameter θ . Interval estimator [ a ( X ) , b ( X ) ] — interval thought to contain the unknown true parameter θ . Many scientific reports quote estimates as ˆ θ ± ² , where ² is an error estimate for ˆ θ , meaning that it is quite probable that θ is within a distance ± ² from ˆ θ . Thus [ ˆ θ- ², ˆ θ + ² ] is an interval estimator for θ . 6.1.2 Data X a ( X ), b ( X ): statistics with a ( X ) ≤ b ( X ) Definition. The random interval [ a ( X ) , b ( X )] is a 100(1- α )% confidence interval for θ if P ( a ( X ) ≤ θ ≤ b ( X ) | θ ) = 1- α. The confidence level of [ a ( X ) , b ( X )] refers to the number 1- α . 6.1.3 Three common types of confidence intervals: (i) 100(1- α )% equal-tailed confidence interval: [ a ( X ) , b ( X )] if P ( θ < a ( X ) | θ ) = P ( θ > b ( X ) | θ ) = α/ 2 , for all θ . (ii) 100(1- α )% upper confidence bound or upper confidence limit : b ( X ) if P ( θ < a ( X ) | θ ) = 0 , P ( θ > b ( X ) | θ ) = α, for all θ . (iii) 100(1- α )% lower confidence bound or lower confidence limit : a ( X ) if P ( θ < a ( X ) | θ ) = α, P ( θ > b ( X ) | θ ) = 0 , for all θ . (ii) and (iii) are one-sided intervals (i.e. all the “exclusion” probability is concentrated on one side). An interval is two-sided if it is not one-sided. 80 6.1.4 Example § 6.1.1 Let X ∼ U [0 ,θ ]. Then X/θ ∼ U [0 , 1], so that P ( X/u ≤ θ ) = u and P ( θ ≤ X/u ) = 1- u, for u ∈ [0 , 1]. Examples of confidence intervals for θ : (i) 100(1- α )% equal-tailed confidence interval: [ X (1- α/ 2)- 1 , X ( α/ 2)- 1 ]; (ii) 100(1- α )% upper confidence bound: Xα- 1 (or 100(1- α )% one-sided confidence interval: [0 , Xα- 1 ]); (iii) 100(1- α )% lower confidence bound: X (1- α )- 1 (or 100(1- α )% one-sided confidence interval: [ X (1- α )- 1 , ∞ )). 6.1.5 If X is observed to be x , the random interval [ a ( X ) , b ( X )] is realized as a fixed interval [ a ( x ) , b ( x )], which either does or does not contain the true θ : P ( a ( x ) ≤ θ ≤ b ( x ) | θ ) = 1 or 0 . The statement that defines the confidence interval [ a ( X ) , b ( X )] is interpreted as if X had not been realized yet. It carries the following frequentist interpretation: If the procedure were to be simulated lots of times under repeated sampling of X , the confidence intervals thereby generated would have contained the true θ about 100(1- α ) % of the times . One would normally fix a small α so that the confidence interval has a high confidence level and is therefore more reliable....
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This note was uploaded on 05/04/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.
- Spring '10